The decision-Diffie-Hellman problem (DDH) is an important computational problem in cryptography. It is known that the Weil and Tate pairings can be used to solve many DDH problems on elliptic curves.â€¦ (More)

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 theâ€¦ (More)

We study the set of isomorphism classes of principal polarizations on abelian varieties of GL2-type. As applications of our results, we construct examples of curves C, C â€²/Q of genus two which areâ€¦ (More)

We present explicit models for non-elliptic genus one Shimura curves X0(D, N) with Î“0(N)-level structure arising from an indefinite quaternion algebra of reduced discriminant D, and Atkin-Lehnerâ€¦ (More)

We study the group of automorphisms of Shimura curves X0(D, N) attached to an Eichler order of square-free level N in an indefinite rational quaternion algebra of discriminant D > 1. We prove that,â€¦ (More)

This article is the first in a series devoted to studying generalised Gross-KudlaSchoen diagonal cycles in the product of three Kuga-Sato varieties and the Euler system properties of the associatedâ€¦ (More)

LetBD be the indefinite quaternion algebra over Q of reduced discriminantD=p1Â· Â· Â· Â· Â·p2r for pairwise different prime numbers pi and let XD/Q be the Shimura curve attached to BD. As it was shown byâ€¦ (More)

In this note we consider several maps that occur naturally between modular Shimura varieties, Hilbert-Blumenthal varieties and the moduli spaces of polarized abelian varieties when forgetting certainâ€¦ (More)

Let E be an elliptic curve over Q, and let %[ and %] be odd two-dimensional Artin representations for which %[ âŠ— %] is self-dual. The progress on modularity achieved in recent decades ensures theâ€¦ (More)

Article history: Received 2 November 2014 Received in revised form 23 June 2015 Accepted 11 July 2015 Available online xxxx Communicated by George E. Andrews