TN  Teoria de Nombres
http://hdl.handle.net/2117/3728
20190321T10:22:51Z

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 .

Elliptic curves of rank two and generalized Kato classes
http://hdl.handle.net/2117/116318
Elliptic curves of rank two and generalized Kato classes
Darmon, Henri; Rotger Cerdà, Víctor
Heegner 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. Yet the fruitful connection between Heegner points and Lseries also accounts for their main limitation, namely that they are torsion in (analytic) rank >1. This partly expository article discusses the generalised Kato classes introduced in Bertolini et al. (J Algebr Geom 24:569–604, 2015) and Darmon and Rotger (J AMS 2016), stressing their analogy with Heegner points but explaining why they are expected to give nontrivial, canonical elements of the idoneous Selmer group in settings where the classical Lfunction (of Hasse–Weil–Artin type) that governs their behaviour has a double zero at the centre. The generalised Kato class denoted ¿(f,g,h) is associated to a triple (f, g, h) consisting of an eigenform f of weight two and classical pstabilised eigenforms g and h of weight one, corresponding to odd twodimensional Artin representations Vg and Vh of Gal(H/Q) with padic coefficients for a suitable number field H. This class is germane to the Birch and SwinnertonDyer conjecture over H for the modular abelian variety E over Q attached to f. One of the main results of Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016) is that ¿(f,g,h) lies in the prop Selmer group of E over H precisely when L(E,Vgh,1)=0, where L(E,Vgh,s) is the Lfunction of E twisted by Vgh:=Vg¿Vh. In the setting of interest, parity considerations imply that L(E,Vgh,s) vanishes to even order at s=1, and the Selmer class ¿(f,g,h) is expected to be trivial when ords=1L(E,Vgh,s)>2. The main new contribution of this article is a conjecture expressing ¿(f,g,h) as a canonical point in (E(H)¿Vgh)GQ when ords=1L(E,Vgh,s)=2. This conjecture strengthens and refines the main conjecture of Darmon et al. (Forum Math Pi 3:e8, 2015) and supplies a framework for understanding the results of Darmon et al. (2015), Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016).
20180416T10:04:01Z
Darmon, Henri
Rotger Cerdà, Víctor
Heegner 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. Yet the fruitful connection between Heegner points and Lseries also accounts for their main limitation, namely that they are torsion in (analytic) rank >1. This partly expository article discusses the generalised Kato classes introduced in Bertolini et al. (J Algebr Geom 24:569–604, 2015) and Darmon and Rotger (J AMS 2016), stressing their analogy with Heegner points but explaining why they are expected to give nontrivial, canonical elements of the idoneous Selmer group in settings where the classical Lfunction (of Hasse–Weil–Artin type) that governs their behaviour has a double zero at the centre. The generalised Kato class denoted ¿(f,g,h) is associated to a triple (f, g, h) consisting of an eigenform f of weight two and classical pstabilised eigenforms g and h of weight one, corresponding to odd twodimensional Artin representations Vg and Vh of Gal(H/Q) with padic coefficients for a suitable number field H. This class is germane to the Birch and SwinnertonDyer conjecture over H for the modular abelian variety E over Q attached to f. One of the main results of Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016) is that ¿(f,g,h) lies in the prop Selmer group of E over H precisely when L(E,Vgh,1)=0, where L(E,Vgh,s) is the Lfunction of E twisted by Vgh:=Vg¿Vh. In the setting of interest, parity considerations imply that L(E,Vgh,s) vanishes to even order at s=1, and the Selmer class ¿(f,g,h) is expected to be trivial when ords=1L(E,Vgh,s)>2. The main new contribution of this article is a conjecture expressing ¿(f,g,h) as a canonical point in (E(H)¿Vgh)GQ when ords=1L(E,Vgh,s)=2. This conjecture strengthens and refines the main conjecture of Darmon et al. (Forum Math Pi 3:e8, 2015) and supplies a framework for understanding the results of Darmon et al. (2015), Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016).

