Victor Rotger - Academia.edu (original) (raw)

Papers by Victor Rotger

Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Ja... more Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over Q and formulate conjectures about their rationality properties. Moreover, if K is a real quadratic field, E is an elliptic curve over Q without complex multiplication and \chi is a ring class character such that L(E/K,\chi,1) is

Let f be a modular eigenform of even weight k>0 and new at a prime p dividing exactly the leve... more Let f be a modular eigenform of even weight k>0 and new at a prime p dividing exactly the level, with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to f a monodromy module D_FM(f) and an L-invariant L_FM(f). The first goal of this paper is building a suitable p-adic integration theory that allows us

We study the set of isomorphism classes of principal polarizations on abelian varieties of GL2-ty... 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'/\Q of genus two which are nonisomorphic over \bar \Q and share isomorphic unpolarized modular Jacobian varieties over \Q ; we also show a method to obtain genus two curves over \Q whose Jacobian

It is conjectured that there exist finitely many isomorphism classes of simple endomorphism algeb... more It is conjectured that there exist finitely many isomorphism classes of simple endomorphism algebras of abelian varieties of GL_2-type over \Q of bounded dimension. We explore this conjecture when particularized to quaternion endomorphism algebras of abelian surfaces by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the

We use rigid analytic uniformization by Schottky groups to give a bound for the order of the abel... more We use rigid analytic uniformization by Schottky groups to give a bound for the order of the abelian subgroups of the automorphism group of a Mumford curve in terms of its genus.

Journal of Algebraic Geometry, 2014

This article is the first in a series devoted to the Euler system arising from p-adic families of... 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 the image of these elements under the p-adic syntomic regulator (as described by Besser [Bes3]) to the special values at the near-central point of Hida's p-adic Rankin Lfunction attached to two Hida families of cusp forms.

Mathematics of Computation, 2015

Let E /Q be an elliptic curve of conductor N and let f be the weight 2 newform on Γ0(N ) associat... more Let E /Q be an elliptic curve of conductor N and let f be the weight 2 newform on Γ0(N ) associated to it by modularity. Building on an idea of S. Zhang, the article [DRS] describes the construction of so-called Chow-Heegner points P T,f ∈ E(Q) indexed by algebraic correspondences T ⊂ X0(N )×X0(N ). It also gives an analytic formula, depending only on the image of T in cohomology under the complex cycle class map, for calculating P T,f numerically via Chen's theory of iterated integrals. The present work describes an algorithm based on this formula for computing the Chow-Heegner points to arbitrarily high complex accuracy, carries out the computation for all elliptic curves of rank 1 and conductor N < 100 when the cycles T arise from Hecke correspondences, and discusses several important variants of the basic construction.

The decision-Diffie-Hellman problem (DDH) is a central computational problem in cryptography. It ... more 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 supersingular elliptic curves. We present an algorithm

An abelian surface A over a field K has potential quaternionic mul- tiplication if the ring End ¯... more An abelian surface A over a field K has potential quaternionic mul- tiplication if the ring End ¯ K(A) of geometric endomorphisms of A is an order in an indefinite rational quaternion algebra. In this brief note, we study the possible structures of the ring of endomorphisms of these surfaces and we provide explicit examples of Jacobians of curves of genus two which show that our result is sharp.

We present explicit models for Shimura curves X_D and Atkin-Lehner quotients X_D/w_m of them of g... more We present explicit models for Shimura curves X_D and Atkin-Lehner quotients X_D/w_m of them of genus 2. We show that several equations conjectured by Kurihara are correct and compute for them the kernel of Ribet's isogeny J_0(D)^{new} --> J_D between the new part of the Jacobian of the modular curve X_0(D) and the Jacobian of X_D.

Journal of the European Mathematical Society, 2012

Mathematical Proceedings of the Cambridge Philosophical Society, 2006

It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebra... more It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of abelian surfaces of GL 2type over Q by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by local and global methods, including Chabauty techniques on explicit equations of Shimura curves.

Bulletin of the London Mathematical Society, 2008

We study the group of automorphisms of Shimura curves X 0 (D, N ) attached to an Eichler order of... more We study the group of automorphisms of Shimura curves X 0 (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, when the genus g of the curve is greater than or equal to 2, Aut(X 0 (D, N )) is a 2-elementary abelian group which contains the group of Atkin-Lehner involutions W 0 (D, N ) as a subgroup of index 1 or 2. It is conjectured that Aut(X 0 (D, N )) = W 0 (D, N ) except for finitely many values of (D, N ) and we provide criteria that allow us to show that this is indeed often the case. Our methods are based on the theory of complex multiplication of Shimura curves and the Cerednik-Drinfeld theory on their rigid analytic uniformization at primes p | D.

… Proceedings of the …, 2006

Abstract. It is conjectured that there exist only finitely many isomorphism classes of endomorphi... more Abstract. It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of ...

Bulletin of the London Mathematical Society, 2008

We use rigid analytic uniformization by Schottky groups to give a bound for the order of the abel... more We use rigid analytic uniformization by Schottky groups to give a bound for the order of the abelian subgroups of the automorphism group of a Mumford curve in terms of its genus.

Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Ja... more Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over Q and formulate conjectures about their rationality properties. Moreover, if K is a real quadratic field, E is an elliptic curve over Q without complex multiplication and \chi is a ring class character such that L(E/K,\chi,1) is

Let f be a modular eigenform of even weight k>0 and new at a prime p dividing exactly the leve... more Let f be a modular eigenform of even weight k>0 and new at a prime p dividing exactly the level, with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to f a monodromy module D_FM(f) and an L-invariant L_FM(f). The first goal of this paper is building a suitable p-adic integration theory that allows us

We study the set of isomorphism classes of principal polarizations on abelian varieties of GL2-ty... 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'/\Q of genus two which are nonisomorphic over \bar \Q and share isomorphic unpolarized modular Jacobian varieties over \Q ; we also show a method to obtain genus two curves over \Q whose Jacobian

It is conjectured that there exist finitely many isomorphism classes of simple endomorphism algeb... more It is conjectured that there exist finitely many isomorphism classes of simple endomorphism algebras of abelian varieties of GL_2-type over \Q of bounded dimension. We explore this conjecture when particularized to quaternion endomorphism algebras of abelian surfaces by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the

We use rigid analytic uniformization by Schottky groups to give a bound for the order of the abel... more We use rigid analytic uniformization by Schottky groups to give a bound for the order of the abelian subgroups of the automorphism group of a Mumford curve in terms of its genus.

Journal of Algebraic Geometry, 2014

This article is the first in a series devoted to the Euler system arising from p-adic families of... 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 the image of these elements under the p-adic syntomic regulator (as described by Besser [Bes3]) to the special values at the near-central point of Hida's p-adic Rankin Lfunction attached to two Hida families of cusp forms.

Mathematics of Computation, 2015

Let E /Q be an elliptic curve of conductor N and let f be the weight 2 newform on Γ0(N ) associat... more Let E /Q be an elliptic curve of conductor N and let f be the weight 2 newform on Γ0(N ) associated to it by modularity. Building on an idea of S. Zhang, the article [DRS] describes the construction of so-called Chow-Heegner points P T,f ∈ E(Q) indexed by algebraic correspondences T ⊂ X0(N )×X0(N ). It also gives an analytic formula, depending only on the image of T in cohomology under the complex cycle class map, for calculating P T,f numerically via Chen's theory of iterated integrals. The present work describes an algorithm based on this formula for computing the Chow-Heegner points to arbitrarily high complex accuracy, carries out the computation for all elliptic curves of rank 1 and conductor N < 100 when the cycles T arise from Hecke correspondences, and discusses several important variants of the basic construction.

The decision-Diffie-Hellman problem (DDH) is a central computational problem in cryptography. It ... more 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 supersingular elliptic curves. We present an algorithm

An abelian surface A over a field K has potential quaternionic mul- tiplication if the ring End ¯... more An abelian surface A over a field K has potential quaternionic mul- tiplication if the ring End ¯ K(A) of geometric endomorphisms of A is an order in an indefinite rational quaternion algebra. In this brief note, we study the possible structures of the ring of endomorphisms of these surfaces and we provide explicit examples of Jacobians of curves of genus two which show that our result is sharp.

We present explicit models for Shimura curves X_D and Atkin-Lehner quotients X_D/w_m of them of g... more We present explicit models for Shimura curves X_D and Atkin-Lehner quotients X_D/w_m of them of genus 2. We show that several equations conjectured by Kurihara are correct and compute for them the kernel of Ribet's isogeny J_0(D)^{new} --> J_D between the new part of the Jacobian of the modular curve X_0(D) and the Jacobian of X_D.

Journal of the European Mathematical Society, 2012

Mathematical Proceedings of the Cambridge Philosophical Society, 2006

It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebra... more It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of abelian surfaces of GL 2type over Q by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by local and global methods, including Chabauty techniques on explicit equations of Shimura curves.

Bulletin of the London Mathematical Society, 2008

We study the group of automorphisms of Shimura curves X 0 (D, N ) attached to an Eichler order of... more We study the group of automorphisms of Shimura curves X 0 (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, when the genus g of the curve is greater than or equal to 2, Aut(X 0 (D, N )) is a 2-elementary abelian group which contains the group of Atkin-Lehner involutions W 0 (D, N ) as a subgroup of index 1 or 2. It is conjectured that Aut(X 0 (D, N )) = W 0 (D, N ) except for finitely many values of (D, N ) and we provide criteria that allow us to show that this is indeed often the case. Our methods are based on the theory of complex multiplication of Shimura curves and the Cerednik-Drinfeld theory on their rigid analytic uniformization at primes p | D.

… Proceedings of the …, 2006

Abstract. It is conjectured that there exist only finitely many isomorphism classes of endomorphi... more Abstract. It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of ...

Bulletin of the London Mathematical Society, 2008

We use rigid analytic uniformization by Schottky groups to give a bound for the order of the abel... more We use rigid analytic uniformization by Schottky groups to give a bound for the order of the abelian subgroups of the automorphism group of a Mumford curve in terms of its genus.