p-adic periods, p-adic L-functions, and the p-adic uniformization of Shimura curves

p-ADIC PERIODS, p-ADIC L-FUNCTIONS, AND THE p-ADIC UNIFORMIZATION OF SHIMURA CURVES

p-ADIC UNIFORMIZATION OF CURVES

Let us presume that we have at our disposal the fully-formed theory of rigidanalytic spaces., as sketched in my last talk. Why would we care to look for uniformizations of algebraic objects in the rigid analytic category? Let’s look at the analogous situation in the complex-analytic category. We know that, for example, abelian varieties are uniformized by spaces of the form C/Λ, where Λ is a fr...

P -adic Uniformization of Unitary Shimura Varieties

Introduction Let Γ ⊂ PGUd−1,1(R) 0 be a torsion-free cocompact lattice. Then Γ acts on the unit ball B ⊂ C by holomorphic automorphisms. The quotient Γ\B is a complex manifold, which has a unique structure of a complex projective variety XΓ (see [Sha, Ch. IX, §3]). Shimura had proved that when Γ is an arithmetic congruence subgroup, XΓ has a canonical structure of a projective variety over some...

Height pairings on Shimura curves and p-adic uniformization

In a recent paper [16] of one of us it was shown that there is a close connection between the value of the height pairing of certain arithmetic 0-cycles on Shimura curves and the values at the center of their symmetry of the derivatives of certain metaplectic Eisenstein series of genus 2. On the one hand, the height pairing can be written as a sum of local height pairings. For example, if the 0...

p-adic Shearlets

The field $Q_{p}$ of $p$-adic numbers is defined as the completion of the field of the rational numbers $Q$ with respect to the  $p$-adic norm $|.|_{p}$. In this paper, we study the continuous and discrete $p-$adic shearlet systems on $L^{2}(Q_{p}^{2})$. We also suggest discrete $p-$adic shearlet frames. Several examples are provided.