Enumerative and Arithmetic Geometry of Logarithmic curves

ongoing

The PI has recently developed new techniques to understand the structure of the double ramification cycle, a geometric object playing a central role in the degeneration of curves and jacobians. In this project we will use these tools and results to count algebraic curves in manifolds, and to count solutions in the rational numbers to polynomial equations. Our counts of curves will be algebraic Gromov-Witten invariants, which play a role in diverse areas including physics (where they are one of the two faces of mirror symmetry) and integrable systems (where their generating functions solve important hierarchies of PDEs). The strongest techniques currently available to understand Gromov-Witten invariants are to break the curve into simpler pieces (a cohomological field theory structure), or break the target into simpler pieces (enhancing to logarithmic Gromov-Witten invariants). This project will build the theoretical foundations needed to combine these two techniques, and explore the delicate combinatorial structures of the resulting invariants. The rational solutions we count will be torsion points on abelian varieties. Abelian varieties are algebraic analogues of compact Lie groups, and play a pivotal role in diverse areas such as cryptography, coding theory, and modular and automorphic forms. The size of the torsion subgroup is one of the key invariants for the conjecture of Birch and Swinnerton-Dyer. We will prove a formal immersion property in dimension 2. This is a major step towards the Torsion Conjecture in dimension 2, which predicts that the number of rational torsion points on abelian surfaces is uniformly bounded.