Splitting of Primes in Extensions

In Algebraic Numbers we discussed how ideals factorize in an algebraic number field (recall that we had to look at factorization of ideals since the elements in the ring of integers of more general algebraic number fields may no longer factorize uniquely). In this post, we develop some more terminology related to this theory, and we also discuss how in the case of a so-called “Galois extension” the Galois group (see Galois Groups) may express information related to the factorization of ideals in an algebraic number field.

Let \mathfrak{p} be a prime ideal of the ring of integers \mathcal{O}_{K} of an algebraic number field K (we will sometimes also refer to \mathfrak{p} as a prime ideal of K – this is common practice and hopefully will not cause any confusion). In an algebraic number field L which contains K (we also say that L is an extension of K, and write L|K), this prime ideal \mathfrak{p} decomposes into a product of prime ideals \mathfrak{P}_{1},\mathfrak{P}_{2}...\mathfrak{P}_{r} in \mathcal{O}_L, with respective exponents e_{1},e_{2}...e_{r}, i.e.

\displaystyle \mathfrak{p}=\mathfrak{P}_{1}^{e_{1}}\mathfrak{P}_{2}^{e_{2}}...\mathfrak{P}_{n}^{e_{r}}.

The exponents e_{1},e_{2}...e_{r} are called the ramification indices of the prime ideals \mathfrak{P}_{1},\mathfrak{P}_{2},...\mathfrak{P}_{r}. If e_{i}=1, and the residue field extension \kappa(\mathfrak{P}_{i})|\kappa(\mathfrak{p}) (see below) is separable, we say that \mathfrak{P}_{i} is unramified over K. If e_{1}=e_{2}=...e_{r}=1, we say that the prime \mathfrak{p} is unramified. If all primes of K are unramified in L, we say that the extension L|K is unramified.

In the rest of this post we will continue to assume the factorization of \mathfrak{p} as shown above. The residue fields \kappa(\mathfrak{P}_{i}) and \kappa(\mathfrak{p}) of \mathcal{O}_{L} and \mathcal{O}_{K} at the primes \mathfrak{P}_{i} and \mathfrak{p} are defined as the quotients \mathcal{O}_{L}/\mathfrak{P}_{i} and \mathcal{O}_{K}/\mathfrak{p}, and the inertia degrees f_{i} are defined as the degrees of the fields \kappa(\mathfrak{P}_{i}) with respect to the field \kappa(\mathfrak{p}) (i.e. the dimensions of the vector spaces \kappa(\mathfrak{P}_{i}) over the field of scalars \kappa(\mathfrak{p})), i.e.

\displaystyle f_{i}=[\kappa(\mathfrak{P}_{i}):\kappa(\mathfrak{p})].

The ramification indices e_{i}, the inertia degrees f_{i}, and the degree n=[L:K] of the field extension L with respect to K are related by the following “fundamental identity“:

\displaystyle \sum_{i=1}^{r}e_{i}f_{i}=n

In order to understand these concepts better, we can look at the following “extreme” cases:

If e_{i}=1 and f_{i}=1 for all i, then r=n, and we say that the prime \mathfrak{p} splits completely in L.

If r=1 and f_{1}=1, then e_{1}=n, and we say that the prime \mathfrak{p} ramifies completely in L.

If r=1 and e_{1}=1, then f_{1}=n, and we say that the prime \mathfrak{p}  is inert in L.

Consider for example, the field \mathbb{Q}(i) as a field extension of the field \mathbb{Q}. The ring of integers of \mathbb{Q}(i) is the ring of Gaussian integers \mathbb{Z}[i] (see The Fundamental Theorem of Arithmetic and Unique Factorization), while the ring of integers of \mathbb{Q} is the ring of ordinary integers \mathbb{Z}. The degree [\mathbb{Q}(i):\mathbb{Q}] is equal to 2. The prime ideal (5) of \mathbb{Z} splits completely as the product (2+i)(2-i) in \mathbb{Z}[i], the prime ideal (2) of \mathbb{Q} ramifies completely as (1+i)^{2} in \mathbb{Z}[i], while the prime ideal (3) of \mathbb{Z} is inert in \mathbb{Z}[i].

We now bring in Galois groups. We assume that L is a Galois extension of K. This means that the order of G(L|K), the Galois group of L over K, is equal to the degree of L over K. In this case, it turns out that we will have

\displaystyle e_{1}=e_{2}=...=e_{r}


\displaystyle f_{1}=f_{2}=...=f_{r}.

The fundamental identity then becomes


This is but the first of many simplifications we obtain whenever we are dealing with Galois extensions.

Given a prime ideal \mathfrak{P} of \mathcal{O}_{K}, we define the decomposition group G_{\mathfrak{P}} as the subgroup of the Galois group G that fixes \mathfrak{P}, i.e.

\displaystyle G_{\mathfrak{P}}=\{\sigma\in G|\sigma\mathfrak{P=\mathfrak{P}}\}.

The elements of L that are fixed by the decomposition group G_{\mathfrak{P}} form what is called the decomposition field of K over \mathfrak{P}, denoted Z_{\mathfrak{P}}:

 \displaystyle Z_{\mathfrak{P}}=\{x\in L|\sigma x=x,\forall\sigma\in G_{\mathfrak{P}}\}

Every element \sigma of G_{\mathfrak{P}} automorphism \bar{\sigma} of \kappa(\mathfrak{P}) sending the element given by a\text{ mod }\mathfrak{P} to the element given by \sigma a\text{ mod }\mathfrak{P}. The residue field of the decomposition field Z_{\mathfrak{P}} with respect to \mathfrak{p} is the same as the residue field of the field K with respect to \mathfrak{p}, which is \kappa(\mathfrak{p}). Therefore we have a surjective homomorphism

\displaystyle G_{\mathfrak{P}}\rightarrow G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))

which sends the element \sigma of G_{\mathfrak{P}} to the element \bar{\sigma} of G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p})). The kernel of this homorphism is called the inertia group of \mathfrak{P} over K. Once again, the elements of L fixed by the inertia group I_{\mathfrak{P}} form what we call the inertia field of K over \mathfrak{P}, denoted T_{\mathfrak{P}}:

 \displaystyle T_{\mathfrak{P}}=\{x\in K|\sigma x=x,\forall\sigma\in I_{\mathfrak{P}}\}

The groups G_{\mathfrak{P}}, I_{\mathfrak{P}}, G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p})) are related by the following exact sequence:

\displaystyle 0\rightarrow I_{\mathfrak{P}}\rightarrow G_{\mathfrak{P}}\rightarrow G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))\rightarrow 0

Meanwhile, the relationship between the fields K, Z_{\mathfrak{P}}, T_{\mathfrak{P}}, and L can be summarized as follows:

\displaystyle K\subseteq Z_{\mathfrak{P}}\subseteq T_{\mathfrak{P}}\subseteq L

The ramification index, inertia degree, and the number of primes in K into which a prime \mathfrak{p} in L splits are given in terms of the degrees of the aforementioned fields as follows:

\displaystyle e=[L:T_{\mathfrak{P}}]

\displaystyle f=[T_{\mathfrak{P}}:Z_{\mathfrak{P}}]

\displaystyle r=[Z_{\mathfrak{P}}:K]

Let \mathfrak{P}_{Z}=\mathfrak{P}\cap Z_{\mathfrak{P}}, and \mathfrak{P}_{T}=\mathfrak{P}\cap T_{\mathfrak{P}}. We also refer to \mathfrak{P}_{Z} (resp. \mathfrak{P}_{T}) as the prime ideal of Z_{\mathfrak{P}} (resp. T_{\mathfrak{P}}) below \mathfrak{P}.

The ramification index of \mathfrak{P} over \mathfrak{P}_{T} is equal to e, and its inertia degree is equal to 1. Meanwhile, the ramification index of \mathfrak{P}_{T} over \mathfrak{P}_{Z} is equal to 1, and its inertia degree is equal to e. Finally, the ramification index and inertia degree of \mathfrak{P}_{Z} over \mathfrak{p} are both equal to 1.

We can therefore see that in the case of a Galois extension, the theory of the splitting of primes becomes simple and elegant. Before we end this post, there is one more concept that we will define. Let \mathfrak{P} be a prime that is unramified over K. Then G_{\mathfrak{P}} is isomorphic to G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p})), it is cyclic, and it is generated by the unique automorphism

\displaystyle \varphi_{\mathfrak{P}}\equiv a^{q}\text{ mod }\mathfrak{P}    for all    \displaystyle a\in \mathcal{O}_{K}

where q=[\kappa(\mathfrak{P}):\kappa(\mathfrak{p})]. The automorphism \varphi_{\mathfrak{P}} is called the Frobenius automorphism, and it is a very important concept that shows up in many aspects of algebraic number theory.


Splitting of Prime Ideals in Galois Extensions on Wikipedia

A Classical Introduction to Modern Number Theory by Kenneth Ireland and Michael Rosen

Number Fields by Daniel Marcus

Algebraic Theory of Numbers by Pierre Samuel

Algebraic Number Theory by Jurgen Neukirch


SEAMS School Manila 2017: Topics on Elliptic Curves

A few days ago, from July 17 to 25, I attended the SEAMS (Southeast Asian Mathematical Society) School held at the Institute of Mathematics, University of the Philippines Diliman, discussing topics on elliptic curves. The school was also partially supported by CIMPA (Centre International de Mathematiques Pures et Appliquees, or International Center for Pure and Applied Mathematics), and I believe also by the Roman Number Theory Association and the Number Theory Foundation. Here’s the official website for the event:

Southeast Asian Mathematical Society (SEAMS) School Manila 2017: Topics on Elliptic Curves

There were many participants from countries all over Southeast Asia, including Indonesia, Malaysia, Philippines, and Vietnam, as well as one participant from Austria and another from India. The lecturers came from Canada, France, Italy, and Philippines.

