Computational Algebra Seminar: Mudigonda -- Finding structure in the randomness of the L-functions of many elliptic curves

**PLEASE NOTE date--rescheduled due to industrial action**

The L-function of a non-CM elliptic curve $E$ is determined by a sequence of integers
$a_p(E)$, where the index p ranges over the prime numbers.  This sequence is typically
thought of as a "random" object - for example, given the values of $a_p(E)$ at all but
one prime, there is no direct procedure for computing the remaining one.  For a given
elliptic curve $E$, the first-order distribution of the (appropriately normalized)
$a_p(E)$ as $p$ grows converges to a semicircular distribution that is well understood.
However, the second-order fluctuations of the $a_p(E)$ are poorly understood - they are
related to the rank of $E$, and are intimately related to the Birch and Swinnerton-Dyer
conjecture.  Recent computations suggest a new approach towards understanding the
sequence $(a_p(E))_p$ for a wide range of elliptic curves $E$.  For such $E$, the
"random" object $(a_p(E))_p$ can be obtained by starting with a periodic (in an
appropriate sense) sequence of numbers associated to a tower of number fields
$L/K/\mathbb{Q}$ and applying a simple "noising" operator based on the theory of Hilbert
modular forms.  In this talk, we discuss computations underway to investigate this
phenomenon, as well as work in progress on understanding the mathematical implications.