Classic Papers in Number Theory, Spring 2018

Plan for the seminar

We'll have a weekly discussion of a classic paper in number theory. Ideally this seminar should be very participatory; read over the paper ahead of time and come prepared with questions for optimal results.

List of papers

Here's the list of papers (in no particular order) that have been suggested; feel free to email me with suggestions if you'd like to hear about/talk about something not on this list.

• Elkies: The existence of infinitely many supersingular primes for every elliptic curve over Q.
• Swinnerton-Dyer: On l-adic representations and congruences for coefficients of modular forms, Serre: Congruences et formes modulaires.
• Lenstra: Factoring integers with elliptic curves.
• Serre: Sur les représentations de degré 2 de Gal(Qbar/Q).
• Stark: papers on the class number 1 problem for imaginary quadratic fields, see A complete determination of the complex quadratic fields of class number one and On the 'gap' in a theorem of Heegner.
• Gross: a tameness criterion for Galois representations associated to modular forms (mod p).
• Fontaine: Il n'y a pas de variété abélienne sur Z.
• Larsen-Pink: papers on compatible systems, see On l-independence of algebraic monodromy groups in compatible systems of representations and Abelian varieties, l-adic representations, and l-independence.
• Serre-Tate: Good reduction of abelian varieties.
• Honda-Tate: see Tate's Classes d'isogénie des variétés abéliennes sur un corps fini (d'après T. Honda) and Honda's Isogeny classes of abelian varieties over finite fields.
• Mazur-Tate: Points of order 13 on elliptic curves.

Schedule

The seminar meets 3:45-5:00 on Tuesdays in the barn (Ryerson 352).

Date	Speaker	Paper
April 10	Karl	Elkies: The existence of infinitely many supersingular primes for every elliptic curve over Q.
April 17		No meeting this week.
April 24	Billy	Swinnerton-Dyer: On l-adic representations and congruences for coefficients of modular forms, Serre: Congruences et formes modulaires.
May 1	John	Serre-Tate: Good reduction of abelian varieties.
May 8	Eric	Tate: The non-existence of certain Galois extensions of Q unramified outside 2.
May 15	Bingjin	Fontaine: Il n'y a pas de variété abélienne sur Z.
May 22	Mathilde	Eskin, Rudnick, Sarnak: A proof of Siegel's weight formula.
May 29	Nazerke	Serre: Sur les représentations de degré 2 de Gal(Qbar/Q).