TN - Teoria de Nombres
http://hdl.handle.net/2117/3728
2024-03-29T12:25:01Z
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Hopf-Galois module structure of quartic Galois extensions of Q
http://hdl.handle.net/2117/367132
Hopf-Galois module structure of quartic Galois extensions of Q
Gil Muñoz, Daniel; Río Doval, Ana
Given a quartic Galois extension L/Q of number fields and a Hopf-Galois structure H on L/Q, we study the freeness of the ring of integers OL as module over the associated order AH in H. For the classical Galois structure Hc, we know by Leopoldt’s theorem that OL is AHc -free. If L/Q is cyclic, it admits a unique non-classical Hopf-Galois structure, whereas if it is biquadratic, it admits three such Hopf-Galois structures. In both cases, we obtain that freeness depends on the solvability in Z of certain generalized Pell equations. We shall translate some results on Pell equations into results on the AH-freeness of OL.
© 2022 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
2022-05-10T09:14:42Z
Gil Muñoz, Daniel
Río Doval, Ana
Given a quartic Galois extension L/Q of number fields and a Hopf-Galois structure H on L/Q, we study the freeness of the ring of integers OL as module over the associated order AH in H. For the classical Galois structure Hc, we know by Leopoldt’s theorem that OL is AHc -free. If L/Q is cyclic, it admits a unique non-classical Hopf-Galois structure, whereas if it is biquadratic, it admits three such Hopf-Galois structures. In both cases, we obtain that freeness depends on the solvability in Z of certain generalized Pell equations. We shall translate some results on Pell equations into results on the AH-freeness of OL.
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Induced Hopf Galois structures and their local Hopf Galois modules
http://hdl.handle.net/2117/366323
Induced Hopf Galois structures and their local Hopf Galois modules
Gil Muñoz, Daniel; Río Doval, Ana
The regular subgroup determining an induced Hopf Galois structure for a Galois extension L/K is obtained as direct product of the corresponding regular groups of the inducing subextensions. We describe here the attached Hopf algebra and Hopf action of an induced structure and we prove that they are obtained by tensoring the corresponding inducing objects. We give a general matrix description of the Hopf action which is useful to compute bases of associated orders. In case of an induced Hopf Galois structures it allows us to decompose the associated order, assuming that inducing subextensions are arithmetically disjoint.
2022-04-26T09:43:04Z
Gil Muñoz, Daniel
Río Doval, Ana
The regular subgroup determining an induced Hopf Galois structure for a Galois extension L/K is obtained as direct product of the corresponding regular groups of the inducing subextensions. We describe here the attached Hopf algebra and Hopf action of an induced structure and we prove that they are obtained by tensoring the corresponding inducing objects. We give a general matrix description of the Hopf action which is useful to compute bases of associated orders. In case of an induced Hopf Galois structures it allows us to decompose the associated order, assuming that inducing subextensions are arithmetically disjoint.
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An inverse Jacobian algorithm for Picard curves
http://hdl.handle.net/2117/366134
An inverse Jacobian algorithm for Picard curves
Lario Loyo, Joan Carles; Somoza Henares, Anna; Vincent, Christelle
We study the inverse Jacobian problem for the case of Picard curves over C. More precisely, we elaborate on an algorithm that, given a small period matrix O¿C3×3 corresponding to a principally polarized abelian threefold equipped with an automorphism of order 3, returns a Legendre–Rosenhain equation for a Picard curve with Jacobian isomorphic to the given abelian variety. Our method corrects a formula obtained by Koike–Weng (Math Comput 74(249):499–518, 2005) which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain equations of all the isomorphism classes of Picard curves with maximal complex multiplication by the maximal order of the sextic CM-fields with class number at most 4. In particular, we obtain the complete list of maximal CM Picard curves defined over Q. In the appendix, Vincent gives a correction to the generalization of Takase’s formula for the inverse Jacobian problem for hyperelliptic curves given in [Balakrishnan–Ionica–Lauter–Vincent, LMS J. Comput. Math., 19(suppl. A):283-300, 2016].
