Mazur's torsion theorem classifies the possible torsion subgroups of the rational points E(Q) of elliptic curves E/Q. It is a short list: Precisely 15 groups can occur, and the maximal order among these is 16.
An important part of Mazur's proof is to show that the modular curve X_0(N) has no rational point for N > 13. This uses results about finite group schemes, Néron models of abelian varieties, and ultimately the geometry of the integral model of X_0(N). Our goal in the seminar is to learn about these connections and to understand the main ideas behind Mazur's proof.
Our seminar follows the Lectures of Andrew Snowden. Further references may be found on the STAGE seminar page from Fall 2023.
Time and place to be decided
| Date | Topic | Contents | Speaker |
|---|---|---|---|
| Introduction | Program introduction, talk distribution | ||
| Talk 1 | Abelian varieties | Background on elliptic curves and abelian varieties (Lectures 2, 3, 4 with focus on the algebraic theory) | |
| Talk 2 | Finite group schemes | Background on finite group schemes (Lectures 5, 6 with focus on the classification of étale ones, Cartier duality, and applications to AVs/ECs) | |
| Talk 3 | Raynaud's theorem | Show that a finite group scheme over the ring of integers O_K of a finite extension K/Q_p is uniquely determined by its generic fiber, assuming that the ramification index satisfies e < p-1 (Lecture 7 with focus on the case F = F_p) | |
| Talk 4 | Elliptic curves over DVRs | Describe the structure of elliptic curves over discrete valuation rings. Present the Néron-Ogg-Shavarevich Criterion and its corollaries (Lecture 8; also [Sil, §VII]) | |
| Talk 5 | Néron models | Define Néron models, give examples for elliptic curves, and describe the reduction types of abelian varieties (Lecture 9; also the introduction of [BLR]) | |
| Talk 6 | Jacobians | Define the Jacobian of a curve and sketch its construction (Lecture 10) | |
| Talk 7 | Criterion for rank 0 | Present and prove the criterion for an abelian variety over Q to have rank 0 (Lecture 11) | |
| Talk 8 | Modular curve, modular forms, Hecke operators | Background on the analytic theory of these objects (Lectures 12 and 13, make sure to have enough time to discuss Hecke operators in detail) | |
| Talk 9 | Integral models for modular curve | Construct an integral model for X_0(N) (Lectures 14 and 15 with focus on the theory over Z) | |
| Talk 10 | Eichler-Shimura | Explain the structure results for modular forms as module under the Hecke algebra. Use them to attach Galois representations and quotients of J_0(N) to newforms (Lectures 16 and 17) | |
| Talk 11 | Criterion for non-existence of rational points | Prove the non-existence of rational points on X_0(N), assuming a suitable map f:X_0(N) --> A to an abelian variety can be constructed (Lecture 18) | |
| Talk 12 | Mazur's theorem | Explain how to construct a map f as in the previous talk. In particular, sketch a proof of Mazur's theorem (Lectures 19, 20 and 21, with focus on Lectures 19 and 20) |
| [BLR] | S. Bosch, W. Lütkebohmert, M. Raynaud; Néron models, Springer-Verlag (1990). |
| [Sil] | J. H. Silverman; The Arithmetic of Elliptic Curves, GTM, Springer-Verlag (2009). |