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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.

Oct 19

Jiuya Wang
Malle's conjecture for compositum of number fields
Abstract: Malle's conjecture is a conjecture on the asymptotic distribution of number fields with bounded discriminant. We propose a general framework to prove Malle's conjecture for compositum of number fields based on known examples of Malle's conjecture and good uniformity estimates. By this method, we prove Malle's conjecture for $S_n\times A$ number fields for $n = 3,4,5$ and $A$ in an infinite family of abelian groups. As a corollary, we show that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ representation is the first counter example of Malle's conjecture given by Kl?\"uners. By a sieve method, we further prove the secondary term for $S_3\times A$ extensions for infinitely many odd abelian groups $A$ over $\mathbb{Q}$.

Nov 2

Carl Wang-Erickson
The rank of the Eisenstein ideal
Abstract: In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. We use deformation theory of pseudorepresentations to study the corresponding Hecke algebra. We will discuss how this method can be used to refine Mazur's results, quantifying the number of Eisenstein congruences. Time permitting, we'll also discuss some partial results in the composite-level case. This is joint work with Preston Wake.

Nov 9

Masahiro Nakahara
Index of fibrations and Brauer-Manin obstruction
Abstract: Let X be a smooth projective variety with a fibration into varieties that either satisfy a condition on representability of zero-cycles or that are torsors under an abelian variety. We study the classes in the Brauer group that never obstruct the Hasse principle for X. We prove that if the generic fiber has a zero-cycle of degree d over the generic point, then the Brauer classes whose orders are prime to d do not play a role in the Brauer--Manin obstruction. As a result we show that the odd torsion Brauer classes never obstruct the Hasse principle for del Pezzo surfaces of degree 2, certain K3 surfaces, and Kummer varieties.

Nov 16

Joseph Gunther
Irrational points on random hyperelliptic curves
Abstract:Let d and g be positive integers with 1 < d < g. If d is odd, we prove there exists B_d such that a positive proportion of odd genus g hyperelliptic curves over Q have at most B_d points of degree d. If d is even, we similarly bound the degree d points not lazily pulled back from degree d/2 points of the projective line. The proofs use tropical geometry work of Park, as well as results of Bhargava and Gross on average ranks of hyperelliptic Jacobians. This is joint work with Jackson Morrow.

Time willing, we'll discuss rich, delicious interactions with work of next week's speaker.

Nov 30

Reed Gordon-Sarney
Zero-Cycles on Torsors under Linear Algebraic Groups
Abstract:In this talk, the speaker will discuss his thesis on the following question of Totaro from 2004: if a torsor under a connected linear algebraic group has index d, does it have a close etale point of degree d? The d = 1 case is an open question of Serre from the `60s. The d > 1 case, surprisingly, has a negative answer.

Dec 7

Rafe Jones
How do you (easily) find the genus of a plane curve?
Abstract: If you’ve ever wanted to show a plane curve has only finitely many rational points, you’ve probably wished you could invoke Faltings’ theorem, which requires the genus of the curve to be at least two. At that point, you probably asked yourself the question in the title of this talk. While the genus is computable for any given irreducible curve, it depends in a delicate way on the singular points. I’ll talk about a much nicer formula that applies to irreducible “variables separated” curves, that is, those given by A(x) = B(y) where A and B are rational functions.

Then I’ll discuss how to use this to resolve the question that motivated me originally: given an integer m > 1 and a rational function f defined over a number field K, does f possess a K-orbit containing infinitely many mth powers of elements of K? The answer turns out to be no unless f has a very special form: for m > 4 the map f must essentially be the mth power of some rational function, while for smaller m other exceptions arise, including maps closely related to multiplication on elliptic curves. If time permits I’ll discuss a connection to an arithmetic dynamical analogue of the Mordell-Lang conjecture.

Dec 14

Robert J. Lemke Oliver
Selmer groups, Tate-Shafarevich groups, and ranks of abelian varieties in quadratic twist families
Abstract: We determine the average size of the $\phi$-Selmer group in any quadratic twist family of abelian varieties having an isogeny $\phi$ of degree 3 over any number field. This has several applications towardsthe rank statistics in such families of quadratic twists. For example, it yields the first known quadratic twist families of absolutely simple abelian varieties over $\mathbb{Q}$, of dimension greater than one, for which the average rank is bounded; in fact, we obtain such twist families in arbitrarily large dimension. In the case that $E/F$ is an elliptic curve admitting a 3-isogeny, we prove that the average rank of its quadratic twists is bounded; if $F$ is totally real, we moreover show that a positive proportion of these twists have rank 0 and a positive proportion have $3$-Selmer rank 1. We also obtain consequences for Tate-Shafarevich groups of quadratic twists of a given elliptic curve. This is joint work with Manjul Bhargava, Zev Klagsbrun, and Ari Shnidman.