Articles de revista
http://hdl.handle.net/2117/3729
20200128T20:15:36Z

BeilinsonFlach elements, Stark units and padic iterated integrals
http://hdl.handle.net/2117/168702
BeilinsonFlach elements, Stark units and padic iterated integrals
Rivero Salgado, Óscar; Rotger Cerdà, Víctor
We study weight one specializations of the Euler system of Beilinson–Flach elements introduced by Kings, Loeffler and Zerbes, with a view towards a conjecture of Darmon, Lauder and Rotger relating logarithms of units in suitable number fields to special values of the Hida–Rankin padic Lfunction. We show that the latter conjecture follows from expected properties of Beilinson–Flach elements and prove the analogue of the main theorem of Castella and Hsieh about generalized Kato classes.
20190925T17:14:57Z
Rivero Salgado, Óscar
Rotger Cerdà, Víctor
We study weight one specializations of the Euler system of Beilinson–Flach elements introduced by Kings, Loeffler and Zerbes, with a view towards a conjecture of Darmon, Lauder and Rotger relating logarithms of units in suitable number fields to special values of the Hida–Rankin padic Lfunction. We show that the latter conjecture follows from expected properties of Beilinson–Flach elements and prove the analogue of the main theorem of Castella and Hsieh about generalized Kato classes.

On the Galois embedding problem associated to the universal central extension of the alternating group of degree 6
http://hdl.handle.net/2117/168376
On the Galois embedding problem associated to the universal central extension of the alternating group of degree 6
Crespo, Teresa; Vela del Olmo, Maria Montserrat
We obtain an explicit expression for the obstruction to the solvability of the Galois embedding problem associated to the universal central extension of the alternating group of degree 6.
© 2013. This manuscript version is made available under the CCBYNCND 4.0 license http://creativecommons.org/licenses/byncnd/4.0/
20190918T15:04:55Z
Crespo, Teresa
Vela del Olmo, Maria Montserrat
We obtain an explicit expression for the obstruction to the solvability of the Galois embedding problem associated to the universal central extension of the alternating group of degree 6.

Fields of definition of elliptic kcurves and the realizability of all genus 2 sato–tate groups over a number field
http://hdl.handle.net/2117/131241
Fields of definition of elliptic kcurves and the realizability of all genus 2 sato–tate groups over a number field
Fite Naya, Francesc; Guitart Morales, Xavier
Let 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 of elliptic Qcurves E is isogenous to a curve defined over a polyquadratic extension of Q. We show that one can adapt Ribet’s methods to study the field of definition of E up to isogeny also in the CM case. We find two applications of this analysis to the theory of Sato–Tate groups: First, we show that 18 of the 34 possible Sato–Tate groups of abelian surfaces over Q occur among at most 51 Qisogeny classes of abelian surfaces over Q; Second, we give a positive answer to a question of Serre concerning the existence of a number field over which abelian surfaces can be found realizing each of the 52 possible Sato–Tate groups of abelian surfaces
20190404T08:06:05Z
Fite Naya, Francesc
Guitart Morales, Xavier
Let 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 of elliptic Qcurves E is isogenous to a curve defined over a polyquadratic extension of Q. We show that one can adapt Ribet’s methods to study the field of definition of E up to isogeny also in the CM case. We find two applications of this analysis to the theory of Sato–Tate groups: First, we show that 18 of the 34 possible Sato–Tate groups of abelian surfaces over Q occur among at most 51 Qisogeny classes of abelian surfaces over Q; Second, we give a positive answer to a question of Serre concerning the existence of a number field over which abelian surfaces can be found realizing each of the 52 possible Sato–Tate groups of abelian surfaces

Del Pezzo surfaces over finite fields and their Frobenius traces
http://hdl.handle.net/2117/124690
Del Pezzo surfaces over finite fields and their Frobenius traces
Banwait, Barinder; Fite Naya, Francesc; Loughran, Daniel
Let 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 treated by Swinnerton–Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton–Dyer's tables on cubic surfaces over finite fields.
20181120T08:41:31Z
Banwait, Barinder
Fite Naya, Francesc
Loughran, Daniel
Let 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 treated by Swinnerton–Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton–Dyer's tables on cubic surfaces over finite fields.

