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By using the relations between supersingular elliptic curves defined over a finite field of characteristic p > 3 and the p-th division polynomial, we present a property about the reduction modulo a prime p > 3 of the p-th division polynomial based on a heuristic argument and numerical evidence. We prove this property determining the p-th division polynomial of supersingular elliptic curves. As a consequence of this result, we present a criterion to discard supersingular elliptic curves.

We consider the binomial random set model $[n]_p$ where each element in $\{1,\dots,n\}$ is chosen independently with probability $p:=p(n)$. We show that for essentially all regimes of $p$ and very general conditions for a matrix $A$ and a column vector $\mathbf{b}$, the count of specific integer solutions to the system of linear equations $A\mathbf{x} = \mathbf{b}$ with the entries of $\mathbf{x}$ in $[n]_p$ follows a (conveniently rescaled) normal limiting distribution. This applies among others to the number of solutions with every variable having a different value, as well as to a broader class of so-called non-trivial solutions in homogeneous strictly balanced systems. Our proof relies on the delicate linear algebraic study both of the subjacent matrices and the corresponding ranks of certain submatrices, together with the application of the method of moments in probability theory.

El lema de Ribet descriu una forma de construir classes de cohomologia a partir de representacions de grups. La primera versió del lema va ser enunciada per Ken Ribet com part de la seva demostració del recíproc del teorema de Herbrand. El seu plantejament, conegut com el mètode de Ribet, es basa en reformular el problema en termes de l'existència d'una certa classe de cohomologia i construir aquesta clase a partir de la representació de Galois associada a una forma modular adient, fent servir el lema de Ribet. Des de la seva aparició, s'han emprat múltiples modificacions d'aquest mètode en la demostració de conjectures tan cèlebres com ara la conjectura principal d'Iwasawa o la conjectura de Gross-Stark, generalitzant degudament el lema de Ribet. La majoria d'aquestes generalitzacions assumeixen que la representació residual factoritza en caràcters no isomorfs, el que anomenem la hipòtesi de distingibilitat residual. No obstant, en el seu recent estudi de la conjectura de Brumer-Stark, Dasgupta i Kakde s'han trobat amb representations que no satisfan aquesta hipòtesi en el cas de característica 2, cosa que ha motivat l'estudi del cas indistingible del lema. L'objectiu d'aquest treball és descriure les limitacions d'aquest cas, donar una idea d'una possible versió general del lema de Ribet, i demostrar-la en alguns casos particulars. El lema de Ribet describe una forma de construir clases de cohomología a partir de representaciones de grupos. La primera versión de este lema fue enunciada por Ken Ribet como parte de su demostración del recíproco del teorema de Herbrand. Su enfoque, comúnmente conocido como el método de Ribet, se basó en reformular el problema en términos de la existencia de una cierta clase de cohomología y construir dicha clase a partir de la representación de Galois asociada a una forma modular apropiada, usando el lema de Ribet. Desde entonces, se han utilizado numerosas modificaciones de su método en la demostración de conjeturas tan célebres como la conjetura principal de Iwasawa o la conjetura de Gross-Stark, generalizando debidamente el lema de Ribet. La mayoría de estas generalizaciones asumen que la representación residual factoriza en carácteres no isomorfos, lo que llamamos la hipótesis de distinguibilidad residual. Sin embargo, en su reciente estudio de la conjetura de Brumer-Stark, Dasgupta y Kakde se encontraron con representationes que no cumplen esta hipótesis en el caso de característica 2, lo que motiva el estudio del caso indistinguible del lema. El objetivo de este trabajo es describir las limitaciones de este caso, dar una idea de una posible versión general del lema de Ribet, y demostrarla en algunos casos particulares. Ribet's lemma describes a way to construct cohomology classes from group representations. The first version of this lemma was stated by Ken Ribet as part of his proof of the converse of Herbrand's theorem. His approach, commonly referred to as Ribet's method, was to reformulate the problem in terms of the existence of a certain cohomology class, and to construct such a class from the Galois representation attached to an appropriate modular form, using Ribet's lemma. Ever since then, modifications of his method have been used in the proof of celebrated conjectures such as the Iwasawa Main Conjecture or the Gross-Stark conjecture, properly generalizing Ribet's lemma. Most of these generalizations assumed that the residual representation factors in characters that are non-isomorphic, what we call the residual distinguishability assumption. However, in their recent work on the Brumer-Stark conjecture, Dasgupta and Kakde dealt with representations that do not satisfy this assumption in the case of characteristic 2, which motivates the study of the indistinguishable case of the lemma. The purpose of this work is to describe the limitations of this case, give an idea of a possible general version of Ribet's lemma, and prove it in some particular cases. Outgoing