2022-04-20T14:58:45Z
Lario Loyo, Joan Carles
Somoza Henares, Anna
Vincent, Christelle
We study the inverse Jacobian problem for the case of Picard curves over C. More precisely, we elaborate on an algorithm that, given a small period matrix O¿C3×3 corresponding to a principally polarized abelian threefold equipped with an automorphism of order 3, returns a Legendre–Rosenhain equation for a Picard curve with Jacobian isomorphic to the given abelian variety. Our method corrects a formula obtained by Koike–Weng (Math Comput 74(249):499–518, 2005) which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain equations of all the isomorphism classes of Picard curves with maximal complex multiplication by the maximal order of the sextic CM-fields with class number at most 4. In particular, we obtain the complete list of maximal CM Picard curves defined over Q. In the appendix, Vincent gives a correction to the generalization of Takase’s formula for the inverse Jacobian problem for hyperelliptic curves given in [Balakrishnan–Ionica–Lauter–Vincent, LMS J. Comput. Math., 19(suppl. A):283-300, 2016].
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Satins, lattices, and extended Euclid's algorithm
http://hdl.handle.net/2117/365868
Satins, lattices, and extended Euclid's algorithm
Brunat Blay, Josep M.; Lario Loyo, Joan Carles
Motivated by the design of satins with draft of period m and step a, we draw our attention to the lattices L(m,a)=¿(1,a),(0,m)¿ where 1=a<m are integers with gcd(m,a)=1. We show that the extended Euclid's algorithm applied to m and a produces a shortest no null vector of L(m, a) and that the algorithm can be used to find an optimal basis of L(m, a). We also analyze square and symmetric satins. For square satins, the extended Euclid's algorithm produces directly the two vectors of an optimal basis. It is known that symmetric satins have either a rectangular or a rombal basis; rectangular basis are optimal, but rombal basis are not always optimal. In both cases, we give the optimal basis directly in terms of m and a.
2022-04-13T11:21:40Z
Brunat Blay, Josep M.
Lario Loyo, Joan Carles
Motivated by the design of satins with draft of period m and step a, we draw our attention to the lattices L(m,a)=¿(1,a),(0,m)¿ where 1=a<m are integers with gcd(m,a)=1. We show that the extended Euclid's algorithm applied to m and a produces a shortest no null vector of L(m, a) and that the algorithm can be used to find an optimal basis of L(m, a). We also analyze square and symmetric satins. For square satins, the extended Euclid's algorithm produces directly the two vectors of an optimal basis. It is known that symmetric satins have either a rectangular or a rombal basis; rectangular basis are optimal, but rombal basis are not always optimal. In both cases, we give the optimal basis directly in terms of m and a.
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Every integer can be written as a square plus a squarefree
http://hdl.handle.net/2117/365735
Every integer can be written as a square plus a squarefree
Jiménez Urroz, Jorge
In the paper we can prove that every integer can be written as the sum of two integers, one perfect square and one squarefree. We also establish the asymptotic formula for the number of representations of an integer in this form. The result is deeply related with the divisor function. In the course of our study we get an independent result about it. Concretely we are able to deduce a new upper bound for the divisor function fully explicit.
© 2022 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
2022-04-12T09:49:56Z
Jiménez Urroz, Jorge
In the paper we can prove that every integer can be written as the sum of two integers, one perfect square and one squarefree. We also establish the asymptotic formula for the number of representations of an integer in this form. The result is deeply related with the divisor function. In the course of our study we get an independent result about it. Concretely we are able to deduce a new upper bound for the divisor function fully explicit.
