NTS ABSTRACT

Return to NTS Spring 2017

Sept 7

David Zureick-Brown

Progress on Mazur’s program B

I’ll discuss recent progress on Mazur’s ”Program B”, including my own recent work with Jeremy Rouse which completely classifies the possibilities for the 2-adic image of Galois associated to an elliptic curve over the rationals. I will also discuss a large number of other very recent results by many authors.

Sept 14

Solly Parenti

Unitary CM Fields and the Colmez Conjecture

Pierre Colmez conjectured a formula for the Faltings height of a CM abelian variety in terms of log derivatives of Artin L-functions arising from the CM type. We will study the relevant class functions in the case where our CM field contains an imaginary quadratic field and use this to extend the known cases of the conjecture.

Sept 21

Chao Li

Goldfeld's conjecture and congruences between Heegner points

Given an elliptic curve E over Q, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (resp. 1). We show this conjecture holds whenever E has a rational 3-isogeny. We also prove the analogous result for the sextic twists of j-invariant 0 curves. For a more general elliptic curve E, we show that the number of quadratic twists of E up to twisting discriminant X of analytic rank 0 (resp. 1) is >> X/log^{5/6}X, improving the current best general bound towards Goldfeld's conjecture due to Ono--Skinner (resp. Perelli--Pomykala). We prove these results by establishing a congruence formula between p-adic logarithms of Heegner points based on Coleman's integration. This is joint work with Daniel Kriz.

Sept 28

Daniel Hast

Rational points on solvable curves over Q via non-abelian Chabauty

By Faltings' theorem, any curve over Q of genus at least two has only finitely many rational points—but the bounds coming from known proofs of Faltings' theorem are often far from optimal. Chabauty's method gives much sharper bounds for curves whose Jacobian has low rank, and can even be refined to give uniform bounds on the number of rational points. This talk is concerned with Minhyong Kim's non-abelian analogue of Chabauty's method, which uses the unipotent fundamental group of the curve to remove the restriction on the rank. Kim's method relies on a "dimension hypothesis" that has only been proven unconditionally for certain classes of curves; I will give an overview of this method and discuss my recent work with Jordan Ellenberg where we prove this dimension hypothesis for any Galois cover of the projective line with solvable Galois group (which includes, for example, any hyperelliptic curve).

Oct 12

Matija Kazalicki

Supersingular zeros of divisor polynomials of elliptic curves of prime conductor and Watkins' conjecture

For a prime number p, we study the mod p zeros of divisor polynomials of elliptic curves E/Q of conductor p. Ono made the observation that these zeros of are often j-invariants of supersingular elliptic curves over F_p. We relate these supersingular zeros to the zeros of the quaternionic modular form associated to E, and using the later partially explain Ono's findings. We notice the curious connection between the number of zeros and the rank of elliptic curve. In the second part of the talk, we briefly explain how a special case of Watkins' conjecture on the parity of modular degrees of elliptic curves follows from the methods previously introduced. This is a joint work with Daniel Kohen.

Oct 19

Andrew Bridy

Arboreal finite index for cubic polynomials

Let K be a global field of characteristic 0. Let f \in K[x] and b \in K, and set K_n = K(f^{-n}(b)). The projective limit of the groups Gal(K_n/K) embeds into the automorphism group of an infinite rooted tree. A major problem in arithmetic dynamics is to find conditions that guarantee the index is finite; a complete answer would give a dynamical analogue of Serre's celebrated open image theorem. I solve the finite index problem for cubic polynomials over function fields by proving a complete list of necessary and sufficient conditions. For number fields, the proof of sufficiency is conditional on both the abc conjecture and a form of Vojta's conjecture. This is joint work with Tom Tucker.