# Joint IAS/PU Number Theory

## The $p$-curvature conjecture and monodromy about simple closed loops

## Congruences between motives and congruences between values of $L$-functions

## Basic loci of Shimura varieties

In mod-$p$ reductions of modular curves, there is a finite set of supersingular points and its open complement corresponding to ordinary elliptic curves. In the study of mod-$p$ reductions of more general Shimura varieties, there is a "Newton stratification" decomposing the reduction into finitely many locally closed subsets, of which exactly one is closed. This closed set is called the basic locus; it recovers the supersingular locus in the classical case of modular curves.

## Galois Representations for the general symplectic group

In a recent preprint with Sug Woo Shin (https://arxiv.org/abs/1609.04223) I construct Galois representations corresponding for cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. In this talk I will explain some parts of this construction that involve the eigenvariety.

## On small sums of roots of unity

Let $k$ be a fixed positive integer. Myerson (and others) asked how small the modulus of a non-zero sum of $k$ roots of unity can be. If the roots of unity have order dividing $N$, then an elementary argument shows that the modulus decreases at most exponentially in $N$ (for fixed $k$). Moreover it is known that the decay is at worst polynomial if $k < 5$. But no general sub-exponential bound is known if $k \geq 5$.