Jerome Dimabayao and Michel Waldschmidt started off the school, introducing the algebraic and analytic aspects of elliptic curves, respectively. We have tackled these subjects in this blog, in Elliptic Curves and The Moduli Space of Elliptic Curves, but the school discussed them in much more detail; for instance, we got a glimpse of how Karl Weierstrass might have come up with the function named after him, which relates the equation defining an elliptic curve to a lattice in the complex plane. This requires some complex analysis, which unfortunately we have not discussed that much in this blog yet.

Francesco Pappalardi then discussed some important theorems regarding rational points on elliptic curves, such as the Nagell-Lutz theorem and the famous Mordell-Weil theorem. Then, Julius Basilla discussed the counting of points of elliptic curves over finite fields, often making use of the Hasse-Weil inequality which we have discussed inThe Riemann Hypothesis for Curves over Finite Fields, and the applications of this theory to cryptography. Claude Levesque then introduced to us the fascinating theory of quadratic forms, which can be used to calculate the class number of a quadratic number field (see Algebraic Numbers), and the relation of this theory to elliptic curves.

Richell Celeste discussed the reduction of elliptic curves modulo primes, a subject which we have also discussed here in the post Reduction of Elliptic Curves Modulo Primes, and two famous problems related to elliptic curves, Fermat’s Last Theorem, which was solved by Andrew Wiles in 1995, and the still unsolved Birch and Swinnerton-Dyer conjecture regarding the rank of the group of rational points of elliptic curves. Fidel Nemenzo then discussed the classical problem of finding “congruent numbers“, rational numbers forming the sides of a right triangle whose area is given by an integer, and the rather surprising connection of this problem to elliptic curves.

On the last day of the school, Jerome Dimabayao discussed the fascinating connection between elliptic curves and Galois representations, which we have given a passing mention to at the end of the post Elliptic Curves. Finally, Jared Guissmo Asuncion gave a tutorial on the software PARI which we can use to make calculations related to elliptic curves.

Participants were also given the opportunity to present their research work or topics they were interested in. I gave a short presentation discussing certain aspects of algebraic geometry related to number theory, focusing on the spectrum of the integers, and a mention of related modern mathematical research, such as Arakelov theory, and the view of the integers as a curve (under the Zariski topology) and as a three-dimensional manifold (under the etale topology).

Aside from the lectures, we also had an excursion to the mountainous province of Rizal, which is a short distance away from Manila, but provides a nice getaway from the environment of the big city. We visited a couple of art museums (one of which was also a restaurant serving traditional Filipino cuisine), an underground cave system, and a waterfall. We used this time to relax and talk with each other, for instance about our cultures, and many other things. Of course we still talked about mathematics, and during this trip I learned about many interesting things from my fellow participants, such as the class field theory problem and the subject of real algebraic geometry .

I believe lecture notes will be put up on the school website at some point by some of the participants of the school. For now, some of the lecturers have put up useful references for their lectures.

SEAMS School Manila 2017 was actually the first summer school or conference of its kind that I attended in mathematics, and I enjoyed very much the time I spent there, not only in learning about elliptic curves but also making new friends among the mathematicians in attendance. At some point I also hope to make some posts on this blog regarding the interesting things I have learned at that school.

Adeles and Ideles

In Valuations and Completions we introduced the p-adic numbers \mathbb{Q}_{p}, which, like the real numbers, are the completion of the rational numbers under a certain kind of valuation. There is one such valuation for each prime number p, and another for the “infinite prime”, which is just the usual absolute value. Each valuation may be thought of as encoding number theoretic information related to the prime p, or to the “infinite prime”, for the case of the absolute value (more technically, the p-adic valuations are referred to as nonarchimedean valuations, while the absolute value is an example of an archimedean valuation).

We can consider valuations not only for the rational numbers, but for more general algebraic number fields as well. In its abstract form, given an algebraic number field K, a (multiplicative) valuation of K is simply any function |\ | from K to \mathbb{R} satisfying the following properties:

(i) |x|\geq 0, where x=0 if and only if x=0

(ii) |xy|=|x||y|

(iii) |x+y|\leq|x|+|y|

If this seems reminiscent of the discussion in Metric, Norm, and Inner Product, it is because a valuation does, in fact, define a metric on K, and by extension, a topology. Two valuations are equivalent if they define the same topology; another way to phrase this statement is that two valuations |\ |_{1} and |\ |_{2} are equivalent if |x|_{1}=|x|_{2}^{s} for some positive real number s, for all x\in K.  The valuation is nonarchimedean if |x+y|\leq\text{max}\{|x|,|y|\}; otherwise, it is archimedean.

Just as in the case of rational numbers, we also have an exponential valuation, defined as a function v from the field K to \mathbb{R}\cup \infty satisfying the following conditions:

(i) v(x)=\infty if and only if x=0

(ii) v(xy)=v(x)+v(y)

(iii) v(x+y)\geq\text{min}\{v(x),v(y)\}

Two exponential valuations v_{1} and v_{2} are equivalent if v_{1}(x)=sv_{2}(x) for some real number s, for all x\in K.

The idea of valuations allows us to make certain concepts in algebraic number theory (see Algebraic Numbers) more abstract. We define a place v of an algebraic number field K as an equivalence class of valuations of K. We write K_{v} to denote the completion of K under the place v; these are the generalizations of the p-adic numbers and real numbers to algebraic number fields other than \mathbb{Q}. The nonarchimedean places are also called the finite places, while the archimedean places are also called the infinite places. To express whether a place v is a finite place or an infinite place, we write v|\infty or v\nmid\infty respectively.

The infinite places are of two kinds; the ones for which K_{v} is isomorphic to \mathbb{R} are called the real places, while the ones for which K_{v} is isomorphic to \mathbb{C} are called the complex places. The number of real places and complex places of K, denoted by r_{1} and r_{2} respectively, satisfy the equation r_{1}+2r_{2}=n, where n is the degree of K over \mathbb{Q}, i.e. n=[K:\mathbb{Q}].

By the way, in some of the literature, such as in the book Algebraic Number Theory by Jurgen Neukirch, “places” are also referred to as “primes“. This is intentional – one may actually think of our definition of places as being like a more abstract replacement of the definition of primes. This is quite advantageous in driving home the concept of primes as equivalence classes of valuations; however, to avoid confusion, we will stick to using the term “places” here, along with its corresponding notation.

When v is a nonarchimedean valuation, we let \mathfrak{o}_{v} denote the set of all elements x of K_{v} for which |x|_{v}\leq 1. It is an example of a ring with special properties called a valuation ring. This means that, for any x in K, either x or x^{-1} must be in \mathfrak{o}_{v}. We let \mathfrak{o}_{v}^{*} denote the set of all elements of \mathfrak{o}_{v} for which |x|_{v}=1, and we let \mathfrak{p}_{v} denote the set of all elements of \mathfrak{o}_{v} for which |x|_{v}< 1. It is the unique maximal ideal of \mathfrak{o}_{v}.

Now we proceed to consider the modern point of view in algebraic number theory, which is to consider all these equivalence classes of valuations together. This will lead us to the language of adeles and ideles.

An adele \alpha of K is a family (\alpha_{v}) of elements \alpha_{v} of K_{v} where \alpha_{v}\in K_{v}, and \alpha_{v}\in\mathfrak{o}_{v} for all but finitely many v. We can define addition and multiplication componentwise on adeles, and the resulting ring of adeles is then denoted \mathbb{A}_{K}. The group of units of the ring of adeles is called the group of ideles, denoted I_{K}. For a finite set of primes S that includes the infinite primes, we let

\displaystyle \mathbb{A}_{K}^{S}=\prod_{v\in S}K_{v}\times\prod_{v\notin S}\mathfrak{o}_{v}


\displaystyle I_{K}^{S}=\prod_{v\in S}K_{v}^{*}\times\prod_{v\notin S}\mathfrak{o}_{v}^{*}.

We denote the set of infinite primes by S_{\infty}. Then \mathfrak{o}_{K}, the ring of integers of the number field K, is given by K\cap\mathbb{A}_{K}^{S_{\infty}}, while \mathfrak{o}_{K}^{*}, the group of units of \mathfrak{o}_{K}, is given by K^{*}\cap I_{K}^{S_{\infty}}.

Any element of K is also an element of \mathbb{A}_{K}, and any element of K^{*} (the group of units of K) is also an element of I_{K}. The elements of I_{K} which are also elements of K^{*} are called the principal ideles. This should not be confused with the concept of principal ideals; however the terminology is perhaps suggestive on purpose. In fact, ideles and fractional ideals are related. Any fractional ideal \mathfrak{a} can be expressed in the form

\displaystyle \mathfrak{a}=\prod_{\mathfrak{p}}\mathfrak{p}^{\nu_{\mathfrak{p}}}.

Therefore, we have a mapping

\displaystyle \alpha\mapsto (\alpha)=\prod_{\mathfrak{p}}\mathfrak{p}^{v_{\mathfrak{p}}(\alpha_v)}

from the group of ideles to the group of fractional ideals. This mapping is surjective, and its kernel is I_{K}^{S_{\infty}}.

