Lectures on Number Theory
Wednesday February 19, 2014
Room 303.B09, University of Auckland

Schedule

 11:00 -- 12:00 Ben Green Bounded gaps between primes 1 12:00 -- 13:00 Lunch 13:00 -- 14:00 Ben Green Bounded gaps between primes 2 14:00 -- 15:00 Tim Trudgian An incomplete version of Wilson's theorem 15:00 -- 15:30 Coffee 15:30 -- 16:30 Daniel Delbourgo Derivatives of p-adic Hasse-Weil L-functions 16:30 -- 17:30 Shaun Cooper Apéry numbers, hypergeometric functions and modular forms

Abstracts

Ben Green (Oxford) Bounded gaps between primes
Abstract: 2013 saw a series of sensational developments on the problem of showing that pairs of primes can be close together infinitely often. Zhang proved in May that there are infinitely many pairs of primes differing by at most 70 million. By October the "Polymath" project had reduced this to around 1000. Then, James Maynard (and independently Terry Tao) came along with a much simpler proof giving a bound of 600, which has subsequently been reduced to around 300 by a further Polymath project. I propose to tell the whole story. There will not be an excess of detail.

Tim Trudgian (ANU) An incomplete version of Wilson's theorem
Abstract: Consider the congruence $(p-1)\cdots(p-r) \equiv -1 \pmod(p)$, where $p$ is an odd prime and $1\leq r \leq p-1$. Clearly this holds for $r=1$. Wilson's theorem is the statement that this congruence holds for $r=p-1$, whence the congruence also holds for $r=p-2$. Call r \equiv 1, -1, -2 \pmod(p) the trivial solutions. What is the proportion of primes $p$ with exactly $N$ non-trivial solutions to this congruence? (This is joint work with David Harvey and Joel Beeren of UNSW).

Daniel Delbourgo (Waikato) Derivatives of p-adic Hasse-Weil L-functions
Abstract: Conjecturally the Mordell-Weil rank of an elliptic curve should be determined by the order of vanishing of its L-function. If one formulates a p-adic version of this statement, an offset to both sides of this statement is required precisely when the elliptic curve has split multiplicative reduction above the prime number p. We then prove a generalisation of a theorem of Greenberg and Stevens to certain non-abelian number fields.

Shaun Cooper (Massey) Apéry numbers, hypergeometric functions and modular forms
Abstract: The numbers $a_n = \sum_{k=0}^n {n \choose k}^2 {n+k \choose k}$ and $b_n= \sum_{k=0}^n {n \choose k}^2 {n+k \choose k}^2$ were used by R. Apéry in his proofs of the irrationality of $\zeta(2)$ and $\zeta(3)$, respectively. The generating functions for the sequences {a_n} and {b_n} are analogues of hypergeometric functions, and they can be uniformized by modular forms. These, and many other similar examples, will be surveyed.

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