Serre famously proved that for elliptic curves $E$ over number fields $k$ without complex multiplication, the galois group $H$ of the field generated over $k$ by all the torsion points $E_{\text{tor}}$ of $E$ is a subgroup of finite index in $G=\displaystyle\lim_{\leftarrow\atop n} \text{GL}_2(\Bbb Z/n\Bbb Z)$. When $k=\Bbb Q$, the smallest the index of $H$ in $G$ can be is 2, and if it is, we say $E$ is a Serre curve over $\Bbb Q$. Now let $E$ be an elliptic curve over $\Bbb Q(t)$. So long as the galois group generated over $\Bbb Q(t)$ by $E_{\text{tor}}$ is all of $G$, ``almost all" specializations $t_0$ of $t$ in $\Bbb Q$ give rise to elliptic curves $E_{t_0}$ which are Serre curves, and if we consider those $t_0$ of height bounded by some $B$, we give bounds for the number of $E_{t_0}$ which are not Serre curves in terms of $B$.

The 17th Southern California Geometric Analysis Seminar

Southern California Number Theory Day, 10am - 5pm

Southern California Analysis and PDE Conference Day 1

Let E be an elliptic curve over an algebraically closed field k of
characteristic p>0. Then the physical p-torsion E[p](k) is either trivial,
and E is called supersingular, or E[p](k) is a group of order p. More
generally, if X/k is an abelian variety of dimension g, then X[p](k)
is isomorphic to (Z/p)^f for some number f, called the p-rank of X.
The p-rank induces a stratification of the moduli space of abelian
varieties; via the Torelli functor, it induces a stratification of the
moduli space of (hyperelliptic) curves.
I'll discuss recent results on the geometry of these strata, with
special attention to their structure at the boundary of the moduli
space. This information yields new applications about the prime-to-p
part of the class group of a quadratic function field with specified geometric
p-rank; the existence of absolutely simple hyperelliptic Jacobians of
every p-rank; and the stratification of the moduli space of curves by
Newton polygon.