Séminaire Mazur

This is the webpage for a learning seminar on Mazur's work on torsion. The goal of the seminar is to discuss several results from the paper

B. Mazur - Modular curves and the Eisenstein ideal - Publ. Math. I.H.E.S. 47 (1977)

We will cultivate a tradition of illustrating (wherever possible) all results with our own explicit examples and computations, so as to help participants build a solid intuition, and to encourage them to explore the arithmetic of modular curves as much as possible in a hands-on way. This topic lends itself exceptionally well to this goal, and speakers are strongly encouraged to design their talks creatively, with the spirit of explicit exploration in mind.

The seminar is organised by Paolo Bordignon and Jan Vonk.

Talks

An overview of the planning of the talks.

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Date	Time	Room	Speaker	Topic
22 July 2024	13:00	BM.2.26	Jan Vonk	Overview of Mazur's theorem
22 July 2024	15:00	BM.2.26	Alex Braat	Hecke algebras and the Eisenstein ideal
24 July 2024	13:00	BW.2.18	Paolo Bordignon	Group schemes I: Raynaud's theorem
24 July 2024	15:00	BW.2.18	Peter Bruin	Group schemes II: Extensions and torsors
25 July 2024	10:30	BM.2.26	Paolo Bordignon	Group schemes III: Admissible group schemes (notes)
25 July 2024	13:00	BM.2.26	Jan Vonk	The Eisenstein ideal: past and present
26 July 2024	13:00	BM.2.26	René Schoof	Non-Eisenstein torsion of modular Jacobians

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Resources

This paper contains a wealth of beautiful and innovative ideas, so it has naturally been been a popular topic of seminars and reading groups for several decades. As such, there are many good resources available that will help us digest this paper. Since the paper has appeared, various technical tools that were confined to highly specialised research literature at the time of Mazur, appeared in recent textbooks and courses. Below is a list of some resources that may be helpful for us in the course of the seminar.

Main papers

The following papers will be our main sources for the seminar.

• B. Mazur - Rational points on abelian varieties with values in towers of number fields - Inventiones Math. 18 (1972)
• B. Mazur, J. Tate - Points of order 13 on elliptic curves - Inventiones Math. 22 (1973)
• B. Mazur - Modular curves and the Eisenstein ideal - Publ. Math. I.H.E.S. 47 (1977)
• B. Mazur - Rational isogenies of prime degree - Inventiones Math. 44 129-162 (1978)

Background material

The following sources may be helpful for some of the background material.

Néron models:

• Silverman - Advanced topics in the arithmetic of elliptic curves
• Liu - Algebraic geometry and arithmetic curves
• Artin - Néron models (in Arithmetic Geometry - Cornell, Silverman eds.)
• Bosch, Lütkebohmert, Raynaud - Néron Models

Group schemes:

• Tate - Finite flat group schemes (in Modular forms and Fermat's Last Theorem - Cornell, Silverman, Stevens eds.)
• Mumford - Abelian varieties
• Oort, Tate - Group schemes of prime order
• Raynaud - Schémas en groupes de type (p, ..., p)

Flat cohomology:

• Milne - Étale cohomology
• Milne - Arithmetic duality theorems
• Bruin - Extensions and torsors for finite group schemes

Models of modular curves:

• Deligne, Rapoport - Les schémas de modules de courbes elliptiques (in Modular Functions of one Variable II)
• Katz, Mazur - Arithmetic moduli of elliptic curves

Learning seminars

Learning seminars on the Eisenstein ideal paper with useful resources:

• Learning seminar at Stanford (2003)
• Course by Andrew Snowden (2013)