John Jones

The tame-wild principle: a users guide

Abstract

In computing tables of number fields, one often has to relate discriminants of various subfields of the normal closure of the desired field. We will explain the tame-wild principle which can make the process simple in some cases. The emphasis of this talk will be computational, i.e., what the tame-wild principle is, cases where it holds, and how one can use it for different types of number field computations.

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David Roberts

A brief introduction to motives

Abstract

I will provide an introduction to motives with a focus on topics of immediate relevance to the LMFDB project.

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Maarten Derickx

The rational group structure of modular Jacobians

Abstract

Let $C/\mathbb{Q}$ be a curve, and assume that one knows that $J(C)(\mathbb{Q})$ is a finite group and one knows explicit generators of $J(C)(\mathbb{Q})$, then it is possible to explicitly find all points of degree $d$ over $\mathbb{Q}$ on $C$ if this list is finite, and give explicit parameterizations if this list is infinite. In this talk I will focus on the case where $C$ is either the modular curve $X_0(N)$, the modular curve $X_1(N)$ or any modular curve lying in between those. If $N$ is a prime then there is already a lot known about the structure of the rational points of their jacobians, but in the composite case there is still a lot that is not known. In this talk I will show the results of some explicit computations about the structure of the modular jacobians $J_0(N)$ and $J_1(N)$, discuss how the things that are known in the prime case might generalize to the composite case based on these results, and show the applications of these results to the question of finding for which pairs $N,d$ there exists an elliptic curve $E$ over a number field $K$ of degree $d$ with a point of order exactly $N$.