This is a report on the construction of $p$-adic $L$-functions attached to ordinary families of holomorphic modular forms on the unitary groups of $n$-dimensional hermitian vector spaces over $CM$ fields. The results have been obtained over a period of nearly 15 years in joint work with Ellen Eischen, Jian-Shu Li, and Chris Skinner. The $p$-adic $L$-functions specialize at classical points to critical values of standard $L$-functions of cohomological automorphic forms on unitary groups, or equivalently of cohomological automorphic forms on $GL(n)$ that satisfy a polarization condition. When $n = 1$ one recovers Katz's construction of $p$-adic $L$-functions of Hecke characters.[-]

Let $E / \mathbb{Q}$ be a non-CM elliptic curve and let $\mathcal{E}$ denote the collection of all elliptic curves geometrically isogenous to $E$. That is, for every $E^{\prime} \in \mathcal{E}$, there exists an isogeny $\varphi: E \rightarrow E^{\prime}$ defined over $\overline{\mathbb{Q}}$. Motivated by ties to Serre's Uniformity Conjecture, we will discuss the problem of identifying minimal torsion curves in $\mathcal{E}$, which are elliptic curves $E^{\prime} \in \mathcal{E}$ attaining a point of prime-power order in least possible degree. Using recent classification results of Rouse, Sutherland, and Zureick-Brown, we obtain an answer to this question in many cases, including a complete characterization for points of odd degree.

This is joint work with Nina Ryalls and Lori Watson.[-]

In this talk we will see that there are only finitely many modular curves that admit a smooth plane model. Moreover, if the degree of the model is greater than or equal to 19, no such curve exists. For modular curves of Shimura type we will show that none can admit a smooth plane model of degree 5, 6 or 7. Further, if a modular curve of Shimura type admits a smooth plane model of degree 8 we will see that it must be a twist of one of four curves.

This is joint work with Samuele Anni and Eran Assaf.[-]

The study of rational points on modular curves has a long history in number theory. Mazur's 1970s papers that describe the possible torsion subgroups and isogeny degrees for rational elliptic curves rest on a computation of the rational points on $X_{0}(N)$ and $X_{1}(N)$, and a large body of work since then continues this tradition.

Modular curves are parameterized by open subgroups $H$ of $\mathrm{GL}_{2}(\hat{\mathbb{Z}})$, and correspondingly parameterize elliptic curves $E$ whose adelic Galois representation $\displaystyle \lim_{ \leftarrow }E[n]$ is contained in $H$. For general $H$, the story of when $X_{H}$ has non-cuspidal rational or low degree points (and thus when there exist elliptic curves with the corresponding level structure) becomes quite complicated, and one of the best approaches we have for understanding it is large-scale computation.

I will describe a new database of modular curves, including rational points, explicit models, and maps between models, along with some of the mathematical challenges faced along the way. The close connection between modular curves and finite groups also arises in other areas of number theory and arithmetic geometry. Most well known are Galois groups associated to field extensions, but one attaches automorphism groups to algebraic varieties and Sato-Tate groups to motives. Building on existing tables of groups, we have added a new finite groups section to the L-functions and modular forms database, which we hope will prove useful both to number theorists and to others who are using and studying finite groups.[-]

We report on progress towards a theory of local Shimura varieties for $p$-adic fields, in parallel with the theory of local shtukas for $\mathbb{F}_q((t))$. Using perfectoid spaces (in particular Scholze's theory of "diamonds"), it is possible to give a unified definition of local shtukas which incorporates both the equal and unequal characteristic cases. Conjecturally, if G is a reductive group, the cohomology of a moduli space of local G-shtukas should realize the Langlands correspondence for G in a systematic way (along the lines described by V. Lafforgue for global stukas). This talk will draw heavily from ideas of Peter Scholze and Laurent Fargues.[-]

I will talk about an application of the differentiability of the arithmetic volume function and an arithmetic Bertini type theorem to classify when one can find a closed point on the generic fiber of an arithmetic variety, whose heights with respect to some finite tuple of arithmetic R-divisors approximate a given tuple of real numbers.This result is used to prove existential closedness of $\mathbb{Q}^{alg}$ as a globally valued field (abbreviated GVF) - it is an arithmetic analogue of the function field case published recently by Ita Ben Yaacov and Ehud Hrushovski.[-]

In our talk we will give a panorama of José Burgos' contributions to various generalizations of the classical arithmetic intersection theory developed by Gillet and Soulé. It starts with the extension of Arakelov geometry allowing to incorporate logarithmically singular metrics with applications to Shimura varieties. Further generalizations include toric varieties as well as the most recent results about arithmetic intersections of arithmetic b-divisors with applications to mixed Shimura varieties including the theory of Siegel-Jacobi forms.[-]