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A study of the separating property in Reed-Solomon codes by bounding the minimum distance
http://hdl.handle.net/2117/365730
A study of the separating property in Reed-Solomon codes by bounding the minimum distance
Fernández Muñoz, Marcel; Jiménez Urroz, Jorge
According to their strength, the tracing properties of a code can be categorized as frameproof, separating, IPP and TA. It is known that, if the minimum distance of the code is larger than a certain threshold then the TA property implies the rest. Silverberg et al. ask if there is some kind of tracing capability left when the minimum distance falls below the threshold. Under different assumptions, several papers have given a negative answer to the question. In this paper, further progress is made. We establish values of the minimum distance for which Reed-Solomon codes do not posses the separating property.
The version of record is available online at: http://dx.doi.org/10.1007/s10623-021-00988-z
2022-04-12T09:27:15Z
Fernández Muñoz, Marcel
Jiménez Urroz, Jorge
According to their strength, the tracing properties of a code can be categorized as frameproof, separating, IPP and TA. It is known that, if the minimum distance of the code is larger than a certain threshold then the TA property implies the rest. Silverberg et al. ask if there is some kind of tracing capability left when the minimum distance falls below the threshold. Under different assumptions, several papers have given a negative answer to the question. In this paper, further progress is made. We establish values of the minimum distance for which Reed-Solomon codes do not posses the separating property.
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Derived Beilinson-Flach elements and the arithmetic of the adjoint of a modular form
http://hdl.handle.net/2117/365445
Derived Beilinson-Flach elements and the arithmetic of the adjoint of a modular form
Rotger Cerdà, Víctor
Kings, Lei, Loeffler and Zerbes constructed in [LLZ], [KLZ1] a three-variable Euler system ¿(g,h) of Beilinson–Flach elements associated to a pair of Hida families (g,h) and exploited it to obtain applications to the arithmetic of elliptic curves, extending the earlier work [BDR]. The aim of this article is to show that this Euler system also encodes arithmetic information concerning the group of units of the associated number fields. The setting becomes specially novel and intriguing when g and h specialize in weight 1 to p-stabilizations of eigenforms such that one is dual to the other. We encounter an exceptional zero phenomenon which forces the specialization of ¿(g,h) to vanish and we are led to study the derivative of this class. The main result we obtain is the proof of the main conjecture of [DLR4] on iterated integrals and the main conjecture of [DR1] for Beilinson–Flach elements in the adjoint setting. The main point of this paper is that the methods of [DLR1], [DLR4] and [CH], where the above conjectures are proved when the weight 1 eigenforms have CM, do not apply to our setting and new ideas are required. In the previous works, a crucial ingredient is a factorization of p-adic L-functions, which in our scenario is not available due to the lack of critical points. Instead we resort to the principle of improved Euler systems and p-adic L-functions to reduce our problems to questions which can be resolved using Galois deformation theory. We expect this approach may be adapted to prove other cases of the Elliptic Stark Conjecture and of its generalizations that appear in the literature.
2022-04-06T12:25:06Z
Rotger Cerdà, Víctor
Kings, Lei, Loeffler and Zerbes constructed in [LLZ], [KLZ1] a three-variable Euler system ¿(g,h) of Beilinson–Flach elements associated to a pair of Hida families (g,h) and exploited it to obtain applications to the arithmetic of elliptic curves, extending the earlier work [BDR]. The aim of this article is to show that this Euler system also encodes arithmetic information concerning the group of units of the associated number fields. The setting becomes specially novel and intriguing when g and h specialize in weight 1 to p-stabilizations of eigenforms such that one is dual to the other. We encounter an exceptional zero phenomenon which forces the specialization of ¿(g,h) to vanish and we are led to study the derivative of this class. The main result we obtain is the proof of the main conjecture of [DLR4] on iterated integrals and the main conjecture of [DR1] for Beilinson–Flach elements in the adjoint setting. The main point of this paper is that the methods of [DLR1], [DLR4] and [CH], where the above conjectures are proved when the weight 1 eigenforms have CM, do not apply to our setting and new ideas are required. In the previous works, a crucial ingredient is a factorization of p-adic L-functions, which in our scenario is not available due to the lack of critical points. Instead we resort to the principle of improved Euler systems and p-adic L-functions to reduce our problems to questions which can be resolved using Galois deformation theory. We expect this approach may be adapted to prove other cases of the Elliptic Stark Conjecture and of its generalizations that appear in the literature.
