TN - Teoria de Nombres
http://hdl.handle.net/2117/3728
2016-02-12T20:52:19ZFrobenius distribution for quotients of Fermat curves of prime exponent
http://hdl.handle.net/2117/81570
Frobenius distribution for quotients of Fermat curves of prime exponent
Fité, Francesc; González Rovira, Josep; Lario Loyo, Joan Carles
Let C denote the Fermat curve over Q of prime exponent l. The Jacobian Jac(C) of C splits over Q as the product of Jacobians Jac(C_k), 0< k < l-1, where C_k are curves obtained as quotients of C by certain subgroups of automorphisms of C. It is well known that Jac(C_k) is the power of an absolutely simple abelian variety B_k with complex multiplication. We call degenerate those pairs (l,k) for which B_k has degenerate CM type. For a non-degenerate pair (l,k), we compute the Sato-Tate group of Jac(C_k), prove the generalized Sato-Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of (l,k) being degenerate or not, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the l-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.
2016-01-15T19:45:28ZFité, FrancescGonzález Rovira, JosepLario Loyo, Joan CarlesLet C denote the Fermat curve over Q of prime exponent l. The Jacobian Jac(C) of C splits over Q as the product of Jacobians Jac(C_k), 0< k < l-1, where C_k are curves obtained as quotients of C by certain subgroups of automorphisms of C. It is well known that Jac(C_k) is the power of an absolutely simple abelian variety B_k with complex multiplication. We call degenerate those pairs (l,k) for which B_k has degenerate CM type. For a non-degenerate pair (l,k), we compute the Sato-Tate group of Jac(C_k), prove the generalized Sato-Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of (l,k) being degenerate or not, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the l-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.Modular forms with large coefficient fields via congruences
http://hdl.handle.net/2117/81139
Modular forms with large coefficient fields via congruences
Dieulefait, Luis; Jiménez Urroz, Jorge; Ribet, Keneth
We use the theory of congruences between modular forms to prove the existence of newforms with square-free level having a fixed number of prime factors such that the degree of their coefficient fields is arbitrarily large. We also prove a similar result for certain almost square-free levels.
2016-01-08T12:27:07ZDieulefait, LuisJiménez Urroz, JorgeRibet, KenethWe use the theory of congruences between modular forms to prove the existence of newforms with square-free level having a fixed number of prime factors such that the degree of their coefficient fields is arbitrarily large. We also prove a similar result for certain almost square-free levels.On the Galois correspondence theorem in separable Hopf Galois theory
http://hdl.handle.net/2117/80381
On the Galois correspondence theorem in separable Hopf Galois theory
Crespo Vicente, Teresa; Río Doval, Ana; Vela del Olmo, Mª Montserrat
In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that the image of the Galois correspondence does not determine the Hopf Galois structure.
2015-12-10T11:13:14ZCrespo Vicente, TeresaRío Doval, AnaVela del Olmo, Mª MontserratIn this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that the image of the Galois correspondence does not determine the Hopf Galois structure.From Galois to Hopf Galois: theory and practice
http://hdl.handle.net/2117/80188
From Galois to Hopf Galois: theory and practice
Crespo Vicente, Teresa; Río Doval, Ana; Vela del Olmo, Mª Montserrat
Hopf Galois theory expands the classical Galois theory by con- sidering the Galois property in terms of the action of the group algebra k [ G ] on K/k and then replacing it by the action of a Hopf algebra. We review the case of separable extensions where the Hopf Galois property admits a group-theoretical formulation suitable for counting and classifying, and also to perform explicit computations and explic it descriptions of all the ingredients involved in a Hopf Galois structure. At the end we give just a glimpse of how this theory is used in the context of Ga lois module theory for wildly ramified extensions
2015-12-04T12:08:29ZCrespo Vicente, TeresaRío Doval, AnaVela del Olmo, Mª MontserratHopf Galois theory expands the classical Galois theory by con- sidering the Galois property in terms of the action of the group algebra k [ G ] on K/k and then replacing it by the action of a Hopf algebra. We review the case of separable extensions where the Hopf Galois property admits a group-theoretical formulation suitable for counting and classifying, and also to perform explicit computations and explic it descriptions of all the ingredients involved in a Hopf Galois structure. At the end we give just a glimpse of how this theory is used in the context of Ga lois module theory for wildly ramified extensionsThe Hopf Galois property in subfield lattices
http://hdl.handle.net/2117/80143
The Hopf Galois property in subfield lattices
Crespo Vicente, Teresa; Río Doval, Ana; Vela del Olmo, Mª Montserrat
Let K/k be a finite separable extension, n its degree and (K) over tilde /k its Galois closure. For n <= 5, Greither and Pareigis show that all Hopf Galois extensions are either Galois or almost classically Galois and they determine the Hopf Galois character of K/ k according to the Galois group (or the degree) of (K) over tilde /k. In this paper we study the case n = 6, and intermediate extensions F/ k such that K subset of F subset of (K) over tilde, for degrees n = 4, 5, 6. We present an example of a non almost classically Galois Hopf Galois extension of (sic) of the smallest possible degree and new examples of Hopf Galois extensions. In the last section we prove a transitivity property of the Hopf Galois condition.
