On the appearance of Eisenstein series through degeneration

Let $\Gamma$ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane $\mathbb H$, and let $M = \Gamma \backslash \mathbb H$ be the associated finite volume hyperbolic Riemann surface. If $\gamma$ is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If $\gamma$ is hyperbolic, then, following ideas due to Kudla-Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If $\gamma \in \Gamma$ corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.Comment: 15 pages, 2 figures. This paper has been accepted for publication in Commentarii Mathematici Helvetic

Strongly secure certificateless key agreement

We introduce a formal model for certificateless authenticated key exchange (CL-AKE) protocols. Contrary to what might be expected, we show that the natural combination of an ID-based AKE protocol with a public key based AKE protocol cannot provide strong security. We provide the first one-round CL-AKE scheme proven secure in the random oracle model. We introduce two variants of the Diffie-Hellman trapdoor the introduced by \cite{DBLP:conf/eurocrypt/CashKS08}. The proposed key agreement scheme is secure as long as each party has at least one uncompromised secret. Thus, our scheme is secure even if the key generation centre learns the ephemeral secrets of both parties

Formal degrees and local theta correspondence

10.1007/s00222-013-0460-5Inventiones Mathematicae1953509-67

Computing CM points on Shimura curves arising from compact arithmetic triangle groups

Let Γ ⊂ P SL2(R) be a cocompact arithmetic triangle group, i.e. a Fuchsian triangle group that arises from the unit group of a quaternion algebra over a totally real number field. The group Γ acts on the upper half-plane H; the quotient XC = Γ \H is a Shimura curve, and there is a map j: XC → P 1 C. We algorithmically apply the Shimura reciprocity law to compute CM points j(zD) ∈ P 1 C and their Galois conjugates so as to recognize them as purported algebraic numbers. We conclude by giving some examples of how this method works in practice