On the bounded generation of arithmetic SL2{\rm SL}_2

Let $K$ be a number field and ${\mathcal O}$ be the ring of $S$-integers in $K$. Morgan, Rapinchuck, and Sury have proved that if the group of units ${\mathcal O}^{\times}$ is infinite, then every matrix in ${\rm SL}_2({\mathcal O})$ is a product of at most $9$ elementary matrices. We prove that under the additional hypothesis that $K$ has at least one real embedding or $S$ contains a finite place we can get a product of at most $8$ elementary matrices. If we assume a suitable Generalized Riemann Hypothesis, then every matrix in ${\rm SL}_2({\mathcal O})$ is the product of at most $5$ elementary matrices if $K$ has at least one real embedding, the product of at most $6$ elementary matrices if $S$ contains a finite place, and the product of at most $7$ elementary matrices in general

Isogeny graphs of superspecial abelian varieties (Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties)

We define three different isogeny graphs of principally polarized superspecial abelian varieties, prove foundational results on them, and explain their role in number theory and geometry. This is background to joint work with Yevgeny Zaytman on properties of these isogeny graphs for dimension g > 1, especially the result that they are connected, but not in general Ramanujan