Bayou Arithmetic Research Days

I will give a demonstration of how to convert decimal approximations into exact algebraic numbers, using the special values of the -function at imaginary quadratic inputs as our key example. We will review elliptic curves with complex multiplication, ideal classes in quadratic fields, modular forms, and modular functions. We will also discuss the shortest vector problem and the LLL lattice basis reduction algorithm.

Zauner’s conjecture posits the existence in every finite dimension of a set of equiangular lines. Recent work by Appleby, Kopp, and myself has attempted a construction of such lines that uses conjectural Stark units. This conjectural construction offers the tantalizing possibility that one can systematically construct numerical SICs. However, the SICs themselves are Galois conjugates of the Stark units, which are only known as values of a transcendental function, rather than as algebraic numbers that would facilitate Galois conjugation. In this talk, I will give a pedagogical overview of the necromancy algorithm recently used to construct SICs from these conjectural “ghost” objects composed of Stark units. I will also describe the open-source Julia package Zauner.jl that implements this algorithm and describe some simplifications and possible improvements.

Thirty years ago, Ekedahl and Serre posed two intriguing questions about the decomposition of Jacobian varieties. They asked if, for every genus, there is a curve whose Jacobian variety decomposes into only elliptic curves and, barring that, if there is an upper bound to the genera for which such a curve exists. Using several different techniques they gave examples up to genus 1297 (but still with many unknown genera less than 1297). I will talk about work that has been done since then, mainly computational, to find many more examples.

-adic fields in the LMFDB

Over the past year, the number of -adic fields included in the LMFDB has increased by a factor of 62; the majority of the new fields are degree 16 and degree 20 extensions of . There were two key improvements allowing for this expansion: a new notion of family of -adic fields and an algorithm for deterministically choosing a defining polynomial for each extension. I will explain some of the underlying theory behind these advances, give a tour of the expanded database within the LMFDB, and discuss some outstanding problems.