Past Junior Number Theory Seminar

In the previous talks we have seen the formulation of the Birch--Swinnerton-Dyer conjecture. This talk will focus on a fundamental question in diophantine geometry. Namely, given an algebraic curve \textit{C} defined over $\mathbb{Q}$ possessing at least one rational point, what is

the probability that \textit{C} has infinitely many rational points?

For curves of genus 0, the answer has been known ever since the ancient Greeks roamed the earth, and for genus > 1 the answer is also known (albeit for a much shorter time). The remaining case is genus 1, and this question has a history filled with tension and

conflict between data and conjecture.

I shall describe the heuristics behind the conjectures, taking into account the Birch--Swinnerton-Dyer Conjecture and the Parity Conjecture. I shall go on to outline the contrary numeric data, both in families of elliptic curves and for all elliptic curves of increasing conductor.

If one instead considers elliptic curves over function fields $\mathbb{F}_{q} (t)$, then, via a conjecture of Artin and Tate, one can compute the rank (and more) of elliptic curves of extremely large discriminant degree. I shall briefly describe the interplay between Artin--Tate and

Birch--Swinnerton-Dyer, and give new evidence finally supporting the conjecture.

This is the second (of two) talks concerning the Birch--Swinnerton-Dyer Conjecture.

This is the first (of two) talks which will be given concerning the Birch--Swinnerton-Dyer Conjecture.

The goal of this talk is to give sufficient conditions for the existence of points on certain varieties defned over finite fields.

This talk will mention methods of testing whether a given integer is prime. Included topics are Carmichael numbers, Fermat and Euler pseudo-primes and results contingent on the Generalised Riemann Hypothesis.

Aside from a few tangential problems, this seminar will include a proof of Ostrowski's Theorem. This states than any norm over the rationals is equivalent to either the Euclidean norm or the $p$-adic norm, for some prime $p$.

This talk will introduce Dirichlet's Theorem on the approximation of real numbers via rational numbers. Once this has been established, a stronger version of the result will be proved, viz Hurwitz's Theorem.

This second 'problem sheet' of the term includes a proof of Jensen's Theorem for the number of zeroes of an analytic function in a disc, the usefulness of which is highlighted by an application to the Riemann zeta-function.

A brief overview of consonance by way of continued fractions and modular arithmetic.

Defined in terms of $\zeta(\frac{1}{2} +it)$ are the Riemann-Siegel functions, $\theta(t)$ and $Z(t)$. A zero of $\zeta(s)$ on the critical line corresponds to a sign change in $Z(t)$, since $Z$ is a real function. Points where $\theta(t) = n\pi$ are called Gram points, and the so called Gram's Law states between each Gram point there is a zero of $Z(t)$, and hence of $\zeta(\frac{1}{2} +it)$. This is known to be false in general and work will be presented to attempt to quantify how frequently this fails.

I will review the construction of algebraic de Rham cohomology, relative de Rham cohomology, and the Gauss-Manin connection. I will then show how we can find a basis for the cohomology and the matrix for the connection with respect to this basis for certain families of curves sitting in weighted projective spaces.