p-adic L-functions are variants of the classical L-functions, with a p-adic domain instead of the complex numbers. There are 2 ways to construct p-adic L-functions. The first one is purely analytic, by interpolation of the special values of the L-function. The second way uses Iwasawa theory and Fontaine's theory of (ϕ, Γ)-modules, and is closely connected with Euler systems. Both constructions can be related using an explicit reciprocity law. We study these constructions in two particular cases: That of the Kubota-Leopoldt zeta function (the p-adic analogue of the Riemann's zeta function), and the case of L-functions attached to modular forms.

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. Both authors were supported by Grant MTM2015-63829-P. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 682152). The first author has also received financial support through “la Caixa” Fellowship Grant for Doctoral Studies (grant LCF/BQ/ES17/ 11600010). The second author gratefully acknowledges Icrea for financial support through an Icrea Academia award.

Aquest treball tracta de relacionar les extensions monògenes amb la teoria de corbes el·líptiques. Determinar si una extensió K/Q de grau n és monògena o no és equivalent a resoldre l'equació diofantina |I(X_1, \dots, X_{n - 1})| = 1, on I(X_1, \dots, X_{n - 1}) és la forma índex del cos. En el cas cúbic, aquesta equació defineix una corba llisa a P^2(Q) donada per C_K: Z^3 = I(X,Y). A aquesta corba li podem associar una corba el·líptica E definida sobre Q, a partir de la qual podem donar condicions suficients per a existència o inexistència de punts racionals a C_K. Durant aquest treball, veurem tots aquests conceptes i acabarem aplicant-los per a donar condicions suficients per a determinar monogeneïtat o no monogeneïtat.