2015-12-03T09:52:31ZCrespo Vicente, TeresaRío Doval, AnaVela del Olmo, Mª MontserratLet K/k be a finite separable extension, n its degree and (K) over tilde /k its Galois closure. For n <= 5, Greither and Pareigis show that all Hopf Galois extensions are either Galois or almost classically Galois and they determine the Hopf Galois character of K/ k according to the Galois group (or the degree) of (K) over tilde /k. In this paper we study the case n = 6, and intermediate extensions F/ k such that K subset of F subset of (K) over tilde, for degrees n = 4, 5, 6. We present an example of a non almost classically Galois Hopf Galois extension of (sic) of the smallest possible degree and new examples of Hopf Galois extensions. In the last section we prove a transitivity property of the Hopf Galois condition.Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields
http://hdl.handle.net/2117/78762
Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields
Darmon, Henri; Lauder, Alan; Rotger Cerdà, Víctor
This article examines the Fourier expansions of certain non-classical p-adic modular forms of weight one: the eponymous generalised eigertforms of the title, so called because they lie in a generalised eigenspace for the Hecke operators. When this generalised eigenspace contains the theta series attached to a character of a real quadratic field K in which the prime p splits, the coefficients of the attendant generalised eigenform are expressed as p-adic logarithms of algebraic numbers belonging to an idoneous ring class field of K. This suggests an approach to
2015-11-04T11:14:19ZDarmon, HenriLauder, AlanRotger Cerdà, VíctorThis article examines the Fourier expansions of certain non-classical p-adic modular forms of weight one: the eponymous generalised eigertforms of the title, so called because they lie in a generalised eigenspace for the Hecke operators. When this generalised eigenspace contains the theta series attached to a character of a real quadratic field K in which the prime p splits, the coefficients of the attendant generalised eigenform are expressed as p-adic logarithms of algebraic numbers belonging to an idoneous ring class field of K. This suggests an approach toAlgorithms for chow-heegner points via iterated integrals
http://hdl.handle.net/2117/78758
Algorithms for chow-heegner points via iterated integrals
Darmon, Henri; Daub, Michael; Lichtenstein, Sam; Rotger Cerdà, Víctor
Let E/Q be an elliptic curve of conductor N and let f be the weight 2 newform on G0(N) associated to it by modularity. Building on an idea of S. Zhang, an article by Darmon, Rotger, and Sols describes the construction of so-called Chow-Heegner points, PT,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 PT,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.
2015-11-04T10:59:38ZDarmon, HenriDaub, MichaelLichtenstein, SamRotger Cerdà, VíctorLet E/Q be an elliptic curve of conductor N and let f be the weight 2 newform on G0(N) associated to it by modularity. Building on an idea of S. Zhang, an article by Darmon, Rotger, and Sols describes the construction of so-called Chow-Heegner points, PT,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 PT,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.Non-isomorphic Hopf Galois structures with isomorphic underlying Hopf algebras
http://hdl.handle.net/2117/28490
Non-isomorphic Hopf Galois structures with isomorphic underlying Hopf algebras
Crespo, Teresa; Río Doval, Ana; Vela del Olmo, Mª Montserrat
We give a degree 8 non-normal separable extension having two non-isomorphic Hopf Galois structures with isomorphic underlying Hopf algebras.
2015-07-01T11:16:43ZCrespo, TeresaRío Doval, AnaVela del Olmo, Mª MontserratWe give a degree 8 non-normal separable extension having two non-isomorphic Hopf Galois structures with isomorphic underlying Hopf algebras.Genetics of polynomials over local fields
http://hdl.handle.net/2117/28206
Genetics of polynomials over local fields
Guàrdia Rubies, Jordi; Nart Vinyals, Enric
Let (K, v) be a discrete valued field with valuation ring O, and let Ov be the completion of O with respect to the v-adic topology. In this paper we discuss the advantages of manipulating polynomials in Ov[x] on a computer by means of OM representations of prime (monic and irreducible) polynomials. An OM representation supports discrete data characterizing the Okutsu equivalence class of the prime polynomial. These discrete parameters are a kind of DNA sequence common to all individuals in the same Okutsu
class, and they contain relevant arithmetic information about the polynomial
and the extension of Kv that it determines
2015-06-05T15:29:48ZGuàrdia Rubies, JordiNart Vinyals, EnricLet (K, v) be a discrete valued field with valuation ring O, and let Ov be the completion of O with respect to the v-adic topology. In this paper we discuss the advantages of manipulating polynomials in Ov[x] on a computer by means of OM representations of prime (monic and irreducible) polynomials. An OM representation supports discrete data characterizing the Okutsu equivalence class of the prime polynomial. These discrete parameters are a kind of DNA sequence common to all individuals in the same Okutsu
class, and they contain relevant arithmetic information about the polynomial
and the extension of Kv that it determinesResidual ideals of MacLane valuations
http://hdl.handle.net/2117/28203
Residual ideals of MacLane valuations
Fernández González, Julio; Guàrdia Rubies, Jordi; Montes Peral, Jesús; Nart Vinyals, Enric
Let K be a field equipped with a discrete valuation v. In a pioneering work,
MacLane determined all valuations on K(x) extending v. His work was recently reviewed
and generalized by Vaqui´e, by using the graded algebra of a valuation. We extend Vaqui´e’s approach by studying residual ideals of the graded algebra as an abstract counterpart of certain residual polynomials which play a key role in the computational applications of
the theory. As a consequence, we determine the structure of the graded algebra of the
discrete valuations on K(x) and we show how these valuations may be used to parameterize
irreducible polynomials over local fields up to Okutsu equivalence
2015-06-05T14:57:13ZFernández González, JulioGuàrdia Rubies, JordiMontes Peral, JesúsNart Vinyals, EnricLet K be a field equipped with a discrete valuation v. In a pioneering work,
MacLane determined all valuations on K(x) extending v. His work was recently reviewed
and generalized by Vaqui´e, by using the graded algebra of a valuation. We extend Vaqui´e’s approach by studying residual ideals of the graded algebra as an abstract counterpart of certain residual polynomials which play a key role in the computational applications of
the theory. As a consequence, we determine the structure of the graded algebra of the
discrete valuations on K(x) and we show how these valuations may be used to parameterize
irreducible polynomials over local fields up to Okutsu equivalence