Arithmetic of Curves and Jacobians

The study of the arithmetic of curves is as old as mathematics itself and takes on many forms.

In some cases, such as Fermat's Last Theorem or Mazur's torsion theorem, one tries to prove that a sequence of curves with growing genus has no interesting rational points.

In other cases, such as the study of rational points in families of elliptic curves, there is no way to classify all solutions, but one tries to understand what is happening on average.

A third approach aims to link the existence of rational points on a given curve to the preponderance of points on the curve modulo larger and larger prime numbers.

This is the idea behind the Birch and Swinnerton-Dyer conjecture, and its generalization, the Beilinson-Bloch conjecture. The proposed research makes progress in each of the three paradigms above.

In corresponding order, we propose a Mazur-type theorem for a family of unitary Shimura curves, by exploiting the Jacquet-Langlands correspondence and a connection with Prym varieties.

A special case of this result would give a classification of torsion points in a family of genus three bielliptic Jacobians.

Second, we propose an approach to the Poonen-Rains heuristics for elliptic curves by combining twisting methods with Bhargava's geometry-of-numbers methods for universal families. Using similar methods, we aim to show that Hilbert's tenth problem has a negative answer over every number field.

Third, we study certain instances of the Beilinson-Bloch conjecture for the degree 3 motive of the Jacobian of a curve with complex multiplication. The strategy involves the construction of an Euler system composed of CM Ceresa cycles. Related work will explore torsion and infinite generation phenomena for Ceresa cycles, as well.