Logarithmic Version Of The Milnor Formula And Derivatives Of Siegel Modular Forms

This thesis consists of two independent parts. Part I consists of Chapter 1 to Chapter5 and this part is about ramification theory in arithmetic geometry. Part II starts from Chapter 6 to Chapter 9. In the second part, we give a new method to construct di?erential operators of Siegel modular forms.In Part I, we propose a logarithmic version of the Milnor formula and prove this formula in the geometric case. Let X be a regular scheme, D a simple normal crossings divisor on X and S a henselian trait of perfect residue field. Let f : X â†' S be a flat morphism of finite type. Let be a prime number invertible on S, ? a locally constant and constructible sheaf of F-modules on U = X- D. Assume that ? is tamely ramified along the boundary D. Then the logarithmic version of the Milnor formula says that, if f has isolated log-singular at a closed point x ∈ D, then the total dimension of vanishing cycles of f for the sheaf ? at x is equal to rank ? times the logarithmic Milnor number of f at x. In chapter 5, we give another interpretation of the logarithmic Milnor formula in terms of characteristic cycle. This implies that P. Deligne’s conjecture about the total dimension of vanishing cycles is true for tamely ramified etale sheaves.In Part II, we introduce a method in di?erential geometry to study the derivative operators of Siegel modular forms. By determining the coe?cients of the Levi-Civita connection on a Siegel upper half plane, and further by calculating the expressions of the di?erential forms under this connection, we get a non-holomorphic derivative operator of the Siegel modular forms. In order to get a holomorphic derivative operator, we introduce a weaker notion, called modular connection, on the Siegel upper half plane. Then we show that on a Siegel upper half plane there exists at most one holomorphic Sp(2g, Z)-modular connection in some sense, and get a possible holomorphic derivative operator of Siegel modular forms.