Allan Perez Murillo

Degeneration of families of curves in the Birch and Swinnerton-Dyer conjecture.

Riemann pioneered the classification of algebraic curves by organising them into geometric spaces of parameters in a natural way. This naturality led to the study of families of curves, and it became clear, for deep geometric reasons, that such families cannot always be smooth. This is the phenomenon of degeneration. Its local study — a basic example of which is reducing a polynomial $y^2 = x^3 + 1$ modulo a prime $p$ — is geometric in origin, yet carries surprising connections to arithmetic which Tate elucidated in 1966 in order to clarify the computations of Birch and Swinnerton-Dyer. I will present a brief exposition of the theory of degeneration of curves, their importance in the BSD conjecture, and how monodromy governs some basic questions about them that remain wide open.