En aquesta tesi introduïm la teoria de cossos de classes i en presentem els teoremes principals: el Teorema de Reciprocitat d’Artin, el Teorema del Conductor, i el Teorema d’Existència. Aplicar el Teorema d’Existència al grup de classes d’ideals en un cos de nombres K porta a la definició del cos de classes radial, el cos de classes ”d’anells” i el cos de classes de Hilbert de K. Presentem el problema de la teoria de classes de cossos explícita, que consisteix a trobar la forma explícita dels cossos de classes mencionats per un cos de nombres K donat. Desenvolupem els únics dos casos pels quals es coneix la solució del problema en aquests moments, el cos dels racionals i els cossos quadràtics imaginaris. Fem servir el teorema de Kronecker-Weber per resoldre el cas dels racionals, i introduïm la teoria de multiplicació complexa per resoldre el cas dels cossos quadràtics imaginaris. Al final, introduïm la teoria de formes quadràtiques binàries i trobem un isomorfisme entre el grup de classes de formes de discriminant D i el grup de classes d’ideals per un ordre de discriminant D. També presentem la teoria de ”genus” de les formes quadràtiques per intentar caracteritzar els primers p de la forma x2 +ny2 per un enter positiu n donat. En esta tesis introduciremos la teoría de cuerpos de clases y presentaremos sus principales teoremas: el Teorema de Reciprocidad de Artin, el Teorema del Conductor, i el Teorema de Existencia. Aplicar el Teorema de Existencia al grupo de clases de ideales en un cuerpo numérico K lleva a la definición del cuerpo de clases radial, el cuerpo de clases de ”anillos” y el cuerpo de clases de Hilbert de K. Presentamos el problema de la teoría de clases de cuerpos explícita, que consiste en encontrar la forma explícita de los cuerpos de clases mencionados para un cuerpo numérico K dado. Desarrollamos los dos únicos casos por los cuales se conoce la solución del problema a día de hoy, el cuerpo de los racionales y los cuerpos cuadráticos imaginarios. Usamos el teorema de Kronecker-Weber para resolver el caso de los racionales, e introducimos la teoría de multiplicación compleja para resolver el caso de los cuerpos cuadráticos imaginarios. Al final, introducimos la teoría de formas cuadráticas binarias y encontramos un isomorfismo entre el grupo de clases de formas de discriminante D y el grupo de clases de ideales para un orden de discriminante D. También presentamos la teoría de ”genus” de las formas cuadráticas para intentar caracterizar los números primos p de la forma x2+ny2 para un entero positivo n dado. In this thesis we introduce class field theory and we present its main theorems: the Artin Reciprocity Theorem, the Conductor Theorem, and the Existence Theorem. Applying the Existence Theorem to the ideal class group in a number field K leads to the definition of the ray class field, the ring class field and the Hilbert class field of K. We present the problem of explicit class field theory, which consists in explicitly finding the mentioned fields for a given number field K. We develop the only two known cases solved so far, the rational numbers and the imaginary quadratic fields. We use the Kronecker-Weber theorem to solve the case of the rational numbers, and we introduce complex multiplication to solve the case of the imaginary quadratic number fields. At the end, we introduce the theory of quadratic forms and find an isomorphism between the form class group of discriminant D and the ideal class group for an order of discriminant D. We also present the genus theory of quadratic forms to try to characterize the primes p of the form x^2 + ny^2 for a given positive integer n. Outgoing

Let A and B be randomly chosen s-subsets of the first n integers such that their sumset \(A+B\) has size at most Ks. We show that asymptotically almost surely A and B are almost fully contained in arithmetic progressions \(P_A\) and \(P_B\) with the same common difference and cardinalities approximately Ks/2. The result holds for \(s = \omega (\log ^3 n)\) and \(2 \le K = o(s/\log ^3 n)\). Our main tool is an asymmetric version of the method of hypergraph containers which was recently used by Campos to prove the result in the special case \(A=B\).

La mitjana aritmètico-geomètrica de Gauss (MAG) neix com a eina per calcular numèricament integrals el·líptiques. Aquesta successió apareix per primer cop en notes de Lagrange, però va ser Gauss qui va descobrir realment la profunditat d'aquesta matèria. L'objectiu d'aquest treball de fi de grau és estudiar amb detall aquesta successió, especialment en el cas complex. Per fer-ho, introduirem des del principi la teoria bàsica de les funcions theta, les funcions theta de Jacobi i les formes modulars. En l escenari complex, l MAG esdevé una funció de múltiples valors i, per tal de determinar la relació entre tots els seus valors, "uniformitzarem" la MAG mitjançant quocients de les funcions theta de Jacobi, que són funcions modulars per a certs subgrups de congruència de nivell 4 per al grup modular SL2(Z).

© 2021 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/ We construct p-adic L-functions associated with triples of finite slope p-adic families of quaternionic automorphic eigenforms over totally real fields on Shimura curves. These results generalize a previous construction, joint work with D.Barrera, performed in the ordinary setting.

