In this talk we will focus on constructing "super-isolated abelian varieties". These are abelian varieties that have isogeny class which contains a single isomorphism class. Their motivation comes from security concerns in elliptic and hyperelliptic curve cryptography. Using a theorem of Honda and Tate, we transfer the problem of finding such varieties to a problem in algebraic number theory. Finding these varieties turns out to be related to finding primes of the form n2 + 1 and to solving Pell's equation.
Wan conjectured that the variation of zeta functions along towers of curves associated to the $p$-adic etale cohomology of a fibration of smooth proper ordinary varieties should satisfy several stabilizing properties. The most basic of these conjectures state that the genera of the curves in these towers grow in a regular way. We state and prove a generalization of this conjecture, which applies to the graded pieces of the slope filtration of an overconvergent $F$-isocrystal. Along the way, we develop a theory of $F$-isocrystals with logarithmic decay and provide a new proof of the Drinfeld-Kedlaya theorem for curves.
Let E be a CM elliptic curve over the rationals with conductor N and p a prime coprime to 6N. If the p^{infty}-Selmer group of E has Z_{p}-corank one, we show that the analytic rank of E is also one (joint with Chris Skinner and Ye Tian). We plan to discuss the setup and strategy in the ordinary case.
A Teichmuller curve is a totally geodesic curve in the moduli space of Riemann surfaces. These curves are defined by polynomials with integer coefficients that are irreducible over C. We will show that these polynomials have surprising factorizations mod p. This is joint work with Keerthi Madapusi Pera.
The class group is a natural abelian group one can associate to a number field, and it is natural to ask how it varies in families. Cohen and Lenstra famously proposed a model for families of quadratic fields based on random matrices of large rank, and this was later generalized by Cohen-Martinet. However, their model was observed by Malle to have issues when the base field contains roots of unity. We study this in detail in the case of function fields using methods of Ellenberg-Venkatesh-Westerland, and based on this we propose a model in the number field setting. Our conjecture is based on keeping track not only of the underlying group structure, but also certain natural pairings one can define in the presence of roots of unity (joint with Lipnowski, Sawin).
We determine the average size of the Φ-Selmer group in any quadratic twist family of abelian varieties having an isogeny Φ of degree 3 over any number field. This has several applications towards the rank statistics in such families of quadratic twists. For example, it yields the first known quadratic twist families of absolutely simple abelian varieties over 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, Robert Lemke Oliver, and Ari Shnidman.
Given a global function field K of characteristic p>0, the fundamental arithmetic invariants include the genus, the class number, the p-rank and more generally the slope sequence of the zeta function of K. In this expository lecture, we explore possible stability of these invariants in a p-adic Lie tower of K. Strong stability is expected when the tower comes from algebraic geometry, but this is already sufficiently interesting and difficult in the case of Zp towers.
Let Y --> X be a branched G-covering of curves over a field k. The genus of X and the genus of Y are related by the famous Hurwitz genus formula. When k is perfect of characteristic p and G is a p-group, one also has the Deuring-Shafarevich formula which relates the p-rank of X to that of Y. In this talk, we will discuss our attempts to find a "motivic" generalization of the Deuring-Shafarevich formula by studying how the p-torsion group schemes of the Jacobians of X and Y are related. In particular, we will explain how to promote the numerical Deuring-Shafarevich formula to an isomorphism of (etale) group schemes. This is ongoing joint work with Rachel Pries.
Since Fermat characterized (without proof) those integers represented by the quadratic form x^2+y^2, number theorists have been extending these results. Recently a paper appeared in Journal of Number Theory answering the question for x^2 ± xy ± y^2. It turns out that this was not news (although JNT refuses to correct or retract). Naively, today's speaker extended these results, through elementary means. This talk will outline these methods, and contrast them with more traditional techniques.