The quotient group I_{K}/K^{*} is called the idele class group of K, and is denoted by C_{K}. Again, this is not to be confused with the ideal class group we discussed in Algebraic Numbers, although the two are related; in the language of ideles, the ideal class group is defined as I_{K}/I_{K}^{S_{\infty}}K^{*}, and is denoted by Cl_{K}. There is a surjective homomorphism C_{K}\mapsto Cl_{K} induced by the surjective homomorphism from the group of ideles to the group of fractional ideals that we have described in the preceding paragraph.

An important aspect of the concept of adeles and ideles is that they can be equipped with topologies (see Basics of Topology and Continuous Functions). For the adeles, this topology is generated by the neighborhoods of 0 in \mathbb{A}_{K}^{S_{\infty}} under the product topology. For the ideles, this topology is defined by the condition that the mapping \alpha\mapsto (\alpha,\alpha^{-1}) from I_{K} into \mathbb{A}_{K}\times\mathbb{A}_{K} be a homeomorphism onto its image. Both topologies are locally compact, which means that every element has a neighborhood which is compact, i.e. every open cover of that neighborhood has a finite subcover. For the group of ideles, its topology is compatible with its group structure, which makes it into a locally compact topological group.

In this post, we have therefore seen how the theory of valuations can allow us to consider a more abstract viewpoint for algebraic number theory, and how considering all the valuations together to form adeles and ideles allows us to rephrase the usual concepts related to algebraic number fields, such as the ring of integers, its group of units, and the ideal class group, in a new form. In addition, the topologies on the adeles and ideles can be used to obtain new results; for instance, because the group of ideles is a locally compact topological (abelian) group, we can use the methods of harmonic analysis (see Some Basics of Fourier Analysis) to study it. This is the content of the famous thesis of the mathematician John Tate. Another direction where the concept of adeles and ideles can take us is class field theory, which relates the idele class group to the other important group in algebraic number theory, the Galois group (see Galois Groups). The language of adeles and ideles can also be applied not only to algebraic number fields but also to function fields of curves over finite fields. Together these fields are also known as global fields.


Adele Ring on Wikipedia

Tate’s Thesis on Wikipedia

Class Field Theory on Wikipedia

Algebraic Number Theory by Jurgen Neukirch

Algebraic Number Theory by J. W. S. Cassels and A. Frohlich

A Panorama of Pure Mathematics by Jean Dieudonne

Some Useful Links on the Hodge Conjecture, Kahler Manifolds, and Complex Algebraic Geometry

I’m going to be fairly busy in the coming days, so instead of the usual long post, I’m going to post here some links to interesting stuff I’ve found online (related to the subjects stated on the title of this post).

In the previous post, An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry, we discussed Calabi-Yau manifolds (which are special cases of Kahler manifolds) and how their interesting properties, namely their Riemannian, symplectic, and complex aspects figure into the branch of mathematics called mirror symmetry, which is inspired by the famous, and sometimes controversial, proposal for a theory of quantum gravity (and more ambitiously a candidate for the so-called “Theory of Everything”), string theory.

We also mentioned briefly a famous open problem concerning Kahler manifolds called the Hodge conjecture (which was also mentioned in Algebraic Cycles and Intersection Theory). The links I’m going to provide in this post will be related to this conjecture.

As with the post An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry, aside from introducing the subject itself, another of the primary intentions will be to motivate and explore aspects of algebraic geometry such as complex algebraic geometry, and their relation to other branches of mathematics.

Here is the page on the Hodge conjecture, found on the website of the Clay Mathematics Institute:

Hodge Conjecture on Clay Mathematics Institute

We have mentioned before that the Hodge conjecture is one of seven “Millenium Problems” for which the Clay Mathematics Institute is offering a million dollar prize. The page linked to above contains the official problem statement as stated by Pierre Deligne, and a link to a lecture by Dan Freed, which is intended for a general audience and quite understandable. The lecture by Freed is also available on Youtube:

Dan Freed on the Hodge Conjecture at the Clay Mathematics Institute on Youtube

Unfortunately the video of that lecture has messed up audio (although the lecture remains understandable – it’s just that the audio comes out of only one side of the speakers or headphones). Here is another set of videos by David Metzler on Youtube, which explains the Hodge conjecture (along with the other Millennium Problems) to a general audience:

Millennium Problem Talks on Youtube

The Hodge conjecture is also related to certain aspects of number theory. In particular, we have the Tate conjecture, which is another conjecture similar to the Hodge conjecture, but more related to Galois groups (see Galois Groups). Alex Youcis discusses it on the following post on his blog Hard Arithmetic:

The Tate Conjecture over Finite Fields on Hard Arithmetic

On the same blog there is also a discussion of a version of the Hodge conjecture called the p-adic Hodge conjecture on the following post:

An Invitation to p-adic Hodge Theory; or How I Learned to Stop Worrying and Love Fontaine on Hard Arithmetic

The first part of the post linked to above discusses the Hodge conjecture in its classical form, while the second part introduces p-adic numbers and related concepts, some aspects of which were discussed on this blog in Valuations and Completions.

A more technical discussion of the Hodge conjecture, Kahler manifolds, and complex algebraic geometry can be found in the following lecture of Claire Voisin, which is part of the Proceedings of the 2010 International Congress of Mathematicians in Hyderabad, India:

On the Cohomology of Algebraic Varieties by Claire Voisin

More about these subjects will hopefully be discussed on this blog at sometime in the future.

Reduction of Elliptic Curves Modulo Primes

We have discussed elliptic curves over the rational numbers, the real numbers, and the complex numbers in Elliptic Curves. In this post, we discuss elliptic curves over finite fields of the form \mathbb{F}_{p}, where p is a prime, obtained by “reducing” an elliptic curve over the integers modulo p (see Modular Arithmetic and Quotient Sets).

We recall that in Elliptic Curves we gave the definition of an elliptic curve as a polynomial equation that we may write as

\displaystyle y^{2}=x^{3}-ax+b

with a and b satisfying the condition that

\displaystyle 4a^{3}+27b^{2}\neq 0.

Still, we claimed that we will not be able to write the equation of the elliptic curve when the coefficients of the elliptic curve are of characteristic equal to 2 or 3, as is the case for the finite fields \mathbb{F}_{2} or \mathbb{F}_{3}, therefore we will give more general forms for the equation of the elliptic curve later, along with the appropriate conditions. To help us with the latter, we will first look at the case of curves over the real numbers, where we can still make use of the equations above, and see what happens when the conditions on a and b are not satisfied.

Let both a and b both be equal to 0, in which case the condition is not satisfied. Then our curve (which is not an elliptic curve) is given by the equation

\displaystyle y^{2}=x^{3}

whose graph in the xy plane is given by the following figure (plotted using the WolframAlpha software):


Next let a=-3 and b=2. Once again the condition is not satisfied. Our curve is given by

\displaystyle y^{2}=x^{3}-3x+2

and whose graph is given by the following figure (again plotted using WolframAlpha):


Note also that in both cases, the right hand side of the equations of the curves are polynomials in x with a double or triple root; for y^{2}=x^{3}, the right hand side, x^{3}, has a triple root at x=0, while for y^{2}=x^{3}-3x+2, the right hand side, x^{3}-3x+2, factors into y^{2}=(x-1)^{2}(x+2) and therefore has a double root at x=1.

The two curves, y^{2}=x^{3} and y^{2}=x^{3}-3x+2, are examples of singular curves. It is therefore a requirement for a curve to be an elliptic curve, that it must be nonsingular.

We now introduce the general form of an elliptic curve, applicable even when the coefficients belong to fields of characteristic 2 or 3, along with the general condition for it to be nonsingular. We note that the elliptic curve has a “point at infinity“; in order to make this idea explicit, we make use of the notion of projective space (see Projective Geometry) and write our equation in homogeneous coordinates X, Y, and Z:

\displaystyle Y^{2}Z+a_{1}XYZ+a_{3}YZ^{2}=X^{3}+a_{2}XZ^{2}+a_{4}X^{2}Z+a_{6}Z^{3}

This equation is called the long Weierstrass equation. We may also say that it is in long Weierstrass form.

We can now define what it means for a curve to be singular. Let

\displaystyle F=Y^{2}Z+a_{1}XYZ+a_{3}YZ^{2}-X^{3}-a_{2}XZ^{2}-a_{4}X^{2}Z-a_{6}Z^{3}

Then a singular point on this curve F is a point with coordinates a, b, and c such that

\displaystyle \frac{\partial F}{\partial X}(a,b,c)=\frac{\partial F}{\partial Y}(a,b,c)=\frac{\partial F}{\partial Z}(a,b,c)=0

It might be difficult to think of calculus when we are considering, for example, curves over finite fields, where there are a finite number of points on the curve, so we might instead just think of the partial derivatives of the curve as being obtained “algebraically” using the “power rule” of basic calculus,

\displaystyle \frac{d(x^{n})}{dx}=nx^{n-1}

and applying it, along with the usual rules for partial derivatives and constant factors, to every term of the curve. Such is the power of algebraic geometry; it allows us to “import” techniques from calculus and other areas of mathematics which we would not ordinarily think of as being applicable to cases such as curves over finite fields.

If a curve has no singular points, then it is called a nonsingular curve. We may also say that the curve is smooth. In order for a curve written in long Weierstrass form to be an elliptic curve, we require that it be a nonsingular curve as well.

