Supersingular elliptic curves have seen a resurgence in the past decade with new post-quantum cryptographic applications. In this talk, we will discover why and how these curves are used in new cryptographic protocol. Supersingular elliptic curve isogeny graphs can be endowed with additional level structure. We will look at the level structure graphs and the corresponding picture in a quaternion algebra.
The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\le X$ such that $p+h$ is prime for any non-zero even integer $h$. While this conjecture remains wide open, Matom\"{a}ki, Radziwi{\l}{\l} and Tao proved that it holds on average over $h$, improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some recent work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \le X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$.
We consider a crystalline universal deformation ring R of an n-dimensional, mod p Galois representation whose semisimplification is the direct sum of two non-isomorphic absolutely irreducible representations. Under some hypotheses, we obtain that R is a discrete valuation ring. The method examines the ideal of reducibility of R, which is used to construct extensions of representations in a Selmer group with specified dimension. This can be used to deduce modularity of representations.
An integral ring R is a ring additively isomorphic to Z^n . The subring zeta function is an important tool in studying subring growth in these rings. One can compute these zeta functions using p-adic integration due to a result of Grunewald, Segal and Smith. I shall talk about computing these zeta functions for Z[t]/(t^n) for small n and describe some results on subring growth and ideal growth for integral rings. This includes joint work with Ramin Takloo-Bighash and Gautam Chinta.
The recent proofs of the Tate conjecture for K3 surfaces over finite fields start by lifting the surface to characteristic 0. Serre showed in the sixties that not every variety can be lifted, but the question whether every motive lifts to characteristic 0 is open. We give a negative answer to a geometric version of this question, by constructing a smooth projective variety that cannot be dominated by a smooth projective variety that lifts to characteristic 0.
