On some arithmetic properties of automorphic forms ofGLmover a division algebra

In this paper we investigate arithmetic properties of automorphic forms on the group G' = GLm/D, for a central division-algebra D over an arbitrary number field F. The results of this article are generalizations of results in the split case, i.e. D = F, by Shimura, Harder, Waldspurger and Clozel for square-integrable automorphic forms and also by Franke and Franke–Schwermer for general automorphic representations. We also compare our theorems on automorphic forms of the group G′ to statements on automorphic forms of its split form using the global Jacquet–Langlands correspondence developed by Badulescu and Badulescu–Renard. Beside that we prove that the local version of the Jacquet–Langlands transfer at an archimedean place preserves the property of being cohomological.

Local Heights of CM Points

This chapter computes the local heights and compares them with the derivatives computed before. It checks the theorem place by place and takes into account all the assumptions on the Schwartz function. According to the reduction of the Shimura curve, the situation is divided to the following four cases: archimedean case, supersingular case, superspecial case, and ordinary case. The treatments in different cases are similar in spirit, except that the fourth case is slightly different. The supersingular case is divided into two subcases: unramified case and ramified case. The chapter also describes local heights of CM points at any archimedean place v. The discussion covers the multiplicity function, the kernel function, unramified quadratic extension, ramified quadratic extension, ordinary components, supersingular components, and superspecial components.

Ordinary reduction of K3 surfaces

Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.

Remarks on Certain Metaplectic Groups

We study metaplectic coverings of the adelized group of a split connected reductive group G over a number field F. Assume its derived group G′ is a simply connected simple Chevalley group. The purpose is to provide some naturally defined sections for the coverings with good properties which might be helpful when we carry some explicit calculations in the theory of automorphic forms on metaplectic groups. Specifically, we1.construct metaplectic coverings of G(A) from those of G′(A);2.for any non-archimedean place v, show the section for a covering of G(Fv) constructed from a Steinberg section is an isomorphism, both algebraically and topologically in an open subgroup of G(Fv);3.define a global section which is a product of local sections on a maximal torus, a unipotent subgroup and a set of representatives for the Weyl group.