Abstracts for Number theory lunch seminar

We call a sequence of rational integers ($a_n$) a divisibility sequence if $a_m$ divides $a_n$ whenever $m$ divides $n$. The divisibility sequence appears in natural way in coordinates of points $nP$ on elliptic curve, where $P$ is a point of infinite order. In general we can define divisibility sequences for arbitrary group scheme over $\mathbb{Z}$. On the lecture we will focus on elliptic case. We will describe main properties and recent results and we will discuss some open problems.

Let $f$ be a holomorphic or Maass cusp form on the upper half plane. We use the slow divergence of the horocycle flow on the upper half plane to get an algorithm to compute $L(f,1/2+iT)$ up to a maximum error $O(T^{-\gamma})$ using $O(T^{7/8+\eta})$ operations. Here $\gamma$ and $\eta$ are any positive numbers and the constants in $O$ are independent of $T$. We thus improve the current approximate functional equation based algorithms which have complexity $O(T^{1+\eta})$.

A (1;e)-curve is a quotient of the upper half plane that is of genus 1 and ramifies above only one point. We explore the finite list, due to Kisao Takeuchi, of arithmetic (1;e)-curves, which are those (1;e)-curves that allow a natural finite-to-one correspondence with a Shimura curve coming from a quaternion algebra over a totally real field.