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On the L-invariant of the adjoint of a weight one modular form
http://hdl.handle.net/2117/365339
On the L-invariant of the adjoint of a weight one modular form
Roset Julià, Martí; Rotger Cerdà, Víctor; Vatsal, Vinayak
The purpose of this article is proving the equality of two natural L -invariants attached to the adjoint representation of a weight one cusp form, each defined by purely analytic, respectively, algebraic means. The proof departs from Greenberg's definition of the algebraic L -invariant as a universal norm of a canonical Z¿ -extension of Q¿ associated to the representation. We relate it to a certain 2×2 regulator of ¿ -adic logarithms of global units by means of class field theory, which we then show to be equal to the analytic L -invariant computed in Rivero and Rotger [J. Eur. Math. Soc., to appear].
This is the accepted version of the following article: Roset, M.; Rotger, V.; Vatsal, V. On the L-invariant of the adjoint of a weight one modular form. "Journal of the London Mathematical Society", 1 Gener 2021, vol. 104, núm. 1, p. 206-232. , which has been published in final form at https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.12428. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited."
2022-04-05T11:30:30Z
Roset Julià, Martí
Rotger Cerdà, Víctor
Vatsal, Vinayak
The purpose of this article is proving the equality of two natural L -invariants attached to the adjoint representation of a weight one cusp form, each defined by purely analytic, respectively, algebraic means. The proof departs from Greenberg's definition of the algebraic L -invariant as a universal norm of a canonical Z¿ -extension of Q¿ associated to the representation. We relate it to a certain 2×2 regulator of ¿ -adic logarithms of global units by means of class field theory, which we then show to be equal to the analytic L -invariant computed in Rivero and Rotger [J. Eur. Math. Soc., to appear].
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Primes represented by quadratic polynomials via exceptional characters
http://hdl.handle.net/2117/361147
Primes represented by quadratic polynomials via exceptional characters
Chamizo Lorente, Fernando; Jiménez Urroz, Jorge
We estimate the number of primes represented by a general quadratic polynomial with discriminant ¿, assuming that the corresponding real character is exceptional.
This is a post-peer-review, pre-copyedit version of an article published in Archiv der Mathematik. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00013-021-01602-3
2022-02-01T07:29:13Z
Chamizo Lorente, Fernando
Jiménez Urroz, Jorge
We estimate the number of primes represented by a general quadratic polynomial with discriminant ¿, assuming that the corresponding real character is exceptional.
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Lazlo Lovász i Avi Widergson: premis Abel 2021
http://hdl.handle.net/2117/359891
Lazlo Lovász i Avi Widergson: premis Abel 2021
Rué Perna, Juan José; Díaz Cort, Josep; Guàrdia Rubies, Jordi; Atserias, Albert; Serra Albó, Oriol
El dia 17 de març l’Acadèmia de Ciències Noruega va anunciar que el Premi Abel 2021 s’atorgava a Laszló Lovász i Avi Widgerson per, segons es llegeix de la laudatio del premi, “. . . les seves contribucions fonamentals a la informàtica teòrica i les matemàtiques discretes, i el seu paper principal en la seva configuració en camps centrals de les matemàtiques modernes.
2022-01-18T10:52:27Z
Rué Perna, Juan José
Díaz Cort, Josep
Guàrdia Rubies, Jordi
Atserias, Albert
Serra Albó, Oriol
El dia 17 de març l’Acadèmia de Ciències Noruega va anunciar que el Premi Abel 2021 s’atorgava a Laszló Lovász i Avi Widgerson per, segons es llegeix de la laudatio del premi, “. . . les seves contribucions fonamentals a la informàtica teòrica i les matemàtiques discretes, i el seu paper principal en la seva configuració en camps centrals de les matemàtiques modernes.