If the coefficients of the curve are not of characteristic equal to 2, we can make a projective transformation of variables to write its equation in a simpler form, known as the short Weierstrass equation, or short Weierstrass form:


In this case the condition for the curve to be nonsingular can be written in the following form:

\displaystyle -4a_{2}^{3}a_{6}+a_{2}^{2}a_{4}^{2}+18a_{4}a_{2}a_{6}-4a_{4}^{3}-27a_{6}^{2}=0

The quantity

\displaystyle D=-4a_{2}^{3}a_{6}+a_{2}^{2}a_{4}^{2}+18a_{4}a_{2}a_{6}-4a_{4}^{3}-27a_{6}^{2}

is called the discriminant of the curve.

We note now, of course, that the usual expressions for the elliptic curve, in what we call affine coordinates x and y, can be recovered from our expression in terms of homogeneous coordinates X, Y, and Z simply by setting x=\frac{X}{Z} and y=\frac{Y}{Z}. The case Z=0 of course corresponds to the “point at infinity”.

We now consider an elliptic curve whose equation has coefficients which are rational numbers. We can make a projective transformation of variables to rewrite the equation into one which has integers as coefficients. Then we can reduce the coefficients modulo a prime p and investigate the points of the elliptic curve considered as having coordinates in the finite field \mathbb{F}_{p}.

It may happen that when we reduce an elliptic curve modulo p, the resulting curve over the finite field \mathbb{F}_{p} is no longer nonsingular. In this case we say that it has bad reduction at p. Consider, for example, the following elliptic curve (written in affine coordinates):

\displaystyle y^{2}=x^{3}-4x^{2}+16

Let us reduce this modulo the prime p=11. Then, since -4\equiv 7 \text{mod }11 and 16\equiv 5 \text{mod }11, we obtain the curve

\displaystyle y^{2}=x^{3}+7x^{2}+5

over \mathbb{F}_{11}. The right hand side actually factors into (x+1)^{2}(x+5) over \mathbb{F}_{11}, which means that it has a double root at x=10 (which is equivalent to x=-1 modulo 11), and has discriminant equal to zero over \mathbb{F}_{11}, hence, this curve over \mathbb{F}_{11} is singular, and the elliptic curve given by y^{2}=x^{3}+7x^{2}+5 has bad reduction at p=11. It also has bad reduction at p=2; in fact, we mentioned earlier that we cannot even write an elliptic curve in the form y^{2}=x^{3}+a_{2}x^{2}+a_{4}x+a_{6} when the field of coefficients have characteristic equal to 2. This is because such a curve will always be singular over such a field. The curve y^{2}=x^{3}+7x^{2}+5 remains nonsingular over all other primes, however; we also say that the curve has good reduction over all primes p except for p=2 and p=11.

In the case that an elliptic curve has bad reduction at p, we say that it has additive reduction if there is only one tangent line at the singular point (we also say that the singular point is a cusp), for example in the case of the curve y^{2}=x^{3}, and we say that it has multiplicative reduction if there are two distinct tangent lines at the singular point (in this case we say that the singular point is a node), for example in the case of the curve y^{2}=x^{3}-3x+2. If the slope of these tangent lines are given by elements of the same field as the coefficients of the curve (in our case rational numbers), we say that it has split multiplicative reduction, otherwise, we say that it has nonsplit multiplicative reduction. We note that since we are working with finite fields, what we describe as “tangent lines” are objects that we must define “algebraically”, as we have done earlier when describing the notion of a curve being singular.

As we have already seen in The Riemann Hypothesis for Curves over Finite Fields, whenever we have a curve over some finite field \mathbb{F}_{q} (where q=p^{n} for some natural number n), our curve will also have a finite number of points, and these points will have coordinates in \mathbb{F}_{q}. We denote the number of these points by N_{q}. In our case, we are interested in the case n=1, so that q=p. When our elliptic curve has good reduction over p, we define a quantity a_{p}, sometimes called the p-defect, or also known as the trace of Frobenius, as

\displaystyle a_{p}=p+1-N_{p}.

We can now define the Hasse-Weil L-function of an elliptic curve E as follows:

\displaystyle L_{E}(s)=\prod_{p}L_{p}(s)

where p runs over all prime numbers, and

\displaystyle L_{p}(s)=\frac{1}{(1-a_{p}p^{-s}+p^{1-2s})}    if E has good reduction at p

\displaystyle L_{p}(s)=\frac{1}{(1-p^{-s})}    if E has split multiplicative reduction at p

\displaystyle L_{p}(s)=\frac{1}{(1+p^{-s})}    if E has nonsplit multiplicative reduction at p

\displaystyle L_{p}(s)=1    if E has additive reduction at p.

The Hasse-Weil L-function encodes number-theoretic information related to the elliptic curve, and much of modern mathematical research involves this function. For example, the Birch and Swinnerton-Dyer conjecture says that the rank of the group formed by the rational points of the elliptic curve (see Elliptic Curves), also known as the Mordell-Weil group, is equal to the order of the zero of the Hasse-Weil L-function at s=1, i.e. we have the following Taylor series expansion of the Hasse-Weil L-function at s=1:

\displaystyle L_{E}(s)=c(s-1)^{r}+\text{higher order terms}

where c is a constant and r is the rank of the elliptic curve.

Meanwhile, the Shimura-Taniyama-Weil conjecture, now also known as the modularity conjecture, central to Andrew Wiles’s proof of Fermat’s Last Theorem, states that the Hasse-Weil L-function can be expressed as the following series:

\displaystyle L_{E}(s)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}

and the coefficients a_{n} are also the coefficients of the Fourier series expansion of some modular form f(E,\tau) (see The Moduli Space of Elliptic Curves):

\displaystyle f(E,\tau)=\sum_{n=1}^{\infty}a_{n}e^{2\pi i \tau}.

For more on the modularity theorem and Wiles’s proof of Fermat’s Last Theorem, the reader is encouraged to read the award-winning article A Marvelous Proof by Fernando Q. Gouvea, which is freely and legally available online. A link to this article (hosted on the website of the Mathematical Association of America) is provided among the list of references below.


Elliptic Curve on Wikipedia

Hasse-Weil Zeta Function on Wikipedia

Birch and Swinnerton-Dyer Conjecture on Wikipedia

Modularity Theorem on Wikipedia

Wiles’s Proof of Fermat’s Last Theorem on Wikipedia

The Birch and Swinnerton-Dyer Conjecture by Andrew Wiles

A Marvelous Proof by Fernando Q. Gouvea

A Friendly Introduction to Number Theory by Joseph H. Silverman

The Arithmetic of Elliptic Curves by Joseph H. Silverman

Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman

Invitation to the Mathematics of Fermat-Wiles by Yves Hellegouarch

A First Course in Modular Forms by Fred Diamond and Jerry Shurman

The Moduli Space of Elliptic Curves

A moduli space is a kind of “parameter space” that “classifies” mathematical objects. Every point of the moduli space stands for a mathematical object, in such a way that mathematical objects which are more similar to each other are closer and those that are more different from each other are farther apart. We may use the notion of equivalence relations (see Modular Arithmetic and Quotient Sets) to assign several objects which are in some sense “isomorphic” to each other to a single point.

We have discussed on this blog before one example of a moduli space – the projective line (see Projective Geometry). Every point on the projective line corresponds to a geometric object, a line through the origin. Two lines which have almost the same value of the slope will be closer on the projective line compared to two lines which are almost perpendicular.

Another example of a moduli space is that for circles on a plane – such a circle is specified by three real numbers, two coordinates for the center and one positive real number for the radius. Therefore the moduli space for circles on a plane will consist of a “half-volume” of some sort, like 3D space except that one coordinate is restricted to be strictly positive. But if we only care about the circles up to “congruence”, we can ignore the coordinates for the center – or we can also think of it as simply sending circles with the same radius to a single point, even if they are centered at different points. This moduli space is just the positive real line. Every point on this moduli space, which is a positive real number, corresponds to all the circles with radius equal to that positive real number.

We now want to construct the moduli space of elliptic curves. In order to do this we will need to first understand the meaning of the following statement:

Over the complex numbers, an elliptic curve is a torus.

We have already seen in Elliptic Curves what an elliptic curve looks like when graphed in the xy plane, where x and y are real numbers. This gives us a look at the points of the elliptic curve whose coordinates are real numbers, or to put it in another way, these are the real numbers x and y which satisfy the equation of the elliptic curve.

When we look at the points of the elliptic curve with complex coordinates, or in other words the complex numbers which satisfy the equation of the elliptic curve, the situation is more complicated. First off, what we actually have is not what we usually think of as a curve, but rather a surface, in the same way that the complex numbers do not form a line like the real numbers do, but instead form a plane. However, even though it is not easy to visualize, there is a function called the Weierstrass elliptic function which provides a correspondence between the (complex) points of an elliptic curve and the points in the “fundamental parallelogram” of a lattice in the complex plane. We can think of “gluing” the opposite sides of this fundamental parallelogram to obtain a torus. This is what we mean when we say that an elliptic curve is a torus. This also means that there is a correspondence between elliptic curves and lattices in the complex plane.

We will discuss more about lattices later on in this post, but first, just in case the preceding discussion seems a little contrived, we elaborate a bit on the Weierstrass elliptic function. We must first discuss the concept of a holomorphic function. We have discussed in An Intuitive Introduction to Calculus the concept of the derivative of a function. Now not all functions have derivatives that exist at all points; in the case that the derivative of the function does exist at all points, we refer to the function as a differentiable function.

