Leírás
Alina C. Cojocaru (University of Illinois at Chicago and Institute of Mathematics of the Romanian Academy, Bucharest): Quantitative bounds related to an isogeny criterion for elliptic curves
Abstract: Let $E_1$ and $E_2$ be elliptic curves defined over a number field $K$, without complex multiplication. Denote by ${\cal{F}}_{E_1, E_2}$ the set of non-zero prime ideals $\mathfrak{p}$ of the ring of integers of $K$, of good reduction for $E_1$ and $E_2$, for which the Frobenius fields $\mathbb{Q}(\pi_{\mathfrak{p}}(E_1))$ and $\mathbb{Q}(\pi_{\mathfrak{p}}(E_2))$ are equal. It is known that the elliptic curves $E_1$ and $E_2$ are potentially isogenous if and only if ${\cal{F}}_{E_1, E_2}$ has a positive upper density within the set of prime ideals of the ring of integers of $K$. Motivated by this result, we investigate the growth of the function ${\cal{F}}_{E_1, E_2}(x)$ counting the prime ideals in ${\cal{F}}_{E_1, E_2}$ of norm at most $x$. In particular, when $E_1$ and $E_2$ are not potentially isogenous, we prove non-trivial upper bounds for ${\cal{F}}_{E_1, E_2}(x)$. This is joint work with Auden Hinz (University of Illinois at Chicago) and Tian Wang (Max Planck Institute for Mathematics, Bonn).
Nathan Jones (University of Illinois at Chicago): Elliptic curves with acyclic reductions modulo primes in arithmetic progressions
Abstract: Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and, for a prime $p$ of good reduction for $E$ let $\tilde{E}_p$ denote the reduction of $E$ modulo $p$. Inspired by an elliptic curve analogue of Artin's primitive root conjecture posed by S. Lang and H. Trotter in 1977, J-P. Serre adapted methods of C. Hooley to prove a GRH-conditional asymptotic formula for the number of primes $p \leq x$ for which the group $\tilde{E}_p(\mathbb{F}_p)$ is cyclic. More recently, Akbal and Gülo$\breve{\text{g}}$lu considered the question of cyclicity of $\tilde{E}_p(\mathbb{F}_p)$ under the additional restriction that $p$ lie in a fixed arithmetic progression. In this note, we study the issue of which elliptic curves $E$ and which arithmetic progressions $a \bmod n$ have the property that, for all but finitely many primes $p \equiv a \bmod n$, the group $\tilde{E}_p(\mathbb{F}_p)$ is not cyclic, answering a question of Akbal and Gülo$\breve{\text{g}}$lu on this issue. This is based on joint work with Sung Min Lee.