Abstract:
Consider a family of smooth elliptic curves degenerating to a singular curve. As two branch points collide, a loop shrinks to a point. Yet moving around the singular parameter produces a nontrivial transformation of the remaining cycles, described by the classical Picard--Lefschetz formula.
In this talk, we will explore this phenomenon through concrete pictures of curves, monodromy, and differential forms. We will then consider the same degeneration over the p-adic integers, where the role of monodromy is played by the inertia group acting on torsion points.
This example leads naturally to arithmetic Picard–Lefschetz theory. I will explain how my thesis generalizes the classical arithmetic formula from ordinary double points to diagonal-type isolated singularities, and describe some consequences for inertia, Frobenius, and the arithmetic of Airy moments.
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