The concept of a holomorphic function in complex analysis (analysis is the term usually used in modern mathematics to refer to calculus and its related subjects) is akin to the concept of a differentiable function in real analysis. The derivative is defined as the limit of a certain ratio as the numerator and the denominator both approach zero; on the real line, there are limited ways in which these quantities can approach zero, but on the complex plane, they can approach zero from several different directions; for a function to be holomorphic, the expression for its derivative must remain the same regardless of the direction by which we approach zero.

In previous posts on topology on this blog we have been treating two different topological spaces as essentially the same whenever we can find a bijective and continuous function (also known as a homeomorphism) between them; similarly, we have been treating different algebraic structures such as groups, rings, modules, and vector spaces as essentially the same whenever we can find a bijective homomorphism (an isomorphism) between two such structures. Following these ideas and applying them to complex analysis, we may treat two spaces as essentially the same if we can find a bijective holomorphic function between them.

The Weierstrass elliptic function is not quite holomorphic, but is meromorphic – this means that it would have been holomorphic everywhere if not for the “lattice points” where there exist “poles”. But it is alright for us, because such a lattice point is to be mapped to the “point at infinity”. All in all, this allows us to think of the complex points of the elliptic curve as being essentially the same as a torus, following the ideas discussed in the preceding paragraph.

Moreover, the torus has a group structure of its own, considered as the direct product group \text{U}(1)\times\text{U}(1) where \text{U}(1) is the group of complex numbers of magnitude equal to 1 with the law of composition given by the multiplication of complex numbers. When the complex points of the elliptic curve get mapped by the Weierstrass elliptic function to the points of the torus, the group structure provided by the “tangent and chord” or “tangent and secant” construction becomes the group structure of the torus. In other words, the Weierstrass elliptic function provides us with a group isomorphism.

All this discussion means that the study of elliptic curves becomes the study of lattices in the complex plane. Therefore, what we want to construct is the moduli space of lattices in the complex plane, up to a certain equivalence relation – two lattices are to be considered equivalent if one can be obtained by multiplying the other by a complex number (this equivalence relation is called homothety). Going back to elliptic curves, this corresponds to an isomorphism of elliptic curves in the sense of algebraic geometry.

Now given two complex numbers \omega_{1} and \omega_{2}, a lattice \Lambda in the complex plane is given by


For example, setting \omega_{1}=1 and \omega_{2}=i, gives a “square” lattice. This lattice is also the set of all Gaussian integers. The fundamental parallelogram is the parallelogram formed by the vertices 0, \omega_{1}, \omega_{2}, and \omega_{1}+\omega_{2}. Here is an example of a lattice, courtesy of used Alvaro Lozano Robledo of Wikipedia:


The fundamental parallelogram is in blue. Here is another, courtesy of user Sam Derbyshire of Wikipedia:


Because we only care about lattices up to homothety, we can “rescale” the lattice by multiplying it with a complex number equal to \frac{1}{\omega_{1}}, so that we have a new lattice equivalent under homothety to the old one, given by



\displaystyle \tau=\frac{\omega_{2}}{\omega_{1}}.

We can always interchange \omega_{1} and \omega_{2}, but we will fix our convention so that the complex number \tau=\frac{\omega_{2}}{\omega_{1}}, when written in polar form \tau=re^{i\theta} always has a positive angle \theta between 0 and 180 degrees. If we cannot obtain this using our choice of \omega_{1} and \omega_{2}, then we switch the two.

Now what this means is that a complex number \omega, which we note is a complex number in the upper half plane \mathbb{H}=\{z\in \mathbb{C}|\text{Im}(z)>0\}, because of our convention in choosing \omega_{1} and \omega_{2}, uniquely specifies a homothety class of lattices \Lambda. However, a homothety class of lattices may not always uniquely specify such a complex number \tau. Several such complex numbers may refer to the same homothety class of lattices.

What \omega_{1} and \omega_{2} specify is a choice of basis (see More on Vector Spaces and Modules) for the lattice \Lambda; we may choose several different bases to refer to the same lattice. Hence, the upper half plane is not yet the moduli space of all lattices in the complex plane (up to homothety); instead it is an example of what is called a Teichmuller space. To obtain the moduli space from the Teichmuller space, we need to figure out when two different bases specify lattices that are homothetic.

We will just write down the answer here; two complex numbers \tau and \tau' refer to homothetic lattices if there exists the following relation between them:

\displaystyle \tau'=\frac{a\tau+b}{c\tau+d}

for integers abc, and d satisfying the identity

\displaystyle ad-bc=1.

We can “encode” this information into a 2\times 2 matrix (see Matrices) which is an element of the group (see Groups) called \text{SL}(2,\mathbb{Z}). It is the group of 2\times 2 matrices with integer entries and determinant equal to 1. Actually, the matrix with entries abc, and d and the matrix with entries -a-b-c, and -d specify the same transformation, therefore what we actually want is the group called \text{PSL}(2,\mathbb{Z}), also known as the modular group, and also written \Gamma(1), obtained from the group \text{SL}(2,\mathbb{Z}) by considering two matrices to be equivalent if one is the negative of the other.

We now have the moduli space that we want – we start with the upper half plane \mathbb{H}, and then we identify two points if we can map one point into the other via the action of an element of the modular group, as we have discussed earlier. In technical language, we say that they belong to the same orbit. We can write our moduli space as \Gamma(1)\backslash\mathbb{H} (the notation means that the group \Gamma(1) acts on \mathbb{H} “on the left”).

When dealing with quotient sets, which are sets of equivalence classes, we have seen in Modular Arithmetic and Quotient Sets that we can choose from an equivalence class one element to serve as the “representative” of this equivalence class. For our moduli space \Gamma(1)\backslash\mathbb{H}, we can choose for the representative of an equivalence class a point from the “fundamental domain” for the modular group. Any point on the upper half plane can be obtained by acting on a point from the fundamental domain with an element of the modular group. The following diagram, courtesy of user Fropuff on Wikipedia, shows the fundamental domain in gray:


The other parts of the diagram show where the fundamental domain gets mapped to by certain special elements, in particular the “generators” of the modular group, which are the two elements where a=0, b=-1, c=1, and d=-1, and a=1, b=1, c=1, and d=0. We will not discuss too much of these concepts for now. Instead we will give a preview of some concepts related to this moduli space. Topologically, this moduli space looks like a sphere with a missing point; in order to make the moduli space into a sphere (topologically), we take the union of the upper half plane \mathbb{H} with the projective line (see Projective Geometry) \mathbb{P}^{1}(\mathbb{Q}). This projective line may be thought of as the set of all rational numbers \mathbb{Q} together with a “point at infinity.” The modular group also acts on this projective line, so we can now take the quotient of \mathbb{H}\cup\mathbb{P}^{1}(\mathbb{Q}) (denoted \mathbb{H}^{*} by the same equivalence relation as earlier; this new space, topologically equivalent to the sphere, is called the modular curve X(1).

The functions and “differential forms” on the modular curve X(1) are of special interest. They can be obtained from functions on the upper half plane (with the “point at infinity”) satisfying certain conditions related to the modular group. These functions are called weakly modular functions; if they are holomorphic everywhere except at the “point at infinity”, they are called modular functions and if they are holomorphic everywhere, including the “point at infinity”, they are called modular forms. Modular forms are an interesting object of study in themselves, and their generalizations, automorphic forms, are a very active part of modern mathematical research.

Moduli Space on Wikipedia

Elliptic Curve on Wikipedia

Weierstrass’s Elliptic Functions on Wikipedia

Fundamental Pair of Periods on Wikipedia

Modular Group on Wikipedia

Fundamental Domain on Wikipedia

Modular Form on Wikipedia

Automorphic Form on Wikipedia

Image by User Alvano Lozano Robledo of Wikipedia

Image by User Sam Derbyshire of Wikipedia

Image by User Fropuff of Wikipedia

Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman

A First Course in Modular Forms by Fred Diamond and Jerry Shurman

Elliptic Curves

An elliptic curve (not to be confused with an ellipse) is a certain kind of polynomial equation which can usually be expressed in the form

\displaystyle y^{2}=x^{3}+ax+b

where a and b satisfy the condition that the quantity

\displaystyle 4a^{3}+27b^{2}

is not equal to zero. This is not the most general form of an elliptic curve, as it will not hold for coefficients of “finite characteristic” equal to 2 or 3; however, for our present purposes, this definition will suffice.

Examples of elliptic curves are the following:

\displaystyle y^{2}=x^{3}-x

\displaystyle y^{2}=x^{3}-x+1

which, for real x and y may be graphed in the “Cartesian” or “xy” plane as follows (image courtesy of user YassineMrabet of Wikipedia):


This rather simple mathematical object has very interesting properties which make it a central object of study in many areas of modern mathematical research.

In this post we focus mainly on one of these many interesting properties, which is the following:

The points of an elliptic curve form a group.

A group is a set with a law of composition which is associative, and the set contains an “identity element” under this law of composition, and every element of this set has an “inverse” (see Groups). Now this law of composition applies whether the points of the elliptic curve have rational numbers, real numbers, or complex numbers for coordinates, and it is always given by the same formula. It is perhaps most visible if we consider real numbers, since in that case we can plot it on the xy plane as we have done earlier. The law of composition is also often called the “tangent and chord” or “tangent and secant” construction.

We now expound on this construction. Given two points on the elliptic curve P and Q on the curve, we draw a line passing through both of them. In most cases, this line will pass through another point R on the curve. Then we draw a vertical line that passes through the point R. This vertical line will pass through another point R' on the curve. This gives us the law of composition of the points of the elliptic curve, and we write P+Q=R'. Here is an image courtesy of user SuperManu of Wikipedia:


The usual case that we have described is on the left; the other three images show other different cases where the line drawn does not necessarily go through three points. This happens, for example, when the line is tangent to the curve at some point Q, as in the second picture; in this case, we think of the line as passing through Q twice. Therefore, when we compute P+Q, the third point is Q itself, and it is through Q that we draw our vertical line to locate Q', which is equal to P+Q.

