Let E be an elliptic curve over Q and K/Q a quadratic extension. We will consider the trace map from E(K) to E(Q) and discuss related questions and implications.

The Serre weight conjectures, as extended by Edixhoven, imply a correspondence between unramified odd irreducible mod $p$ Galois representations and weight 1 mod $p$ modular forms. This correspondence has a natural generalization to Hilbert modular varieties, where we now consider the higher mod $p$ coherent cohomology of Hilbert modular varieties. I will talk about a proof of one direction of this correspondence, namely that the Galois representation associated to the mod $p$ coherent cohomology of Hilbert modular varieties in parallel weight 1 is unramified.

The main topic of this talk is how the L-polynomials of abelian varieties vary, where we either fix a prime and vary the abelian variety, or fix the abelian variety and vary the prime. This L-polynomial determines the p-adic valuations of the normalized Weil numbers of the Frobenius morphism. I will start by discussing the case of elliptic curves, where the main distinction is whether the elliptic curve is ordinary or supersingular. Then we will switch to the case of curves that are cyclic covers of the projective line; the moduli spaces of these cyclic covers can be mapped to unitary Shimura varieties by the Torelli morphism. I will describe joint work with Li, Mantovan, and Tang, where we used this connection to obtain new results about L-polynomials of curves. First, for a fixed prime p, we prove cases of a conjecture of Oort for abelian varieties of arbitrarily large dimension. Second, for certain abelian varieties of dimension 4, we generalize a result of Elkies about infinitely many primes of non-ordinary reduction.

Joint work with Li, Mantovan, Tang

In this talk I’ll introduce some harmonic analysis on local systems over graphs, including an explicit construction of Green's function. We will also discuss its application towards a Gross-Zagier formula for CM cycles over Shimura curves. More specifically, using the Green’s functions we compute suitable integral extensions of the CM cycles for the arithmetic intersection pairing.