p-adic zeta function, Hodge theory and hyperplane arrangements

In 1988, Igusa observed a mysterious relationship between the poles of the p-adic zeta function and the roots of the Bernstein-Sato polynomial. This relationship was later formulated precisely by Denef and Loeser and is now known as the Strong Monodromy Conjecture. In the special case of hyperplane arrangements, Budur, Mustațǎ and Teitler proposed the n/d conjecture in 2009, which asserts that if a polynomial defines a central, essential, and indecomposable hyperplane arrangement of degree d in C^n, then -n/d must be a root of its b-function. They showed that the n/d conjecture implies the Strong Monodromy Conjecture for hyperplane arrangements.

In this talk, I will discuss my recent joint work with Dougal Davis on a proof of the n/d conjecture, which draws on the theory of complex mixed Hodge modules of Sabbah and Schnell, as well as our new ”wall-crossing” theory for V-filtrations of holonomic D-modules along local complete intersections. The latter is inspired by the recent breakthrough by Davis-Vilonen on the Schmid-Vilonen conjecture, which characterizes the unitarity of a representation of a real Lie group via Hodge theory. Furthermore, we also prove that the pole order of the Igusa zeta function is less than or equal to the multiplicity of the b-function along the real part of the pole. If time permits, I will discuss how to extend this idea to prove the Strong Monodromy Conjecture for multi-arrangements, as well as the multivariate n/d conjecture, both proposed by Budur in 2015.