The second picture also shows another computation, that of Q+Q, or 2Q. Again, since this necessitates taking a line that passes through the point Q twice, this means that the line must be tangent to the elliptic curve at Q. The third point that it passes through is the point P, and we draw the vertical line through P to find the point P', which is equal to 2Q.

Now we discuss the case described by the third picture, where the line going through the two points P and Q which we want to “add” is a vertical line. To explain what happens, we need the notion of a “point at infinity” (see Projective Geometry). We write the point at infinity as 0, expressing the idea that it is the identity element of our group. We cannot find this point at infinity in the xy plane, but we can think of it as the third point that the vertical line passes through aside from P and Q. In this case, of course, there is no need to draw another vertical line – we simply write P+Q=0.

Finally we come to the case described by the fourth picture; this is simply a combination of the earlier cases we have described above. The vertical line is tangent to the curve at the point P, so we can think of it as passing through P twice, and the third point is passes through is the point at infinity 0, so we can write 2P=0.

We will not prove explicitly that the points form a group under this law of composition, i.e. that the conditions for a set to form a group are satisfied by our procedure, but it is an interesting exercise to attempt to do so; readers may try it out for themselves or consult the references provided at the end of the post. It is worth mentioning that our group is also an abelian group, i.e. we have P+Q=Q+P, and hence we have written our law of composition “additively”.

Now, to make the group law apply even when x and y are not real numbers, we need to write this procedure algebraically. This is a very powerful approach, since this allows us to operate with mathematical concepts even when we cannot visualize them.

Let x_{P} and y_{P} be the x and y coordinates of a point P, and let x_{Q} and y_{Q} be the x and y coordinates of another point Q. Let

\displaystyle m=\frac{y_{Q}-y_{P}}{x_{Q}-x_{P}}

be the slope of the line that connects the points P and Q. Then the point P+Q has x and y coordinates given by the following formulas:

\displaystyle x_{P+Q}=m^{2}-x_{P}-x_{Q}

\displaystyle y_{P+Q}=-y_{P}-m(x_{P+Q}-x_{P})

In the case that Q is the same point as P, then we define the slope of the tangent line to the elliptic curve at the point P using the formula

\displaystyle m=\frac{3x_{P}^{2}+a}{2y_{P}}

where a is the coefficient of x in the formula, of the elliptic curve, i.e.

\displaystyle y^{2}=x^{3}+ax+b.

Then the x and y coordinates of the point 2P are given by the same formulas as above, appropriately modified to reflect the fact that now the points P and Q are the same:

\displaystyle x_{2P}=m^{2}-2x_{P}

\displaystyle y_{2P}=-y_{P}-m(x_{2P}-x_{P})

This covers the first two cases in the image above; for the third case, when P and Q are distinct points and y_{P}=-y_{Q}, we simply set P+Q=0. For the fourth case, when P and Q refer to the same point, and y_{P}=0, we set 2P=0. The point at infinity itself can be treated as a mere point and play into our computations, by setting P+0=P, reflecting its role as the identity element of the group.

The group structure on the points of elliptic curves have practical applications in cryptography, which is the study of “encrypting” information so that it cannot be deciphered by parties other than the intended recipients, for example in military applications, or when performing financial transactions over the internet.

On the purely mathematical side, the study of the group structure is currently a very active field of research. An important theorem called Mordell’s theorem states that even though there may be an infinite number of points whose coordinates are given by rational numbers (called rational points), these points may all be obtained by performing the “tangent and chord” or “tangent and secant” construction on a finite number of points. In more technical terms, the group of rational points on an elliptic curve is finitely generated.

There is a theorem concerning finitely generated abelian groups stating that any finitely generated abelian group G is isomorphic to the direct sum of r copies of the integers and a finite abelian group called the torsion subgroup of G. The number r is called the rank of G. The famous Birch and Swinnerton-Dyer conjecture, which currently carries a million dollar prize for its proof (or disproof), concerns the rank of the finitely generated abelian group of rational points on an elliptic curve.

Another thing that we can do with elliptic curves is use them to obtain representations of Galois groups (see Galois Groups). A representation of a group G on a vector space V over a field K is a homomorphism from G to GL(V), the group of bijective linear transformations of the vector space V to itself. We know of course from Matrices that linear transformations of vector spaces can always be written as matrices (in our case the matrices must have nonzero determinant to ensure that the linear transformations are bijective). Representation theory allows us to study the objects of abstract algebra using the methods of linear algebra.

To any elliptic curve we can associate a certain algebraic number field (see Algebraic Numbers). The elements of these algebraic number fields are “generated” by the algebraic numbers that provide the coordinates of “p-torsion” points of the elliptic curve, i.e. those points P for which pP=0 for some prime number p.

The set of p-torsion points of the elliptic curve is a 2-dimensional vector space over the finite field \mathbb{Z}/p\mathbb{Z} (see Modular Arithmetic and Quotient Sets), also written as \mathbb{F}_{p}. Among other things this means that we can choose two p-torsion points P and Q of the elliptic curve such that any other p-torsion point can be written as aP+bQ for integers a and b between 0 and p-1. When an element of the Galois group of the algebraic number field generated by the coordinates of the p-torsion points of the elliptic curve permutes the elements of the algebraic number field, it also permutes the p-torsion points of the elliptic curve. This permutation can then be represented by a 2\times 2 matrix with coefficients in \mathbb{F}_{p}.

The connection between Galois groups and elliptic curves is a concept that is central to many developments and open problems in mathematics. It plays a part, for example in the proof of the famous problem called Fermat’s Last Theorem. It is also related to the open problem called the Kronecker Jugendtraum (which is German for Kronecker’s Childhood Dream, and named after the mathematician Leopold Kronecker), also known as Hilbert’s Twelfth Problem, which seeks a procedure for obtaining all field extensions of algebraic number fields whose Galois group is an abelian group. This problem has been solved only in the special case of imaginary quadratic fields, and the solution involves special kinds of “symmetries” of elliptic curves called complex multiplication (not to be confused with the multiplication of complex numbers). David Hilbert, who is one of the most revered mathematicians in history, is said to have referred to the theory of complex multiplication as “…not only the most beautiful part of mathematics but of all science.”


Elliptic Curve on Wikipedia

Mordell-Weil Theorem on Wikipedia

Birch and Swinnerton-Dyer Conjecture on Wikipedia

Wiles’ Proof of Fermat’s Last Theorem on Wikipedia

Hilbert’s Twelfth Problem on Wikipedia

Complex Multiplication on Wikipedia

Image by User YassineMrabet of Wikipedia

Image by User SuperManu of Wikipedia

Fearless Symmetry: Exposing the Hidden Patterns of Numbers by Avner Ash and Robert Gross

Elliptic Tales: Curves, Counting, and Number Theory by Avner Ash and Robert Gross

Rational Points on Elliptic Curves by Joseph H. Silverman

Valuations and Completions

In ordinary everyday life, there are several notions of closeness. There is for example, a physical notion of distance, and we say, for instance, that we are close to our next-door neighbors. But there is another sense of closeness, such that we can say that we are “close” to our relatives, or to our friends, even though physically they may be far away.

There is also a similar notion of “closeness” between numbers. The most basic method is provided by the familiar “absolute value“. Given three numbers x, x_{1}, and x_{2}, to say that x is closer to x_{1} than to x_{2} means that |x-x_{1}|<|x-x_{2}|. So for example, since |(-1)-(2)|=3 and |(8)-(2)|=6, we therefore say that the number 2 is “closer” to the number -1 than to the number 8. In other words, the smaller the value of |x-y|, the closer x and y are to each other.

But there are also other notions of “closeness” for numbers, just as we have explained above, that with our relatives or friends we may be “close” even if we are far away from each other. Consider the numbers 1 and 10001. Simply by looking, they can perhaps be said to be “relatives” or “friends”, which makes them in some way closer than, say, 1 and 18. The same may be said for 5 and 3000005, that they are perhaps members of the same “family”. This is, of course, because their difference is divisible by a large power of 10, and since we use the decimal system to write our numbers, there is some sort of visual cue that these numbers are “family members”.

But in number theory, 10 is not really very special. Perhaps it just so happens that we have 10 fingers which we use for counting, so we used 10 as a base for our number system. What is really special in number theory are the prime numbers. So for our notion of closeness we choose a prime, and define our measure of closeness so that two numbers are closer together whenever their difference is divisible by a large power of that prime number. For our chosen prime p, we want an analogue of the absolute value, which we will call the p-adic absolute value, and written |x-y|_{p}, which is smaller if the difference of x and y is divisible by a large power of p. The “ordinary” absolute value will now be denoted by |x-y|_{\infty}.

We want to define this for rational numbers as follows. Given a rational number a, we express it as


such that b, c, and p are mutually prime, i.e. they have no factors in common except 1. Then we set


We can see that this definition gives us the properties we are looking for – the value of |a|_{p} is indeed smaller if a is divisible by a large power of p.

The absolute value (both the “ordinary” absolute value and the p-adic absolute value) is also called the multiplicative valuation. There is also a related notion called the exponential valuation, which, in the p-adic case, we denote by v_{p}(a) for a rational number a. The exponential valuation is obtained from the multiplicative valuation by setting


In the case above, where a=p^{m}\frac{b}{c} and b, c, and p are mutually prime, we simply have


For the ordinary absolute value, we just set


where \text{ln } of course stands for the natural logarithm.

