In Galois Representations we mentioned briefly that Galois representations can be obtained from modular forms. In this post we elaborate more on this construction, in the case that the modular form is a weight eigenform (a weight cusp form that is a simultaneous eigenfunction for all Hecke operators not dividing the level ). This specific case is also known as the Shimura construction, after Goro Shimura.
Let be a weight Hecke eigenform, of some level (this also works with other level structures). We want to construct a -adic Galois representation associated to this Hecke eigenform, such that the two are going to be related in the following manner. For every prime not dividing and not equal to , the characteristic polynomial of the image of the Frobenius element associated to under this Galois representation will be of the form
where is the eigenvalue of the Hecke operator and is a Dirichlet character associated to another kind of Hecke operator called the diamond operator . This diamond operator acts on the argument of the modular form by an upper triangular element of whose bottom right entry is mod . This action is the same as the action of a Dirichlet character . The above polynomial is also known as the Hecke polynomial. All of this comes from what is known as the Eichler-Shimura relation, which relates the Hecke operators and the Frobenius.
The first thing that we will need is the identification of the weight cusp forms with the holomorphic differentials on the modular curve (as mentioned in Modular Forms in the case of , although this is can be done more generally).
The second thing that we will need is the Jacobian. One can think of the Jacobian as the space given by the equivalence classes of all path integrals on a curve (in general we can do this for any algebraic curve, not just modular curves), where two path integrals are to be considered equivalent if they differ by integration along a loop. Since path integration can be considered as a linear functional from holomorphic differentials to the complex numbers, we consider such path integrals as the dual space to the space of holomorphic differentials. However, the loops we wanted to quotient out by can also be expressed as elements of the homology group of the curve (see also Homology and Cohomology)!
Therefore we now define the Jacobian of a curve as
where denotes the holomorphic differentials on . The notation denotes the dual to , since as we said the path integrals form the dual to the holomorphic differentials. The Jacobian can also described in other ways – for instance it is also the connected component of the Picard group (see also Divisors and the Picard Group), and the connection to the description given here is an important classical theorem called the Abel-Jacobi theorem.
The Jacobian is a higher-dimensional complex torus, and actually more is true – it is also an abelian variety, i.e. a projective variety whose points form a group (and hence a generalization of elliptic curves). Note that every complex torus is an elliptic curve, but this is not true in higher dimensions – only certain special kinds of higher dimensional complex tori (namely those with a polarization) are abelian varieties. In this vein the Jacobian of a curve has yet another description – it is “universal” among abelian varieties in that, if there is a morphism from a curve to any abelian variety, it can be expressed as a morphism from the curve to its Jacobian, followed by a morphism to that other abelian variety.
Now we go back to the case of modular curves. Denoting by the space of cusp forms of weight two for the level structure , which as discussed above is isomorphic to the space of holomorphic differentials on the corresponding modular curve , we can now define the Jacobian as
The third ingredient that we need is a certain ideal of the Hecke algebra (the ring of endomorphisms of generated by the actions of the Hecke operators and diamond operators) corresponding to the weight Hecke eigenform (let us denote this ideal by that we want to obtain our Galois representation from. This ideal is defined to be the one generated by all elements of the Hecke algebra whose eigenvalue when acting on is zero.
Since the Hecke operators and diamond operators act on the Jacobian (we can see this this way – since the Jacobian is the quotient of the linear functionals on , the action is obtained by first applying the Hecke operator or diamond operator to the weight eigenform, then applying the linear functional), we can use the ideal to cut down a quotient of the Jacobian which is another abelian variety :
Finally, we can take the Tate module of , and this will give us precisely the Galois representation that we want. The abelian variety will have dimension equal to the degree of the number field generated by the eigenvalues of the Hecke operators.
If the eigenvalues are all rational, then will actually be an elliptic curve – in other words, given an eigenform of weight whose Hecke eigenvalues are all rational, we can always use it to construct an elliptic curve! This also gives us a map from the modular curve to this elliptic curve, called a modular parametrization. The resulting elliptic curve will have the property that its L-function, built from point counts when it is reduced modulo primes, is the same as the L-function of the modular form which is built from its Fourier coefficients! This is because the Frobenius and the Fourier coefficients (which are also the eigenvalues of the Hecke operators) are related, as discussed above. The question of whether, given an elliptic curve, it comes from a modular form in this way, is another restatement of the question of modularity. The affirmative answer to this question, at least for certain elliptic curves over , led to the proof of Fermat’s Last Theorem.
This theory, which is only very roughly sketched here, is just a very special case – one can also obtain, for instance, Galois representations from modular forms which are not of weight . We leave this for the future.
Jacobian variety on Wikipedia
Abel-Jacobi map on Wikipedia
Modularity theorem on Wikipedia
Course on Mazur’s Theorem Lecture 10: Jacobians by Andrew Snowden
Course on Mazur’s Theorem Lecture 17: Eichler-Shimura by Andrew Snowden
A First Course in Modular Forms by Fred Diamond and Jerry Shurman