Number Theory Seminar
Frederick Vincent Saia
University of Georgia
A volcanic approach to CM points on Shimura curves
Abstract: A CM component of the $\ell$-isogeny graph of elliptic curves has
a particular structure, that of an $\ell$-volcano,
at least away from certain CM orders.
The structure of “isogeny volcanoes’’ has seen much use in the study
of CM elliptic curves over finite fields,
originating with the 1996 PhD thesis work of Kohel.
Recent work of Clark—Saia leverages infinite depth versions of
these graphs to study moduli of isogenies of CM elliptic curves
over $\overline{\mathbb{Q}}$.
We will discuss an analogue of this work for abelian surfaces with
quaternionic multiplication. A main result includes an algorithm
to compute the $\mathfrak{o}$-CM locus on the Shimura curve
$X_0^D(N)$ over $\mathbb{Q}$, for $\mathfrak{o}$ any imaginary
quadratic order and $\text{gcd}(D,N) = 1$.
As an application, we give an explicit list of pairs $(D,N)$ for
which the Shimura curves $X_0^D(N)$ and $X_1^D(N)$ may fail to have
a sporadic CM point.
The talk will be held on zoom:
https://uic.zoom.us/j/88173268700?pwd=aEhmTGpSOVhidWE4L1VWUnNhNVlvUT09
Friday April 14, 2023 at 12:00 PM in 612 SEO