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.