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Thesis defences

PhD Oral Exam - Antoine Comeau-Lapointe, Mathematics

Dirichlet Twists of Elliptic Curves over Function Fields

DATE & TIME
Tuesday, August 23, 2022 (all day)
COST

This event is free

ORGANIZATION

School of Graduate Studies

CONTACT

Daniela Ferrer

WHERE

Online

When studying for a doctoral degree (PhD), candidates submit a thesis that provides a critical review of the current state of knowledge of the thesis subject as well as the student’s own contributions to the subject. The distinguishing criterion of doctoral graduate research is a significant and original contribution to knowledge.

Once accepted, the candidate presents the thesis orally. This oral exam is open to the public.

Abstract

Some of the most fundamental questions about $L$-functions are concerned with the location of their zeros, in particular at the central point, or on the critical line. Following the work of Montgomery and then of Katz and Sarnak, number theorists have learned that it is very fruitful to study families of $L$-functions rather than individual $L$-functions. Given a family of $L$-functions, it is common to classify it according to its symmetry type. The symmetry type can be either symplectic, orthogonal, or unitary, which refers to the corresponding ensemble from random matrix theory that models most accurately the distribution of the zeros of the family.

This thesis presents two papers studying the zeros of the family of Dirichlet twists of the $L$-function of an elliptic curve $E$ over $\mathbb{F}_q[t]$. In the first paper (Chapter 2), we show that the one-level density (the study of the low-lying zeros) for this family agrees with the conjecture of Katz and Sarnak based on random matrix theory, for test functions with limited support on the Fourier transform. For quadratic twists, the support of the Fourier transforms of the test functions is restricted to the interval (-1,1), and we observe an orthogonal symmetry. For higher order twists, the support is restricted to (-1/2,1/2) and we observe a unitary symmetry. For quadratic twists, the support is large enough to obtain a positive proportion of $L$-functions which do not vanish at the central point.

In the second paper (Chapter 3), we are taking the opposite point of view, and we construct certain elliptic curves $\mathbb{F}_q[t]$ with infinitely many twists of high order vanishing at the central point, generalizing a construction of Li and Donepudi-Li for Dirichlet $L$-functions. This construction only works when $E/\mathbb{F}_q[t]$ is a constant curve. In the general case where $E$ is not a constant curve, we performed extensive numerical computations that are compatible with the conjectures of David-Fearnley-Kisilevsky and Mazur-Rubin over number fields, which predict that such vanishing should be rare.

The last chapter presents an algorithm to construct the factorizations of the monic polynomials, a description of the zeros of $L(E\otimes\chi,u)$, and a family of quadratic twists such that the rank of $L(\chi,u)$ goes to infinity.

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