On the rank and the convergence rate toward the SatoTate measure
http://hdl.handle.net/2117/114584
On the rank and the convergence rate toward the SatoTate measure
Fite Naya, Francesc; Guitart Morales, Xavier
Let 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 the convergence rate of any virtual character of G. We find that this convergence rate is dictated by several arithmetic invariants of A, such as its rank or its Sato–Tate group G. The results are consonant with some previous experimental observations, and we also provide additional numerical evidence consistent with them. The techniques that we use were introduced by Sarnak in a letter to Mazur, in order to explain the bias in the sign of the Frobenius traces of an elliptic curve without complex multiplication defined over Q. We show that the same methods can be adapted to study the convergence rate of the characters of its Sato–Tate group, and that they can also be employed in the more general case of abelian varieties over number fields. A key tool in our analysis is the existence of limiting distributions for automorphic Lfunctions, which is due to Akbary, Ng, and Shahabi.
20180228T09:52:07Z
Fite Naya, Francesc
Guitart Morales, Xavier
Let 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 the convergence rate of any virtual character of G. We find that this convergence rate is dictated by several arithmetic invariants of A, such as its rank or its Sato–Tate group G. The results are consonant with some previous experimental observations, and we also provide additional numerical evidence consistent with them. The techniques that we use were introduced by Sarnak in a letter to Mazur, in order to explain the bias in the sign of the Frobenius traces of an elliptic curve without complex multiplication defined over Q. We show that the same methods can be adapted to study the convergence rate of the characters of its Sato–Tate group, and that they can also be employed in the more general case of abelian varieties over number fields. A key tool in our analysis is the existence of limiting distributions for automorphic Lfunctions, which is due to Akbary, Ng, and Shahabi.

Stark points and the HidaRankin padic Lfunction
http://hdl.handle.net/2117/114369
Stark points and the HidaRankin padic Lfunction
Casazza, Daniele; Rotger Cerdà, Víctor
This 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 Lseries L(E,¿1¿¿2,s) of an elliptic curve E/Q twisted by the tensor product ¿1¿¿2 of two odd 2dimensional Artin representations, when the order of vanishing is two. The main ingredient of this formula is a 2×2 padic regulator involving the padic formal group logarithm of suitable Stark points on E. This conjecture was proved by Darmon (Forum Math Pi 3:e8, 2015) in the setting where ¿1 and ¿2 are induced from characters of the same imaginary quadratic field K. In this note, we prove a refinement of this result that was discovered experimentally by Darmon (Forum Math Pi 3:e8, 2015, [Remark 3.4]) in a few examples. Namely, we are able to determine the algebraic constant up to which the main theorem of Darmon (Forum Math Pi 3:e8, 2015) holds in a particular setting where the Hida–Rankin padic Lfunction associated to a pair of Hida families can be exploited to provide an alternative proof of the same result. This constant encodes local and global invariants of both E and K.
The final publication is available at Springer via http://dx.doi.org/10.1007/s111390169824y
20180222T12:37:51Z
Casazza, Daniele
Rotger Cerdà, Víctor
This 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 Lseries L(E,¿1¿¿2,s) of an elliptic curve E/Q twisted by the tensor product ¿1¿¿2 of two odd 2dimensional Artin representations, when the order of vanishing is two. The main ingredient of this formula is a 2×2 padic regulator involving the padic formal group logarithm of suitable Stark points on E. This conjecture was proved by Darmon (Forum Math Pi 3:e8, 2015) in the setting where ¿1 and ¿2 are induced from characters of the same imaginary quadratic field K. In this note, we prove a refinement of this result that was discovered experimentally by Darmon (Forum Math Pi 3:e8, 2015, [Remark 3.4]) in a few examples. Namely, we are able to determine the algebraic constant up to which the main theorem of Darmon (Forum Math Pi 3:e8, 2015) holds in a particular setting where the Hida–Rankin padic Lfunction associated to a pair of Hida families can be exploited to provide an alternative proof of the same result. This constant encodes local and global invariants of both E and K.