The goal of this work is to develop a computational method able to optimize the design process of a car structure and provide a tool which can support designers during the decision-making phase. The design of a car body-in-white (BIW) structure is the process which goes from the initial idea to the final approved model. During this phase, which represents the most time-consuming part of the whole development process, designers have to deal with very complex parametric problems where material and geometric characteristics of the car components are the unknown. Any change in these parameters might significantly affect the global behaviour of the car. A target which is very sensitive to small variations of the parameters is the noise and vibration response of the vehicle (NVH test), which strictly depends on the global static and dynamic stiffness. In order to find the optimal solution, a lot of configurations exploring all the possible parametric combinations need to be tested. Standard numerical methods are computationally very expensive when applied to this kind of multidimensional problems. An alternative is represented by reduced order models (ROM), which are based on the idea that the essential behaviour of complex systems can be accurately described by simplified low-order models. In this work, the encapsulated proper generalized decomposition (Encapsulated-PGD) toolbox, based on the PGD Least-Squares approximation [3] is proposed. As a main advantage, this ROM technique requires only one offline computation. The latter provides a separable solution which depends explicitly on an a-priori unknown number of parametric and mechanic modes or snapshots. Then, during an online stage, the solution can be particularized in realtime for any set of the parameters. In a previous work [4], a coupling of the PGD method with the Inertia Relief technique was implemented in order to perform the parametric static analysis of an unconstrained structures. A novel algebraic approach allowed to incorporate both material and complex geometric parameters and to perform shape optimization. Here, the method is extended to the case of a parametric generalized eigenvalue problem, in order to identify how a variation of user-defined parameters affects the dynamic response of the structure in terms of dominant eigenmodes and related natural frequencies. Moreover, thanks to the nonintrusive format of the toolbox, an interaction with commercial software is possible, which makes it particularly interesting for real industrial applications.

© 2021 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/ 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.

The regular subgroup determining an induced Hopf Galois structure for a Galois extension $L/K$ is obtained as the direct product of the corresponding regular groups of the inducing subextensions. We describe here the associated Hopf algebra and Hopf action of an induced structure and we prove that they are obtained by tensoring the corresponding inducing objects. In order to deal with their associated orders we develop a general method to compute bases and free generators in terms of matrices coming from representation theory of Hopf modules. In the case of an induced Hopf Galois structure it allows us to decompose the associated order, assuming that inducing subextensions are arithmetically disjoint.

We prove that, for all [Formula: see text] and for all invertible residue classes [Formula: see text] modulo [Formula: see text], there exists a natural number [Formula: see text] that is congruent to [Formula: see text] modulo [Formula: see text] and that is the product of exactly three primes, all of which are below [Formula: see text]. The proof is further supplemented with a self-contained proof of the special case of the Kneser Theorem we use.

Preprint sotmès a publicació. We prove a new (conditional) result towards the subconvexity problem for certain automorphic L-functions for GL2xGL3. This follows from the computation of new SL2-period integrals associated with newforms f and g of even weight and odd squarefree level. The same computations lead to a central value formula for degree 6 comples L-series of the form L(f x Ad(g),s), extending previous work.

This paper gives a sketch of proof of Mazur's Theorem classifying the possible rational torsion subgroups of elliptic curves defined over rational numbers. We will prove Mazur's theorem by using two main lemmas. The sketch of proof of these two lemmas will be the goal of all the work. We will provide the tools needed to understand the proofs, trying to make the work as self-contained as possible although we must avoid some technical parts. We will present the basic notions of the theory of elliptic curves, discuss about Weierstrass equations, the Mordell-Weil group, isogenies, endomorphisms, the basics of the theory of group schemes, Néron models for elliptic curves, and modular curves.

In this work we shall relate the factorization of polynomials modulo p with the splitting of primes in number fields, and we shall study in which cases the different possibilities of factorization or splitting can be explained by the coefficients of the q-expansion of a certain modular form.

We study the special values of the triple product $p$-adic $L$-function constructed by Darmon and Rotger at all classical points outside the region of interpolation. We propose conjectural formulas for these values that can be seen as extending the Elliptic Stark Conjecture, and we provide theoretical evidence for them by proving some particular cases., Comment: Corrected some typographical errors and some references

This is a post-peer-review, pre-copyedit version of an article published in Journal of scientific computing. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10915-019-00917-2 Tensor representations allow compact storage and efficient manipulation of multi-dimensional data. Based on these, tensor methods build low-rank subspaces for the solution of multi-dimensional and multi-parametric models. However, tensor methods cannot always be implemented efficiently, specially when dealing with non-linear models. In this paper, we discuss the importance of achieving a tensor representation of the model itself for the efficiency of tensor-based algorithms. We investigate the adequacy of interpolation rather than projection-based approaches as a means to enforce such tensor representation, and propose the use of cross approximations for models in moderate dimension. Finally, linearization of tensor problems is analyzed and several strategies for the tensor subspace construction are proposed.

