Victor Rotger

Easy Decision Diffie-Hellman Groups

The decision Diffie-Hellman problem (DDH) is a central computational problem in cryptography. It is known that the Weil and Tate pairings can be used to solve many DDH problems on elliptic curves. Distortion maps are an important tool for solving DDH problems using pairings, and it is known that distortion maps exist for all super-singular elliptic curves. An algorithm is presented here to construct suitable distortion maps. The algorithm is efficient on the curves that are usable in practice, and hence all DDH problems on these curves are easy. The issue of which DDH problems on ordinary curves are easy is also discussed.

Beilinson-Flach elements and Euler systems I: syntomic regulators and p-adic Rankin L-series

This article is the first in a series devoted to the Euler system arising from p-adic families of Beilinson-Flach elements in the first K-group of the product of two modular curves. It relates the image of these elements under the p-adic syntomic regulator (as described by Besser (2012)) to the special values at the near-central point of Hidas p-adic Rankin L-function attached to two Hida families of cusp forms.

Equations of Shimura Curves of Genus Two

LetBD be the indefinite quaternion algebra overQ of reduced discriminantD=p1· · · · ·p2r for pairwise different prime numbers pi and let XD/Q be the Shimura curve attached to BD. As it was shown by Shimura [23], XD is the coarse moduli space of abelian surfaces with quaternionic multiplication by BD. Let W = {ωm : m | D} ⊆ Aut Q(XD) be the group of Atkin-Lehner involutions. For any m | D, we will denote X D = XD/〈ωm〉, the quotient of the Shimura curve XD, by ωm. The importance of the curves X D is enhanced by their moduli interpretation as curves embedded in Hilbert-Blumenthal surfaces and Igusa’s threefold A2 (cf. [21, 22]). The classical modular case arises when D = 1. In this case, automorphic forms of these curves admit Fourier expansions around the cusp of infinity and we know explicit generators of the field of functions of such curves. Also, explicit methods are known to determine bases of the space of their regular differentials, which are used to compute equations for quotients of modular curves. When D = 1, the absence of cusps has been an obstacle for explicit approaches to Shimura curves. Explicit methods to handle functions and regular differential forms on these curves are less accessible and we refer the reader to [3] for progress in this regard. For this reason, at present, few equations of Shimura curves are known, all of them of genus 0 or 1 (cf. [6, 11, 13]). In addition, in a later work, Kurihara conjectured equations for all Shimura curves of genus two and for several curves of genera three and five, though he was not able to give a proof for his guesses (cf. [14]).

On rigid analytic uniformizations of Jacobians of Shimura curves

The main goal of this article is to give an explicit rigid analytic uniformization of the maximal toric quotient of the Jacobian of a Shimura curve over

On the Group of Automorphisms of Shimura Curves and Applications

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Modular Shimura varieties and forgetful maps

at a prime dividing exactly the level. This result can be viewed as complementary to the classical theorem of \v{C}erednik and Drinfeld which provides rigid analytic uniformizations at primes dividing the discriminant. As a corollary, we offer a proof of a conjecture formulated by M. Greenberg in his paper on Stark-Heegner points and quaternionic Shimura curves, thus making Greenbergs construction of local points on elliptic curves over

Beilinson-Flach elements and Euler systems II: The Birch-Swinnerton-Dyer conjecture for Hasse-Weil-Artin -series

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On finiteness conjectures for endomorphism algebras of abelian surfaces

unconditional.

The Birch and Swinnerton-Dyer conjecture for Q-curves and Oda's period relations

AbstractLet VD be the Shimura curve over