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Fields of definition of elliptic kcurves and the realizability of all genus 2 sato–tate groups over a number field
(20180701)
Article
Open AccessLet A/Q be an abelian variety of dimension g = 1 that is isogenous over Q to Eg, where E is an elliptic curve. If E does not have complex multiplication (CM), by results of Ribet and Elkies concerning fields of definition ... 
Del Pezzo surfaces over finite fields and their Frobenius traces
(20180410)
Article
Open AccessLet S be a smooth cubic surface over a finite field q. It is known that #S( q) = 1 + aq + q2 for some a ¿ {2, 1, 0, 1, 2, 3, 4, 5, 7}. Serre has asked which values of a can arise for a given q. Building on special cases ... 
Heegner points on HijikataPizerShemanske curves and the Birch and SwinnertonDyer conjecture
(20180101)
Article
Open AccessWe study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rather general type of quaternionic orders. We address several questions ... 
Computation of numerical semigroups by means of seeds
(20180901)
Article
Open AccessFor the elements of a numerical semigroup which are larger than the Frobenius number, we introduce the definition of seed by broadening the notion of generator. This new concept allows us to explore the semigroup tree in ... 
Stark points and padic iterated integrals attached to modular forms of weight one
(Cambridge University Press, 20150101)
Article
Open AccessLet be an elliptic curve over , and let and be odd twodimensional Artin representations for which is selfdual. The progress on modularity achieved in recent decades ensures the existence of normalized eigenforms , , and ... 
BeilinsonFlach elements and Euler systems II: padic families and the Birch and SwinnertonDyer conjecture
(20150323)
Article
Open AccessLet E be an elliptic curve over Q and let % be an odd, irreducible twodimensional Artin representation. This article proves the Birch and SwinnertonDyer conjecture in analytic rank zero for the HasseWeilArtin Lseries ... 
GrossStark units and padic iterated integrals attached to modular forms of weight one
(20160801)
Article
Open AccessThis article can be read as a companion and sequel to the authors’ earlier article on Stark points and padic iterated integrals attached to modular forms of weight one, which proposes a conjectural expression for the ... 
Elliptic curves of rank two and generalized Kato classes
(20160824)
Article
Open AccessHeegner points play an outstanding role in the study of the Birch and SwinnertonDyer conjecture, providing canonical Mordell–Weil generators whose heights encode first derivatives of the associated Hasse–Weil Lseries. ... 
On the rank and the convergence rate toward the SatoTate measure
(20171016)
Article
Open AccessLet A be an abelian variety defined over a number field and let G denote its Sato–Tate group. Under the assumption of certain standard conjectures on L functions attached to the irreducible representations of G, we study ... 
Stark points and the HidaRankin padic Lfunction
(201802)
Article
Open AccessThis article is devoted to the elliptic Stark conjecture formulated by Darmon (Forum Math Pi 3:e8, 2015), which proposes a formula for the transcendental part of a padic avatar of the leading term at s=1 of the Hasse–Weil–Artin ... 
On the SatoTate conjecture for nongeneric abelian surfaces
(201701)
Article
Open AccessWe prove many nongeneric cases of the SatoTate conjecture for abelian surfaces as formulated by Fité, Kedlaya, Rotger and Sutherland, using the potential automorphy theorems of BarnetLamb, Gee, Geraghty and Taylor. 
Diagonal cycles and Euler systems II: the Birch and SwinnertonDyer conjecture for HasseWeilArtin Lfunctions
(20170701)
Article
Open AccessThis article establishes new cases of the Birch and SwinnertonDyer conjecture in analytic rank 0, for elliptic curves over $ \mathbb{Q}$ viewed over the fields cut out by certain selfdual Artin representations of dimension ...