Efficiently distinguishing prime and composite numbers is one of the fundamental problems in number theory. In 1987, Gordon introduced analogues of Fermat pseudoprimes and Carmichael numbers for elliptic curves with complex multiplication (CM): elliptic pseudoprimes, strong elliptic pseudoprimes and elliptic Carmichael numbers.

Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication by an order in $\mathbb{Q}(\sqrt{-d})$ and let $P \in E(\mathbb{Q})$ have infinite order. A composite number $N$ is called an elliptic pseudoprime if $(\frac{-d}{N}) = -1$, $N$ is coprime to $\Delta(E)$, and $N$ satisfies $(N+1)P\equiv \mathcal{O}\pmod{N}$.

In 2012, Silverman extended Gordon’s notion of elliptic pseudoprimes and elliptic Carmichael numbers to arbitrary elliptic curves.

Let $N \in \mathbb{Z}$, let $E/\mathbb{Q}$ be an elliptic curve, and let $P \in E(\mathbb{Z}/N\mathbb{Z})$. Write the $L$-series of $E/\mathbb{Q}$ as $L(E/\mathbb{Q},s) = \sum_n \frac{a_n}{n^s}$. Then $N$ is an elliptic pseudoprime for $(E,P)$ if $N$ has at least two distinct prime factors, $E$ has good reduction at every prime $p$ dividing $N$, and $(N+1-a_N) P \equiv \mathcal{O}\pmod{N}$.

In this talk we present results that provide bounds on the number of points on a given elliptic curve for which an odd integer $N$ is a strong elliptic pseudoprime and probabilistic results for a given odd integer $N$ being a strong elliptic pseudoprime for a randomly chosen point on a randomly chosen elliptic curve. In addition, we present similar results for strong elliptic pseudoprimes.

We will outline a technique that uses a hyperelliptic fibration on a surface to determine two-torsion elements in the Brauer group of a surface. Of particular note is that this technique can also find transcendental Brauer elements, which remain non-trivial upon extending the base field to its algebraic closure.

As some examples:

(i) It is of interest to understand qualitative features, in the form of tight defining inequalities for the Diophantine approximation sets that are defined as an application of the asymptotic theory of linear sections, with respect to a given linear system, that arise via the Diophantine arithmetic exceptional sets of the Subspace Theorem.

(ii) It is of interest to understand the extent to which algebraic points of a given bounded degree in a given projective variety, and more generally Deligne-Mumford stack, accumulate along proper subvarieties.

(iii) It is of interest to understand the extent to which defining equations and higher syzygies of embedded projective varieties govern questions about effectivity and complexity for calculation of local Weil and height functions (and twisted variants thereof).

As I will explain, there are several points of departure for these inter-related themes.

Finally, I intend to report on the recent and very interesting joint work, with Chatchai Noytaptim, in which we give criteria for non-Zariski density of (D,S)-integral points in forward orbits. This is achieved as an application of Schmidt's Subspace Theorem.

1)How explicit descriptions and computations of global sections of toric vector bundles can be employed to study heights, arithmetic dynamics, and potentially moduli spaces of morphisms.\newline 2) How to carry out these computations using Macaulay2, along with a report on recent progress and applications to the Kawaguchi-Silverman conjecture in arithmetic dynamics.

In this talk we present results on these structural aspect of the quadratic integer ring $\mathbb{Z}(\sqrt{N})$.