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.

The well--known Freiman--Ruzsa Theorem provides a structural description of a set $A$ of integers with $|2A|\le c|A|$ as a subset of a $d$--dimensional arithmetic progression $P$ with $|P|\le c'|A|$, where $d$ and $c'$ depend only on $c$. The estimation of the constants $d$ and $c'$ involved in the statement has been the object of intense research. Freiman conjectured in 2008 a formula for the largest volume of such a set. In this paper we prove the conjecture for a general class of sets called chains.

In this article, we give a group cohomological interpretation to the Eichler–Shimura isomorphism. For any quaternion algebra A over a totally real field with multiplicative group G , we interpret a weight ( k 1 , k 2 , ⋯ , k d ) -automorphic form of G as a G ( F ) -invariant homomorphism of ( G ∞ , K ∞ ) -modules. Then the Eichler–Shimura isomorphism is given by the connection morphism provided by the natural exact sequences defining the ( G ∞ , K ∞ ) -module of discrete series of weight ( k 1 , k 2 , ⋯ , k d ) .

We construct a projective Galois representation attached to an abelian L-surface with quaternionic multiplication, describing the Galois action on its Tate module. We prove that such representation characterizes the Galois action on the isogeny class of the abelian L-surface, seen as a set of points of certain Shimura curves.

In the main part of this paper we present polynomial expressions for the cardinalities of some sets of interest of the nice distance-layer structure of the well-known De Bruijn and Kautz digraphs. More precisely, given a vertex v, let S i ⋆ (v) be the set of vertices at distance i from v. We show that | S i ⋆ ( v ) | = d i − a i − 1 d i − 1 − ⋯ − a 1 d − a 0 , where d is the degree of the digraph and the coefficients a k ∈ { 0 , 1 } are explicitly calculated. Analogously, let w be a vertex adjacent from v such that S i ⋆ ( v ) ∩ S j ⁎ ( w ) ≠ ∅ for some j. We prove that | S i ⋆ ( v ) ∩ S j ⁎ ( w ) | = d i − b i − 1 d i − 1 − … − b 1 d − b 0 , where the coefficients b t ∈ { 0 , 1 } are determined from the coefficients a k of the polynomial expression of | S i ⋆ ( v ) | . An application to deflection routing in De Bruijn and Kautz networks serves as motivation for our study. It is worth-mentioning that our analysis can be extended to other families of digraphs on alphabet or to general iterated line digraphs.

We give archimedean and non-archimedean constructions of Darmon points on modular abelian varieties attached to automorphic forms over arbitrary number fields and possibly non-trivial central character. An effort is made to present a unifying point of view, emphasizing the automorphic nature of the construction., Comment: Final version. Corrected typos. Accepted for publication in Indiana University Mathematics Journal

Dissertation about the Gross-Zagier formulae and Shimura varieties.

For a set $A$ of points in the plane, not all collinear, we denote by ${\rm tr}(A)$ the number of triangles in any triangulation of $A$; that is, ${\rm tr}(A) = 2i+b-2$ where $b$ and $i$ are the numbers of points of $A$ in the boundary and the interior of $[A]$ (we use $[A]$ to denote "convex hull of $A$"). We conjecture the following analogue of the Brunn-Minkowski inequality: for any two point sets $A,B \subset {\mathbb R}^2$ one has \[ {\rm tr}(A+B)^{\frac12}\geq {\rm tr}(A)^{\frac12}+{\rm tr}(B)^{\frac12}. \] We prove this conjecture in several cases: if $[A]=[B]$, if $B=A\cup\{b\}$, if $|B|=3$, or if none of $A$ or $B$ has interior points., Comment: 30 pages

Article number 20; International audience; Structural Health Monitoring is of major interest in many areas of structural mechanics. This paper presents a new approach based on the combination of dimensionality reduction and data-mining techniques able to differentiate damaged and undamaged regions in a given structure. Indeed, existence, severity (size) and location of damage can be efficiently estimated from collected data at some locations from which the fields of interest are completed before the analysis based on machine learning and dimensionality reduction techniques proceed.