The concept of “closeness” between numbers, even just the “ordinary” one, was used to discover something interesting about the number line. If it was merely composed of the rational numbers, then there would be “gaps” in the line. To make a “true” number line, one must fill in these gaps, and this lead to the construction of the real numbers by the mathematician Richard Dedekind in the 19th century.

We elaborate on the nature of these “gaps”, following closely the idea behind Dedekind’s construction. Consider the real number \sqrt{2}. It is known from ancient times that this number cannot be written as a ratio of two integers and is therefore not a rational number. However, we can construct an infinite sequence of rational numbers such that every successive rational number in the sequence is “closer” to \sqrt{2}, compared to the one before it.

The mathematician Leopolod Kronecker once claimed, “God made the integers, all else is the work of man.” We know how to construct the rational numbers from the integers (for those who would like to think of the natural numbers as being even more basic than the integers, it is also easy to construct the integers from the natural numbers), by taking pairs of integers, and considering sets of equivalence classes (see Modular Arithmetic and Quotient Sets) of these pairs; for example, we set \frac{1}{2} and \frac{2}{4} as equivalent, because “cross multiplication” on the numerators and denominators gives us the same result. So the rational numbers are really equivalence classes of pairs of integers.

The problem we face now is how to construct the real numbers from the rational numbers. We have seen that we can construct sequences which “converge” in some sense to some value that is not a rational number. By “converge”, we mean that successive terms become closer and closer to each other late in the sequence. Technically, we do not refer to such a sequence as a convergent sequence, since it is a sequence of rational numbers but it does not converge to a rational number. Instead, we refer to it as a Cauchy sequence.

And this gives us a possible solution to our problem above – we could simply define the real numbers as the set of all Cauchy sequences. Those that converge to a rational number “represent” that rational number, and those that do not “represent” an irrational number such as \sqrt{2}. However, there is still one more problem that we have to take care of. There may be more than one Cauchy sequence that “represents” a certain rational or irrational number.

Consider, for instance, the sequence

\displaystyle 5,5,5,5,5,...

which obviously converges to the rational number 5, and consider another sequence

\displaystyle 6,5,5,5,5,...

which is different in the first term but similarly converges to the rational number 5. They are different sequences, but they “represent” the same rational number. We would like to have a method of “identifying” these two sequences under some equivalence relation. In order to do this, we consider the “difference” of these two sequences:

\displaystyle 1,0,0,0,0,...

We see that it converges to 0. Such a sequence is called a nullsequence, and this gives us our equivalence relation – two Cauchy sequences are to be considered equivalent if they differ by a nullsequence. The set of real numbers \mathbb{R} is then defined as the set of equivalence classes of Cauchy sequences under this equivalence relation.

The process of “filling in” the “gaps” between the rational numbers is called completion. Note that a notion of “closeness” is important in the process of completion. If we had a different notion of closeness, for example, by using the p-adic absolute value instead of the ordinary absolute value, we would obtain a different kind of completion. Instead of the real numbers \mathbb{R}, we would have instead the p-adic numbers \mathbb{Q}_{p}. The p-adic numbers play an important role in number theory, as they encode information related to primes.


Valuation on Wikipedia

Complete Metric Space on Wikipedia

p-adic Number on Wikipedia

Algebraic Number Theory by Jurgen Neukirch

Algebraic Number Theory by J. W. S. Cassels and A. Frohlich

Divisors and the Picard Group

In this post, once again focusing on the subject of algebraic geometry, we will consider a “curve”, which, confusingly, refers what we usually think of as a surface. The reason for this is that if we are considering complex numbers x and y, an equation such as y^{2}=x^{3}-x, which we would normally think of as a “curve” if x and y were real numbers, actually refers to something that looks like a surface, in the same way the real numbers form a line and complex numbers form a plane. We will rely on this intuition and leave the more formal definitions of curves, surfaces, and dimension to the references for now.

A divisor is a finite “linear combination” of points on the curve, with integer coefficients. For example, if we have points P_{1} and P_{2} on the curve, we can have something like

\displaystyle 5P_{1}-3P_{2}.

The degree of a divisor is the sum of its coefficients. For the example above, the degree is equal to 2.

A special kind of divisor called a principal divisor comes from  so-called “rational functions” (which, despite the name, may not really be “functions” in the set-theoretic sense but merely expressions involving a “fraction” whose numerator and denominator are both polynomials) on the curve. We let the coefficients of each point denote the “order of vanishing” of the function. For instance, the function

\displaystyle \frac{x(x-1)^{2}}{(x-3)^5}

gives rise to the principal divisor


where P_{1} is the point x=0P_{2} is the point x=1, and P_{3} is the point x=3.

The Picard group of a curve is a group (whose law of composition is given by addition – see also Groups) obtained from the divisors by considering two divisors D and D' equivalent (see Modular Arithmetic and Quotient Sets) if their difference D-D' is a  principal divisor. An element of the Picard group is also called a divisor class.

The Picard group of a curve can say a lot of things about the curve. For instance, it can be used to prove that on the curve y^{2}=x^{3}-x, which is an example of what is called an elliptic curve, the points form a group. The group structure on the elliptic curve, along with other properties such as its being a Riemann surface (a surface which “locally” looks like the complex plane), makes it one of the most interesting objects in mathematics.

The Picard group is also important because its elements, the divisor classes on the curve, correspond to line bundles (vector bundles of dimension 1 – see Vector Fields, Vector Bundles, and Fiber Bundles – but do keep in mind our discussion earlier regarding complex numbers and how this changes our conventions regarding dimension, as in the case of the line and the plane, and curves and surfaces) on the curve. Line bundles are also related to sheaves, in particular those called “locally free sheaves of rank 1” (more general vector bundles correspond to locally free sheaves of finite rank). There is, therefore, a relation between the concept of divisors, the concept of vector bundles, and the concept of sheaves.

We now relate the theory of divisors and the Picard group to number theory. We have mentioned in Localization that we can obtain a scheme out of the integers \mathbb{Z}; the points of this scheme are the prime ideals of \mathbb{Z}, and the set of all these points (prime ideals) we call \text{Spec }\mathbb{Z}. As we can make a scheme out of a more general ring, we can therefore make a scheme out of the ring of integers \mathcal{O}_{K} of an algebraic number field K (see Algebraic Numbers); its points will be the prime ideals of \mathcal{O}_{K}, and the “rational functions” on this scheme will be the elements of K.

In this case, the divisors are made up of “linear combinations” of prime ideals. The principal divisors, which come from rational functions, then correspond, accordingly, to principal fractional ideals, ideals which are generated by a single element of K, which as we have mentioned above correspond to the rational functions. Finally, the Picard group is none other than the ideal class group, which “measures” the failure of unique factorization in an algebraic number field!

More explicitly, an example of a divisor may be written in this way:

\displaystyle 5\mathfrak{p}_{1}-3\mathfrak{p}_{2}

for prime ideals \mathfrak{p}_{1} and \mathfrak{p}_{2}, which as we have said correspond to points. For a principal divisor, we may have, for example, the following element of the rational numbers \mathbb{Q}

\displaystyle \frac{63}{64}

which generates the principal fractional ideal

\displaystyle (\frac{63}{64})=\{...,-\frac{189}{64},-\frac{126}{64},-\frac{63}{64},0,\frac{63}{64},\frac{126}{64},\frac{189}{64},...\}

which in turn gives us the principal divisor

\displaystyle 2\mathfrak{p}_{1}+\mathfrak{p}_{2}-6\mathfrak{p}_{3}

where \mathfrak{p}_{1}=(3)\mathfrak{p}_{2}=(7), and \mathfrak{p}_{3}=(2), the principal ideals generated by 3, 7, and 2 respectively. Note that if we “factorize” the numerator and denominator of \frac{63}{64}, we obtain

\displaystyle \frac{63}{64}=\frac{3^{2}\cdot 7}{2^{6}}.

More generally, we should “factorize” in terms of ideals, in case we don’t have unique factorization:

\displaystyle (\frac{63}{64})=\frac{(3)^{2}(7)}{(2)^{6}}.

The coefficients of a principal divisor, measuring “how much” of a certain prime is in the factorization of the principal fractional ideal it corresponds to, are called valuations. The theory of valuations offers us another way to develop the entire field of algebraic number theory under a new perspective.


Divisor on Wikipedia

Picard Group on Wikipedia

Algebraic Geometry by Robin Hartshorne

Algebraic Number Theory by Jurgen Neukirch

Etale Cohomology of Fields and Galois Cohomology

In Cohomology in Algebraic Geometry we have introduced sheaf cohomology and Cech cohomology as well as the concept of etale morphisms, and the Grothendieck topology (see More Category Theory: The Grothendieck Topos) that it defines. In this post, we give one important application of these ideas, related to the ideas discussed in Galois Groups.

Let K be a field (see Rings, Fields, and Ideals). A field has only two ideals: (0) and (1), the latter of which is the unit ideal and is therefore the entire field itself as well. Its only prime ideal (which is also a maximal ideal) is (0); recall that in algebraic geometry (see Basics of Algebraic Geometry), the “points” of the mathematical object we call a scheme correspond (locally, at least) to the prime ideals of a ring R, and we refer to this set of “points” as \text{Spec }R. Therefore, for the field K, \text{Spec }K=(0), in other words, \text{Spec }K is made up of a single point.

