Let K be a function field with a constant field of size q. If E is an elliptic curve over K with nonconstant j-invariant then its L-function L(T,E/K) is a polynomia.orgl in 1 + T Z[T]. Inspired by the algorithms of Schoof and Pila for computing zeta functions of curves over finite fields, we consider the problem of computing the reduction of L(T,E/K) modulo an integer without first computing the whole L-function. Doing so for a large enough integer which is coprime with q completely determines L(T,E/K). The existing literature on this problem could be summarized as follows: Under the assumption that the Mordell-Weil group E(K) has a subgroup of order N ≥ 2, with N coprime with q, Chris Hall gave an explicit formula for the reduction L(T,E/K) mod N. We present novel theorems going beyond Hall's. https://arxiv.org/abs/2110.12156

Let G be a finite group.  A representation V of G is said to be unisingular if det(1-g) = 0 for all g in G.  Unisingular representations arise naturally in arithmetic via point counts on curves over finite fields and l-adic representations on abelian varieties.

In this talk we will survey recent work on properties of elliptic curves and higher dimensional abelian varieties with unisingular l-adic representations with an emphasis on explicit calculation and construction.   Some of this work is joint with John Voight, Jeff Yelton, and Meagan Kenney.

Supersingular elliptic curves have seen a resurgence in the past decade with new post-quantum cryptographic applications. In this talk, we will discover why and how these curves are used in new cryptographic protocol. Supersingular elliptic curve isogeny graphs can be endowed with additional level structure. We will look at the level structure graphs and the corresponding picture in a quaternion algebra.

Chebyshev famously observed empirically that more often than not, there are more primes of the form 3 mod 4 up to x than primes of the form 1 mod 4. This was confirmed theoretically much later by Rubinstein and Sarnak in a logarithmic density sense. Our understanding of this is conditional on the generalized Riemann Hypothesis as well as Linear Independence of the zeros of L-functions. 

We investigate similar questions for sums of two squares in arithmetic progressions. We find a significantly stronger bias than in primes, which happens for almost all integers in a natural density sense. Because the bias is more pronounced, we do not need to assume Linear Independence of zeros, only a Chowla-type Conjecture on non-vanishing of L-functions at 1/2.

We'll aim to be self-contained and define all the notions mentioned above during the talk. We shall review the origin of the bias in the case of primes and the work of Rubinstein and Sarnak. We'll explain the main ideas behind the proof of the bias in the sums-of-squares setting.