Study group on Mazur's Torsion Theorem

Schedule for the talks

Here is a link for the recorded sessions, and a programme for the second half of the study group.

Date	Speaker	Title	Description	Notes
21/03	Pedro	Introduction	We give an introduction to Mazur's theorem and we roughly sketch Mazur's proof. We then describe the plan for the study group. If time permits it, we also discuss some related results and open problems, namely Serre's uniformity conjecture and Merel's uniform boundedness theorem.	Notes
21/03	Pedro	Reminders about Elliptic Curves	We cover some useful basic facts about elliptic curves. We begin with a brief review of algebraic curves, divisors and Riemann-Roch. We then define elliptic curves, and talk about their group structure and defining equations. Then there is the theory of isogenies, including the important fact that degree map is "quadratic". Next is the complex point of view : elliptic curves are one-dimensional tori C/L for some lattice L in C. We then talk about the Tate module and the Weil pairing. A good reference for these notions can be Silverman's book.	Notes
28/03	Elvira	Analytic theory of Abelian Varieties	We talk about abelian varieties over C from the analytic point of view. A complex abelian variety comes naturally with the structure of a compact complex Lie group. In fact, one can show that it is a complex torus, together with an additional datum which is called a polarisation. In order to introduce polarisations of tori, we discuss line bundles over them, and their sections. This leads to the notion of a dual torus associated to a complex torus, which is once again a complex abelian variety. A standard reference is Birkenhake and Lange's "Complex Abelian Varieties".	Notes
28/03	Sofian	Algebraic theory of Abelian Varieties	We cover the basic theory of abelian varieties over arbitrary fields. We begin with the basic results such as commutativity and the structure of torsion. Then we discuss the dual abelian variety and compare its construction to the one in the complex analytic case. After that we prove the weak Mordell-Weil theorem, as the ideas of the proof will be relevant for the upcoming talks. Lastly, we discuss Poincaré irreducibility, and its consequence : abelian varieties together with isogenies form a semisimple category. The main reference is Milne's notes.	Notes
04/04	Diana	Group Schemes 1	This is the first of four talks on group schemes. We begin by introducing the idea of a group object in a category. We then define what a group scheme is, and explain the connection to Hopf algebras. This is followed by several important examples.	Notes
04/04	Swann	Group Schemes 2	In this talk, we discuss Cartier duality and give the basic examples. Then, we get into the theory over finite fields (Frobenius, Verschiebung...).	Notes
11/04	Swann	Group Schemes 3	This is the third talk about group schemes. We first give the classification of height 1 group schemes, and we use it to classify the simple group schemes over an algebraically closed field. We also briefly discuss Dieudonné modules. We then apply the theory of group schemes to the of study abelian varieties. We relate abelian variety duality and Cartier duality. Then we characterize ordinary and supersingular elliptic curves using their p-torsion. Finally, we give a tight bound on the p-torsion of an abelian variety.	Notes
18/04	Alex	Raynaud's Theorem	This is the last talk about group schemes, which is about Raynaud's theorem : in mixed characteristic and low ramification, a group scheme is determined by its generic fiber. The proof proceeds by first reducing the statement to a special class of group schemes, the Raynaud F-module schemes. Next, we classify these schemes. Finally, one verifies the theorem for Raynaud F-module schemes using the classification obtained before.	Notes
25/04	Alvaro	Elliptic Curves over DVRs	We discuss elliptic curves over DVRs. We define the various types of reduction (good, multiplicative, additive), and study their behavior under extensions. We then move to the behaviour of torsion under extensions. Finally, we prove the Néron-Ogg-Shafarevich theorem.	Notes
25/04	Samuel	Néron Models 1	We begin by discussing quasi-finite étale group schemes over DVRs: these are the sorts of things that occur as the prime-to-p torsion of Néron models. Then, before going to the general theory, we discuss Néron models of elliptic curves, especially their relationship to Weierstrass and minimal regular models (a good reference in this case is Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves"). We provide some simple examples. Finally, we introduce Néron models of general abelian varieties (see the book of Raynaud, Lütkebohmert and Bosch for a complete exposition). Two applications are discussed: the Néron-Ogg-Shafarevich criterion, and Grothendieck's generalization. The semi-stable reduction theorem is also discussed.	Notes
02/05	Samuel	Néron Models 2	We begin by discussing quasi-finite étale group schemes over DVRs: these are the sorts of things that occur as the prime-to-p torsion of Néron models. Then, before going to the general theory, we discuss Néron models of elliptic curves, especially their relationship to Weierstrass and minimal regular models (a good reference in this case is Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves"). We provide some simple examples. Finally, we introduce Néron models of general abelian varieties (see the book of Raynaud, Lütkebohmert and Bosch for a complete exposition). Two applications are discussed: the Néron-Ogg-Shafarevich criterion, and Grothendieck's generalization. The semi-stable reduction theorem is also discussed.	Notes