Now we need to define sheaves on \text{Spec }K. Using ordinary concepts of topology will not be very productive, since our topological space consists only of a single point; therefore, we will not be able to obtain any interesting open covers out of this topological space. However, using the ideas in More Category Theory: The Grothendieck Topos, we can “expand” our idea of open covers. Instead of inclusions of open sets, we will instead make use of etale morphisms, as we have discussed in Cohomology in Algebraic Geometry.

Let K\rightarrow L be an etale morphism. This also means that L is an etale K-algebra (see also The Hom and Tensor Functors for the definition of algebra in our context). It is a theorem that an etale K-algebra is a direct product of finitely many separable field extensions of K (see Algebraic Numbers).

The definition of presheaf and sheaf remains the same, however the sheaf conditions can be restated in our case as the following (perhaps easier to understand) statement, which we copy verbatim from the book Etale Cohomology and the Weil Conjecture by Eberhard Freitag and Reinhardt Kiehl:

The elements s\in \mathcal{F}(B) correspond one-to-one to families of elements

s_{i}\in\mathcal{F}(B_{i}), i\in I

having the property

\text{Image }(s_{i})=\text{Image }(s_{j}), in (B_{i}\otimes_{B}B_{j})

This condition must also hold for i=j!

A separable closure \bar{K} of K is a separable field extension of K (see Cohomology in Algebraic Geometry) that is a subfield of the algebraic closure of K. The algebraic closure of K is an algebraic extension (see Algebraic Numbers) of K which is algebraically closed, i.e., it contains all the roots of polynomials with coefficients in this algebraic extension. Both the algebraic closure and the separable closure of K are unique up to isomorphism. In the case of the field of rational numbers \mathbb{Q}, the separable closure and the algebraic closure coincide and they are both equal to the field of algebraic numbers.

Given the separable closure \bar{K} of K, we define \mathcal{F}(\bar{K}) as the stalk (see Localization) of the sheaf at \bar{K}. It is also written using the language of direct limits (also called an inductive limit):

\displaystyle \mathcal{F}(\bar{K})=\varinjlim\mathcal{F}(L)

We digress slightly in order to explain what this means. The language of direct limits and inverse limits (the latter are also called projective limits) are ubiquitous in abstract algebra, algebraic geometry, and algebraic number theory, and are special cases of the notion of limits we have discussed in Even More Category Theory: The Elementary Topos.

A directed set I is an ordered set in which for every pair i,j there exists k such that i\leq k,j\leq k. A direct, resp. inverse system over I is a family \{A_{i},f_{ij}|i,j\in I,i\leq j\} of objects A_{i} and morphisms f_{ij}: A_{i}\rightarrow A_{j}, resp. f_{ij}: A_{j}\rightarrow A_{i} such that

\displaystyle f_{ii} is the identity map of A_{i}, and

\displaystyle f_{ik}=f_{jk}\circ f_{ij} resp. f_{ik}=f_{ij}\circ f_{jk}

for all i\leq j\leq k .

The direct limit of a direct system is then defined as the quotient

\displaystyle \varinjlim_{i\in I} A_{i}=\coprod_{i\in I} A_{i}/\sim

where two elements x_{i}\in A_{i} and x_{j}\in A_{j} are considered equivalent, x_{i}\sim x_{j} if there exists k such that f_{ik}(x_{i})=f_{jk}(x_{j}).

Meanwhile, the inverse limit of an inverse system is the subset

\displaystyle \varprojlim_{i\in I} A_{i}=\{(x_{i})_{i\in I}\in \prod_{i\in I}A_{i}|f_{ij}(x_{j})=x_{i}\text{ for }i\leq j \}

of the product \displaystyle \prod_{i\in I}A_{i}.

The classical definition of stalk, for a sheaf \mathcal{F} can then also be expressed as the direct limit of the direct system given by the sets (or abelian groups, or modules, etc.) \mathcal{F}(U) and the restriction maps \rho_{UV}: \mathcal{F(U)}\rightarrow \mathcal{F}(V) for open sets V\subseteq U. In our case, of course, instead of inclusion maps V\subseteq U we instead have more general maps induced by etale morphisms.

An example of an etale sheaf over \text{Spec }K is given by the following: Let

\displaystyle \mathcal{G}_{m}(B)=B^{*} where B^{*} is the multiplicative group of the etale K-algebra B.

In this case we have \mathcal{F}(\bar{K})=\bar{K}^{*}, the multiplicative group of the separable closure \bar{K} of K. We note that the multiplicative group of a field F is just the group F-\{0\}, with the law of composition given by multiplication.

In order to make contact with the theory of Galois groups, we now define the concept of G-modules, where G is a group. A left G-module is given by an abelian group M and a map \rho: G\times M\rightarrow M such that

\displaystyle \rho(e,a)=x,

\displaystyle \rho (gh,a)=\rho(g,\rho(h,a)),


\displaystyle \rho(g,(ab))=\rho(g,a)\rho(g,b).

Instead of \rho(g,a) we usually just write g\cdot a. A right G-module may be similarly defined, and may be obtained from a left G-module by defining a\cdot g=g^{-1}\cdot a.

The abelian group \mathcal{F}(\bar{K}) has the structure of a G-module, where G is the Galois group \text{Gal}(\bar{K}/K) (also written as G(\bar{K}/K)), the group of field automorphisms of \bar{K} that keep K fixed.

We see now that there is a connection between Galois theory and etale sheaves over a field. More generally, there is a connection between the Etale cohomology of a field and “Galois cohomology“, an important part of algebraic number theory that we now define. Galois cohomology is the derived functor (see More on Chain Complexes and The Hom and Tensor Functors) of the fixed module functor.

First we construct the standard resolution of the the Galois module (a G-module where G is the Galois group of some field extension) A. It is given by X^{n}(G,A), the abelian group of all functions from the direct product G^{n+1} to A, and the coboundary map

\displaystyle \partial^{n}: X^{n-1}\rightarrow X^{n}

given by

\displaystyle \partial^{n}x(\sigma_{0},...,\sigma{n})=\sum_{i=0}^{n}(-1)^{i}x(\sigma_{0},...,\hat{\sigma_{i}},...,\sigma_{n})

where \hat{\sigma_{i}} signifies that \sigma_{i} is to be omitted.

We now apply the fixed module functor to obtain the cochain complex

\displaystyle C^{n}(G,A)=X^{n}(G,A)^{G}.

The elements of C^{n}(G,A) are the functions x: G^{n+1}\rightarrow A such that

\displaystyle x(\sigma\sigma_{0},...,\sigma\sigma_{n})=\sigma x(\sigma_{0},...,\sigma_{n})

for all \sigma\in G.

The Galois cohomology groups H^{n}(G,A) are then obtained by taking the cohomology of this cochain complex, i.e.

\displaystyle H^{n}(G,A)=\text{Ker }\partial^{n+1}/\text{Im }\partial^{n}

Note: We have adopted here the notation of the book Cohomology of Number Fields by Jurgen Neukirch, Alexander Schmidt, and Kay Wingberg. Some references use a different notation; for instance X_{n} may be defined as the abelian group of functions from G^{n} to A instead of from G^{n+1} to A. This results in different notation for the cochain complexes and their boundary operators; however, the Galois cohomology groups themselves will remain the same.

It is a basic result of Galois cohomology that H^{0}(G,A) gives A^{G}, the subset of A such that \sigma\cdot a=a for all \sigma\in G. In other words, A^{G} is the subset of A that is fixed by G.

We have the following connection between Etale cohomology for fields and Galois cohomology:

\displaystyle H^{n}(K,\mathcal{F})=H^{n}(G,\mathcal{F}(\bar{K}))

We now mention some other basic results of the theory. In analogy with sheaf cohomology, the group H^{0}(K,\mathcal{F}) is just the set of “global sections” \Gamma(K,\mathcal{F})=\mathcal{F}(K) of \mathcal{F}. Letting \mathcal{F}=\mathcal{G}_{m} which we have defined earlier, we have

\displaystyle H^{0}(K,\mathcal{G}_{m})=\mathcal{G}_{m}(K)=K^{*}

In the language of Galois cohomology,

\displaystyle H^{0}(G,\mathcal{G}_{m}(\bar{K}))=(\bar{K}^{*})^{G}=K^{*}

Meanwhile, for H^{1}, we have the following result, called Hilbert’s Theorem 90:

\displaystyle H^{1}(K,\mathcal{G}_{m})=H^{1}(G,\bar{K}^{*})=\{1\}.

The group H^{2}(K,\mathcal{G}_{m})=H^{2}(G,\bar{K}^{*}) is called the Brauer group and also plays an important part in algebraic number theory. The etale cohomology of fields, or equivalently, Galois cohomology, are the topic of famous problems in modern mathematics such as the Milnor conjecture and its generalization, the Bloch-Kato conjecture, which was solved by Vladimir Voevodsky in 2009. They also play an important part in the etale cohomology of more general rings.


Etale Cohomology on Wikipedia

Stalk on Wikipedia

Direct Limit on Wikipedia

Inverse Limit on Wikipedia

Hilbert’s Theorem 90 on Wikipedia

Group Cohomology on Wikipedia

Galois Cohomology on Wikipedia

Milnor Conjecture on Wikipedia

Norm Residue Isomorphism Theorem

Etale Cohomology and the Weil Conjecture by Eberhard Freitag and Reinhardt Kiehl

Cohomology of Number Fields by Jurgen Neukirch, Alexander Schmidt, and Kay Wingberg