On the Generalized Arithmetic Picard--Lefschetz Formula
This dissertation establishes an explicit arithmetic Picard--Lefschetz formula for isolated singularities of diagonal type. We investigate proper and flat families of varieties defined over a henselian discrete valuation ring of mixed characteristic. When the smooth generic fiber degenerates into a special fiber with isolated diagonal singularities, we construct an explicit eigenbasis for the tame inertia action on the local vanishing cycles. This construction yields a complete spectral decomposition, from which we deduce explicit formulas for the intersection pairing and the variation morphism. Furthermore, by computing the action of the geometric Frobenius operator on the subspace of inertia invariants, we establish an exact trace formula expressed in terms of Jacobi sums. We apply this theoretical framework to study the arithmetic of the symmetric powers of the Airy sheaf. By relating the compactly supported cohomology to degenerating affine hypersurfaces, we completely determine the local Galois representation for the $\ell$-adic realization of motives associated to Airy moments. This application provides a precise direct sum decomposition of the Galois module, a clear identification of the inertia invariants, and the explicit characteristic polynomial of the Frobenius action.