Statistiques Arithmétiques et Stabilité Diophantienne / Arithmetic Statistics and Diophantine Stability
Séminaire organisé par Barry Mazur (Harvard University, Cambridge, USA) et Karl Rubin (University of California, Irvine, USA) du 23 au 28 juillet 2018
Alexander Best, Francesca Bianchi, Maarten Derickx, Wei Ho, Chi Yun Hsu, Nathan Kaplan, Barry Mazur, Karl Rubin, Arul Shankar, Nicholas Triantafillou, Arnav Tripathy, Jan Vonk, Myungjuy Yu
Et, pour la Fondation des Treilles, Alain Peyraube (Président du conseil scientifique) et Bernard Teissier (Membre du bureau du conseil d’administration)
Compte-rendu (en anglais)
The broad objective of our workshop was to understand rational points on curves in a theoretical and computational way—with an eye toward statistics that govern their behavior. One can study these concepts directly by algebraic methods, and one can also approach this phenomenon (often only conjecturally) by analytic means, specifically by a basic tool called L-functions. The vanishing of certain values of such L-functions indicates that there are many rational points; and non-vanishing indicates the reverse.
In addition to the two senior organizers, there were 11 junior participants: 4 graduate students, 4 postdoctoral researchers, and 3 Assistant Professors.
The workshop was designed so that our participants, in separate or overlapping groups, might focus on particular projects within the broad framework described above. One hope was to promote research directions and collaborations that might continue even after the workshop ends. Here is a brief description of the projects:
(i) Important databases are already available that are relevant for work on curves (esp. elliptic curves) and modular forms. They offer data over Q, the field of rational numbers, or—via modular symbols (equivalently “theta-elements”)—over abelian extensions of Q but not over more general number fields. In consequence, any analysis of statistics regarding the size—or vanishing—of special values of L functions is conveniently served by these databases only for abelian extensions of Q.
One project in our workshop was to broaden the range of the programs for the computation of ’theta-elements’ and expand the database to (abelian extensions of) fields other that Q. Maarten Derickx and Alex Best had already begun this project before the workshop and have significantly expanded it during our workshop—already accumulating data for (some abelian extensions of) Q(√5). They will be continuing this collaboration (already scheduled as a topic in yet another workshop in Cambridge MA this August).
(ii) Studying the statistics of the shapes of lattices that arise from the group of units in a number field, and the (Mordell-Weil) group of rational points in an elliptic curve. Curiously, this is a very classical topic—clearly important—and yet essentially no substantial data has systematically been computed, or analyzed. This project, initiated by Wei Ho, Nathen Kaplan, and Arul Shankar for our workshop has already revealed striking (nonuniform) phenomena. This deserves close inspection, and will be continued.
(iii) Rational points on a fundamental family of ‘modular curves.’ Namely, the family—for p ranging through prime numbers—denoted X0(p)+. Without going into any detail or definition of ’modular curve,’ it may suffice to say that any point on such a curve has ‘importance” and corresponds to some interesting mathematical structure (the type of the structure depending on the modular curve). The project in our workshop pursued by Francesca Bianchi, Nicholas Triantafillou, and Jan Vonk was to initiate a classification of rational points on the hyperelliptic curves that are member of the family just mentioned, developing a streamlined program using a new method due to Chabauty-Coleman-Kim. (Certain aspects of this project was also worked on by Wei Ho, Nathan Kaplan, and Maarten Derickx.) This collaboration will also be vigorously continued.
(iv) Given a number field extension L/K and a curve V defined over K, say that V is Diophantine unstable for L/K if V has more L-rational points than K-rational points. In certain contexts, Diophantine unstability is quite rare. For example, it is conjectured that for prime numbers p > 5 an elliptic curve over Q is Diophantine unstable for only finitely many (Galois, cyclic) extensions L/Q of degree p. So, p = 5 is a particularly interesting case: can one show, for an elliptic curve E, that there are infinitely many (Galois, cyclic) extensions L/Q of degree 5 for which E is Diophantine unstable? Arnav Tripathy (and both of us: B.M & K.R.) considered this problem—not resolving it, but understanding, a bit more, aspects of it—and
all three of us worked on an interesting idea of Arnav’s regarding the question of actually identifying curves of genus one by the field extensions L/Q for which they obtain L-rational points. This, too, is a collaboration that will continue.
(v) Two of our participants Chi-Yun Hsu and Myungjun Yu each worked on their individual projects in discussion with us.