Seminar on Recent Advances in Iwasawa Theory and the Arithmetic of Elliptic Curves
Spring 2020, Thursdays 1-3 pm
The main goal of this seminar is to go over, in detail, my recent paper, which yields two interesting arithmetic applications: it proves Sylvester's 1879 conjecture on sums of cubes, resolves the congruent number problem in 100% of cases, and establishes Goldfeld's conjecture for the congruent number family. The main novelty of my approach is to formulate and prove a new Rubin-type main conjecture for imaginary quadratic fields K in the "height 2" case when p is inert or ramified in K. This involves developing a new interplay between p-adic Hodge theory and the machinery of Coleman power series in order to construct p-adic L-functions from the Euler system of elliptic units. For supersingular elliptic curves over Q with CM by K, one can formulate a new Perrin-Riou-type main conjecture involving Heegner points, and exploit a factorization of the relevant Galois representation to reduce this main conjecture from a product of the aforementioned Rubin-type main conjectures.
Later on, I hope to also go over new ongoing joint work with M. Zanarella on Kolyvagin's conjecture, generalizing W. Zhang's previous work.
Ideally the seminar will be as self-contained as possible, but I will assume knowledge of class field theory, particularly local class field theory (i.e. Lubin-Tate formal groups), structure of unit group of number fields, basic properties of elliptic curves and their associated Galois representations, modular forms. Knowledge of Heegner points and the geometry of Shimura curves would also be useful in the second half (post Spring Break). It might be helpful if you have seen Iwasawa main conjectures before (see for example the notes from the 2018 Arizona Winter School), but not necessary as I will go over the set-up in my situation.
ScheduleHere is a (tentative) schedule of the seminar. Some talks might change or move around, but in any case I will keep this updated. Unless otherwise noted, I will be speaking.
For any comments, questions, or if you want to be added to the seminar mailing list, contact me at firstname.lastname@example.org.
- 2/6: No meeting, I am out of town
- 2/13: Overview of the paper and approach: converse theorems, Heegner point main conjecture, Rubin-type main conjecture, plan of proof, notes
- 2/20: Relative Lubin-Tate groups, Coleman map, p-adic measures (and how they are power series), some p-adic Hodge theory, notes
- 2/27: Measures from local units continued, notes
- 3/5: Finishing up measures from local units, recap of where we are, notes
- 3/12: The GL_1 explicit reciprocity law: p-adic L-functions from elliptic units
- 3/19: Formulation of the Rubin-type main conjecture (RMC), computation of cyclotomic algebraic mu-invariant of Y, proof of Rubin-type main conjecture
- 3/26: Spring Break
- 4/2: Wiles explicit reciprocity law, relating RMC to Selmer groups of elliptic curves with CM
- 4/9: Rank 0 converse theorems from RMC, and Wiles explicit reciprocity law
- 4/16 Factorizations of p-adic Rankin L-functions and Selmer groups, + families of Heegner points
4/17: + families of Heegner points + Selmer groups continued, big logarithm and GL_2 Coleman maps
- 4/23: explicit reciprocity law relating Lambda-adic Heegner class to holomorphic p-adic Rankin-Selberg p-adic L-function, via Wiles reciprocity law and p-adic Waldspurger formula
- 4/30: Calculations in Kummer theory, relating + local condition and kernel of log-Coleman map (from the talk on 2/20)
- 5/7: Formulation of Heegner point main conjecture, proof via global duality and explicit reciprocity law, control theorems, rank 1 converse theorem