In the early 1990s Ribet observed that the classical mod l multiplicity one results for modular curves, which are a consequence of the q-expansion principle, fail to generalize to Shimura curves. Specifically he found examples of Galois representations which occur with multiplicity 2 in the mod l cohomology of a Shimura curve with discriminant pq and level 1.

I will describe a new approach to proving multiplicity statements for Shimura curves, using the Taylor-Wiles-Kisin patching method (which was shown by Diamond to give an alternate proof of multiplicity one in certain cases), as well as specific computations of local Galois deformation rings done by Shotton. This allows us to re-interpret and generalize Ribet's result. I will prove a mod l "multiplicity 2^k" statement in the minimal level case, where k is a number depending only on local Galois theoretic data. This proof also yields additional information the Hecke module structure of the cohomology of a Shimura curve, which among other things has applications to the study of congruence modules.

If an algebraic curve over a field of characteristic 0 admits a finite
map to the projective line ramified only over three points, then it must
be definable over some number field. This fact has a famous converse due
to Belyi: any curve over a number field admits such a finite map over
its field of definition.

Similarly, if an algebraic curve over a field of characteristic p>0
admits a finite *tamely ramified* map to the projective line ramified
only over three points, then it must be definable over some finite
field. We prove the converse: any curve over a finite field admits such
a finite map over its field of definition.

A construction of Saidi shows that this reduces to the existence of a
single tamely ramified map. This is easy to establish over an infinite
field of odd characteristic, and only slightly harder (using
Poonen-style probabilistic techniques) over a finite field of odd
characteristic. To handle the case of a finite field of characteristic
2, we use a construction of Sugiyama-Yasuda that they used to establish
existence of tame morphisms over an algebraically closed field of
characteristic 2.

Joint work with Daniel Litt (Georgia) and Jakub Witaszek (Michigan).

We study experimentally the Hermite factor of BKZ2.0 on uSVP lattices, with the motivation of understanding the concrete security of LWE in the setting of homomorphic encryption. We run experiments by generating instances of LWE in small dimensions, where we consider secrets sampled from binary, ternary or discrete Gaussian distributions. We convert each LWE instance into a uSVP instance and run the BKZ2.0 algorithm to find an approximation to the shortest vector. When the attack is successful, we can deduce a bound on the Hermite factor achieved for the given blocksize. This allows us to give concrete values for the Hermite factor of the lattice generated for the uSVP instance. We compare the values of the Hermite factors we find for these lattices with estimates from the literature and find that the Hermite factor may be smaller than expected for blocksizes 30, 35, 40, 45. Our work also demonstrates that the experimental and estimated values of the Hermite factor trend differently as we increase the dimension of the lattice, highlighting the importance of a better theoretical understanding of the performance of BKZ2.0 on uSVP lattices.