Hopf Galois structures on symmetric and alternating extensions
http://hdl.handle.net/2117/124086
Hopf Galois structures on symmetric and alternating extensions
Río Doval, Ana; Vela del Olmo, Maria Montserrat; Crespo Vicente, Teresa
By using a recent theorem by Koch, Kohl, Truman and Underwood on normality, we determine that some types of Hopf Galois structures do not occur on Galois extensions with Galois group isomorphic to alternating or symmetric groups. Our theory of induced Hopf Galois structures allows us to obtain the whole picture of types of Hopf Galois structures on A4extensions, S4extensions, and S5extensions. Combining it with a result of Carnahan and Childs, we obtain a complete count of the Hopf Galois structures on S5extensions.
20181113T10:22:53Z
Río Doval, Ana
Vela del Olmo, Maria Montserrat
Crespo Vicente, Teresa
By using a recent theorem by Koch, Kohl, Truman and Underwood on normality, we determine that some types of Hopf Galois structures do not occur on Galois extensions with Galois group isomorphic to alternating or symmetric groups. Our theory of induced Hopf Galois structures allows us to obtain the whole picture of types of Hopf Galois structures on A4extensions, S4extensions, and S5extensions. Combining it with a result of Carnahan and Childs, we obtain a complete count of the Hopf Galois structures on S5extensions.

Heegner points on HijikataPizerShemanske curves and the Birch and SwinnertonDyer conjecture
http://hdl.handle.net/2117/123220
Heegner points on HijikataPizerShemanske curves and the Birch and SwinnertonDyer conjecture
Longo, Matteo; Rotger Cerdà, Víctor; Vera Piquero, Carlos de
We 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 arising from the Birch and SwinnertonDyer (BSD) conjecture in this general context. In particular, under mild technical conditions, we show the existence of nontorsion Heegner points on elliptic curves in all situations in which the BSD conjecture predicts their existence.
20181030T11:05:25Z
Longo, Matteo
Rotger Cerdà, Víctor
Vera Piquero, Carlos de
We 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 arising from the Birch and SwinnertonDyer (BSD) conjecture in this general context. In particular, under mild technical conditions, we show the existence of nontorsion Heegner points on elliptic curves in all situations in which the BSD conjecture predicts their existence.

Computation of numerical semigroups by means of seeds
http://hdl.handle.net/2117/118339
Computation of numerical semigroups by means of seeds
Bras Amorós, Maria; Fernández González, Julio
For 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 an alternative efficient way, since the seeds of each descendant can be easily obtained from the seeds of its parent. The paper is devoted to presenting the results which are related to this approach, leading to a new algorithm for computing and counting the semigroups of a given genus.
20180622T08:57:35Z
Bras Amorós, Maria
Fernández González, Julio
For 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 an alternative efficient way, since the seeds of each descendant can be easily obtained from the seeds of its parent. The paper is devoted to presenting the results which are related to this approach, leading to a new algorithm for computing and counting the semigroups of a given genus.

