# Past Junior Number Theory Seminar

18 June 2009
16:00
Abstract
Suppose a power series $f(x):= \sum_{n=0}^{\infty} a_{n} x^{n}$ has radius of convergence equal to $1$ and that $lim_{x\rightarrow 1}f(x) = s$. Does it therefore follow that $\sum_{n=0}^{\infty} a_{n} = s$? Tauber's Theorem answers in the affirmative, \textit{if} one imposes a certain growth condition (a \textit{Tauberian Condition}) on the coefficients $a_{n}$. Without such a condition it is clear that this cannot be true in general - take, for example, $f(x) = \sum_{n=0}^{\infty} (-1)^{n} x^{n}.$
• Junior Number Theory Seminar
1 June 2009
16:00
Abstract
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.
• Junior Number Theory Seminar
25 May 2009
16:00
Abstract
This is the second (of two) talks concerning the Birch--Swinnerton-Dyer Conjecture.
• Junior Number Theory Seminar
18 May 2009
16:00
Sebastian Pancratz
Abstract
This is the first (of two) talks which will be given concerning the Birch--Swinnerton-Dyer Conjecture.
• Junior Number Theory Seminar
9 March 2009
16:00
Abstract
The goal of this talk is to give sufficient conditions for the existence of points on certain varieties defned over finite fields.
• Junior Number Theory Seminar
2 March 2009
16:00
Sebastian Pancratz
Abstract
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.
• Junior Number Theory Seminar
23 February 2009
16:00
Jahan Zahid
Abstract
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 $p$.
• Junior Number Theory Seminar