09/05	Robin	Jacobians	We provide an exposition to the theory of Jacobians of curves. We begin with the case where the curve C is defined over the complex numbers : the Jacobian is thus defined as the quotient of a vector space (the dual of the space of global holomorphic 1-forms on C) by a lattice (the first homology group of C). We show that this is a complex torus together with a polarisation, hence a complex abelian variety. We then move to the theory over arbitrary fields. We first discuss representability issues of the functor of points. Then, we sketch Weil's construction of the Jacobian variety via the Picard functor and say a word about the relative situation.	Notes
16/05	Håvard	Criterion for rank 0	In this talk, we thoroughly prove the Theorem B which was discussed by Pedro in the introduction. We recall that it is a criterion for an abelian variety A to have rank 0, i.e. for its Mordell-Weil group to be a torsion group. The idea of the proof is similar to the one provided by Sofian for the weak Mordell-Weil theorem : for each prime p we embed the p-primary torsion of the Néron model of A into an inverse limit of fppf-cohomology groups whose orders (finite in each level by construction) do not depend on the corresponding powers of p. Taking cohomology on the fppf site (which is possible by the existence of the Néron model) allows us to control the ramification of the cohomology classes much more carefully than in the Galois case. Most of the work of the proof goes into understanding a certain class of group schemes (the admissible ones) very precisely.	Notes
-	-	No talk	Summer Break!	-
19/10	Lucas	Moduli Problems I	In this talk we discuss moduli problems for elliptic curves as described in Katz and Mazur. We first recall the construction of the modular curve Y(1) parametrizing isomorphism classes of complex elliptic curves and its level N avatars. The goal is to motivate the consideration of modular curves in Mazur’s paper. Then we focus on the study of modular curves over Q. We show that moduli problems are representable in level larger than 3 (and we give explicit examples of schemes representing these functors), and explain why the moduli problem in level 1 cannot be represented by a scheme.	Notes
26/10	Lucas	The language of stacks	In this talk we provide an introduction to stacks : their definition, classes of stacks (Deligne-Mumford, algebraic), and why you should care about them for the purposes of this study group. In particular we show that the moduli functor for Γ(1)-structures and Γ_0(N)-structures (N ≥1) cannot be represented by a scheme but have a natural structure of smooth Deligne-Mumford stack, respectively over Z and Z[1/N].	Notes
02/11	Martin	Moduli Problems II	In this talk, we go further in the study of modular curves and describe the coarse space M_0(N) which is our "prototype" scheme for the modular curve X_0(N). The issue is that this curve is not proper, and it is furthermore defined over Z[1/N]. In order to obtain a proper curve over the integers, we will compactify it by extending the moduli functor to generalised elliptic curves, which roughly corresponds to adding the cusps.	Notes
09/11	Arshay	Structure of the Hecke Algebra	In this talk, we will reinterpret S_2(N) as the space of global sections of the sheaf of 1-differential forms on the modular curve X_0(N). We will recall some basic facts about the Hecke operators and detail the structure of the Hecke algebra by using Hecke correspondences. Finally, we will introduce the concept of the Atkin-Lehmer involution as a first step towards the Eichler-Shimura theorem.	Notes
16/11	Robin	Eichler-Shimura and the Shimura representation	In this talk, we will begin by studying maps from the Jacobian J_0(N) induced by Hecke correspondences and proceed to prove the Eichler-Shimura theorem. After that, we will study the trace and determinant of the Frobenius element, and define the Shimura variety A_f associated to any weight two cusp form f. Then, and after studying these two objects and their relations, we will prove the existence and uniqueness of the Shimura Galois representation. As a corollary to this result, we will see that two newforms which agree as T_p-eigenforms for all p in a set of Dirichlet density 1 are necessarily equal.	Notes
23/11	Florian	Proof of Theorem A	In this talk, we will prove one of the main theorems in the study group, Theorem A, which provides a criterion for the non-existence of torsion points. We will prove this by recollecting and putting together many of the results in the study group, e.g. the connected-étale sequence, the Néron mapping property and Raynaud’s theorem.	Notes
30/11	Lucas	Toric reduction of J_0(N)	In this talk, we show that one of the key conditions for applying theorem A+B (general criterion for non-existence of torsion points) comes for free, namely that J_0(N) and its quotients all have completely toric reduction at N (we will first see that they have good reduction away from N). We will proceed the following way. Firstly, we will invoke a theorem of Michèle Raynaud in order to compute the identity component of the Néron model of the Jacobian of X_0(N) via the Picard scheme of its minimal regular model. Secondly, we will construct this minimal regular model very explicitly and study its special fibre at N. Finally, we will combine these steps in order to obtain that the special fibre of the Néron model of J_0(N) is the Picard scheme of a nodal curve (an arithmetic surface) whose irreducible components are rational curves, and we will prove that it is a torus.	Notes
07/12	Alvaro	The Eisenstein ideal	In this talk, we will begin the proof of Mazur's theorem by introducing the concept of the Eisenstein quotient. For this purpose, we will study the action of Hecke operators and define an appropiate quotient of the Jacobian of an abelian variety A, and study how the Jordan-Hölder condition interacts with this quotient.	Notes
21/12	Pedro	Proof of Mazur’s theorem	In this talk, we will combine results of previous talks in order to finish proving Mazur’s theorem. We will adapt the proof of Theorem B in the first part of the study group by computing certain quantities related to the abelian variety A and its fppf cohomology groups, which will allow us to prove that A(Q) is finite.	Notes