Stark points and padic iterated integrals attached to modular forms of weight one
http://hdl.handle.net/2117/116342
Stark points and padic iterated integrals attached to modular forms of weight one
Darmon, Henri; Lauder, Alan; Rotger Cerdà, Víctor
Let 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 of respective weights two, one, and one, giving rise to , , and via the constructions of Eichler and Shimura, and of Deligne and Serre. This article examines certain adic iterated integrals attached to the triple , which are adic avatars of the leading term of the Hasse–Weil–Artin series when it has a double zero at the centre. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on —referred to as Stark points—which are defined over the number field cut out by . This formula can be viewed as an elliptic curve analogue of Stark’s conjecture on units attached to weightone forms. It is proved when and are binary theta series attached to a common imaginary quadratic field in which splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell–Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark–Heegner points attached to Shintanitype cycles on ), and extensions of with Galois group a central extension of the dihedral group or of one of the exceptional subgroups , , and of
This article has been published in a revised form in Forum of Mathematics Pi, http://dx.doi.org/10.1017/fmp.2015.7. This version is free to view and download for private research and study only. Not for redistribution, resale or use in derivative works. © 2015.
20180416T13:25:12Z
Darmon, Henri
Lauder, Alan
Rotger Cerdà, Víctor
Let 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 of respective weights two, one, and one, giving rise to , , and via the constructions of Eichler and Shimura, and of Deligne and Serre. This article examines certain adic iterated integrals attached to the triple , which are adic avatars of the leading term of the Hasse–Weil–Artin series when it has a double zero at the centre. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on —referred to as Stark points—which are defined over the number field cut out by . This formula can be viewed as an elliptic curve analogue of Stark’s conjecture on units attached to weightone forms. It is proved when and are binary theta series attached to a common imaginary quadratic field in which splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell–Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark–Heegner points attached to Shintanitype cycles on ), and extensions of with Galois group a central extension of the dihedral group or of one of the exceptional subgroups , , and of

BeilinsonFlach elements and Euler systems II: padic families and the Birch and SwinnertonDyer conjecture
http://hdl.handle.net/2117/116321
BeilinsonFlach elements and Euler systems II: padic families and the Birch and SwinnertonDyer conjecture
Bertolini, Massimo; Darmon, Henri; Rotger Cerdà, Víctor
Let 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 L(E, %, s), namely, the implication L(E, %, 1) 6= 0 ¿ (E(H) ¿ %) Gal(H/Q) = 0, where H is the finite extension of Q cut out by %. The proof relies on padic families of global Galois cohomology classes arising from BeilinsonFlach elements in a tower of products of modular curves.
20180416T10:28:28Z
Bertolini, Massimo
Darmon, Henri
Rotger Cerdà, Víctor
Let 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 L(E, %, s), namely, the implication L(E, %, 1) 6= 0 ¿ (E(H) ¿ %) Gal(H/Q) = 0, where H is the finite extension of Q cut out by %. The proof relies on padic families of global Galois cohomology classes arising from BeilinsonFlach elements in a tower of products of modular curves.

GrossStark units and padic iterated integrals attached to modular forms of weight one
http://hdl.handle.net/2117/116319
GrossStark units and padic iterated integrals attached to modular forms of weight one
Darmon, Henri; Lauder, Alan; Rotger Cerdà, Víctor
This 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 socalled p adic iterated integrals attached to a triple (f, g, h) of classical eigenforms of weights (2, 1, 1). When f is a cusp form, this expression involves the padic logarithms of socalled Stark points: distinguished points on the modular abelian variety attached to f, defined over the number field cut out by the Artin representations attached to g and h. The goal of this paper is to formulate an analogous conjecture when f is a weight two Eisenstein series rather than a cusp form. The resulting formula involves the padic logarithms of units and punits in suitable number fields, and can be seen as a new variant of Gross’s padic analogue of Stark’s conjecture on Artin Lseries at s=0 .
The final publication is available at Springer via http://dx.doi.org/10.1007/s4031601500426
20180416T10:20:51Z
Darmon, Henri
Lauder, Alan
Rotger Cerdà, Víctor
This 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 socalled p adic iterated integrals attached to a triple (f, g, h) of classical eigenforms of weights (2, 1, 1). When f is a cusp form, this expression involves the padic logarithms of socalled Stark points: distinguished points on the modular abelian variety attached to f, defined over the number field cut out by the Artin representations attached to g and h. The goal of this paper is to formulate an analogous conjecture when f is a weight two Eisenstein series rather than a cusp form. The resulting formula involves the padic logarithms of units and punits in suitable number fields, and can be seen as a new variant of Gross’s padic analogue of Stark’s conjecture on Artin Lseries at s=0 .