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Thesis Format

Monograph

Degree

Doctor of Philosophy

Program

Mathematics

Supervisor

Chris Hall

Abstract

Let $\mathbb{F}_q$ be a finite field of size $q$, where $q$ is a power of a prime $p \geq 5$. Let $C$ be a smooth, proper, and geometrically connected curve over $\mathbb{F}_q$. Consider an elliptic curve $E$ over the function field $K$ of $C$ with nonconstant $j$-invariant. One can attach to $E$ its $L$-function $L(T,E/K)$, which is a generating function that contains information about the reduction types of $E$ at the different places of $K$. The $L$-function of $E/K$ was proven to be a polynomial in $\mathbb{Z}[T]$.

In 1985, Schoof devised an algorithm to compute the zeta function of an elliptic curve over a finite field by directly computing its numerator modulo sufficiently many primes $\ell$. By analogy with Schoof, we consider an elliptic curve $E$ over $K$ with nonconstant $j$-invariant and study the problem of directly computing the reduction modulo $\ell$ of $L(T,E/K)$. In this work, we obtain results in two directions. Firstly, given an integer $N$ different from $p$ and an elliptic curve $E$ with $K$-rational $N$-torsion, we give a formula for the reduction modulo $N$ of the $L$-function of certain quadratic twists, extending a result from Chris Hall. We also give a formula relating the $L$-functions modulo $2$ of any two quadratic twists of $E$, without any assumption on the $K$-rational $2$-torsion. Secondly, given a prime $\ell \neq p$, we give, under some relatively general conditions, formulas for the reduction of $L(T,E/K)$ modulo $\ell$.

Summary for Lay Audience

The study of elliptic curves is a major area of research in the branch of number theory in mathematics. They are featured in the proof by Andrew Wiles of the famous Fermat's Last Theorem and are used in cryptography. They are smooth curves which can be represented inside some mathematical ambient space. Quite often, they can be essentially described by equations of the form $y^2 = x^3 + a \cdot x + b$, with $x$ and $y$ variables and $a$ and $b$ constants, for example $y^2 = x^3 + t \cdot x$. By replacing the variable $t$ in the previous equation with different integers, we describe different kinds of curves. Most of the time, we get again an elliptic curve, but sometimes the curve is singular. For example, if we replace $t$ by $0$, we get the singular curve $y^2 = x^3$ with singularity at the point $(x,y) = (0,0)$.

An important tool to study elliptic curves and which takes into account, for example, all the aforementioned substitutions of $t$ by integer values in the equation $y^2 = x^3 + t \cdot x$ at the same time, is the $L$-function of this elliptic curve. A very famous and unsolved problem in number theory is the conjecture of Birch and Swinnerton-Dyer, which claims a profound relation between the points of an elliptic curve and its $L$-function. In practice, it is a very difficult task to even compute explicitly these $L$-functions. We are interested in those $L$-functions of the form $1 + a_1 \cdot T + \cdots + a_n \cdot T^n$, where $a_1, \cdots, a_n$ are integers. In some branch of number theory it is common to have $L$-functions of this form.

We explain how to directly compute algorithmically, in many interesting cases, and without knowing these integers to begin with, the remainders of the division by prime numbers of the $a_1,\cdots,a_n$. By doing so for enough prime numbers, we obtain the precise values of the $a_1, \cdots, a_n$ using the so-called Chinese remainder theorem. This thesis builds on the pioneering work of Chris Hall in 2003 and is inspired by an algorithm developed by Ren\'e Schoof in 1985 to compute zeta functions of elliptic curves.

Creative Commons License

Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

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