Given a family of algebraic varieties over an irreducible scheme, a natural question to ask is what type of properties of the generic fiber, and how do those properties extend to other fibers. For example, the Hilbert irreducibility theorem states that a dominant map from an irreducible variety X defined over a number field to some projective space which is generically of degree d provides a Zariski dense set of degree d points on X. One can also get quantitative estimates for size of the complement which does not carry the generic property.

We will explore this topic from an arithmetic point of view by looking at several scenarios. For instance, suppose we have a 1-dimensional family of pairs of elliptic curves over a number field,  with the generic fiber of this family being a pair of non-isogenous elliptic curves. One may ask how does the property of "being (non-)isogenous" extends to the special fibers. Can we give a quantitative estimation for the number of specializations of height at most B, such that the two elliptic curves at the specializations are isogenous? 

The cokernel of a random p-adic matrix can be used to study the distribution of objects that arise naturally in number theory. For example, Cohen and Lenstra suggested a conjectural distribution of the p-parts of the ideal class groups of imaginary quadratic fields. Friedman and Washington proved that the distribution of the cokernel of a random p-adic matrix is the same as the Cohen–Lenstra distribution. Recently, Wood generalized the result of Friedman–Washington by considering a far more general class of measure on p-adic matrices. In this talk, we explain a further generalization of Wood’s work. This is joint work with Dong Yeap Kang and Jungin Lee.

Complex Hodge theory equips the rational singular cohomology of smooth projective variety with a Hodge filtration, and period maps measure how these filtrations vary in families. Period maps are highly transcendental in nature, and the transcendence properties of these maps reflect interesting aspects of the geometry; for example a theorem of Cohen and Shiga-Wolfhart show that a complex abelian variety A has CM if and only both A and the Hodge filtration are defined over a number field.

In this talk, we describe a similar situation in p-adic Hodge theory, and discuss joint work with Sean Howe that proves an analogue of the theorem of Cohen and Shiga-Wolfhart. The first part of the talk will contain a brief introduction to Hodge theory, so no previous experience with Hodge theory (complex or p-adic) is required!

Gauss's quadratic reciprocity law has been extensively generalized in multiple directions within number theory. This talk will begin with explicit examples of reciprocity laws, including an interpretation of the proof of Fermat’s Last Theorem by Wiles and Taylor-Wiles as a consequence of a reciprocity law. As part of this discussion, I will introduce modular forms, elliptic curves, and Galois representations, leading to an overview of the Langlands reciprocity. I will then discuss the role of congruences in the study of reciprocity laws, with a particular focus on the Serre weight conjectures. I will conclude by outlining the proof of the Serre weight conjectures for GSp4. This is partly based on joint work with Daniel Le and Bao Le Hung.