Preprint sotmès a publicació. In this note we give a detailed construction of a Lambda-adic d-th Shintani lifting. We derive a p-adic version of Kohnen's formula relating Fourier coefficients of half-integral weight modular forms and special values of twisted L-series. As a by-product we obtain a mild generalization of such classical formula.

OD matrices that describe mobility patterns provide major input to most transport analysis models. Since OD matrices are not yet directly observable, they are usually estimated indirectly. Data from new sources such as mobile phone records and GPS traces from mobile apps are emerging alternatives that allow for cheaper and timely estimates. However, they are still hindered by weaknesses that must be studied for use as input to transportation models. This paper presents a case study using mobility data from mobile phone records, with the goal of establishing a methodology for validating the obtained OD matrices and generating the appropriate input for traffic assignment models. Spatial and temporal consistency of OD data elaborated from mobile records has been proved to be useful for transportation modelling needs.

The main goal of this thesis is the study of elliptic curves over finite fields and the number of points of them. A study of interest is to analyze and find the cases when the difference #E(F_q^n )-#E(F_q) vanishes, where #E(F_q) denotes the number of points of an elliptic curve E over the finite field F_q. Moreover we can show that the sequence a_n=#E(F_q^n ) - #E(F_q)=q^n-q for n odd, q prime of the form q=4k+3 where k is a nonnegative integer and any elliptic curve of the form E : y^2=x^3+tx over F_q. Also a_n=#E(F_q^n ) - #E(F_q)=q^n-q for q prime of the form 3k+2 where k is a positive integer, n odd and any elliptic curve of the form E: y^2=x^3+b over F_q.

A Theorem of Hou, Leung and Xiang generalised Kneser's addition Theorem to field extensions. This theorem was known to be valid only in separable extensions, and it was a conjecture of Hou that it should be valid for all extensions. We give an alternative proof of the theorem that also holds in the non-separable case, thus solving Hou's conjecture. This result is a consequence of a strengthening of Hou et al.'s theorem that is a transposition to extension fields of an addition theorem of Balandraud., 17 pages

In this thesis we examine the proof of a theorem due to Scholz and Reichardt in 1937. It states that any odd p-group occurs as a Galois group over the rationals.

We study the problem of rotating a simple polygon to contain the maximum number of elements from a given point set in the plane. We consider variations of this problem where the rotation center is a given point or lies on a line segment, a line, or a polygonal chain. We also solve an extension to 3D where we rotate a polyhedron around a given point to contain the maximum number of elements from a set of points in the space., 25 pages, 12 figures, submitted to journal in February 2017

Aquest treball és una introducció a la teoria clàssica d'Iwasawa. La primera part del treball està dedicada a demostrar un teorema d'estructura pels mòduls sobre l'anell de sèries de potències amb coeficients enters p-àdics i a demostrar el teorema de control d'Iwasawa, que estableix el comportament del grup de classes en torres d'extensions de cossos ciclotòmics. A la segona part del treball, enunciem i donem una estratègia de prova per la conjectura principal de la teoria d'Iwasawa, que prediu la connexió entre les Zp-extensions de cossos i les funcions L p-àdiques. Aquesta prova es basa en tres resultats importants, dels quals en demostrem dos, incloent el teorema que estableix l'existència d'un anàleg p-àdic a la funció zeta de Riemann.

In this degree thesis, we present some of the theory of integer binary quadratic forms, namely the classification by discriminant, proper equivalence relation, the composition laws and their connection to the ideal class groups. We will also see how quadratic forms are intrinsically connected to some other areas of mathematics, such as the continued fraction of quadratic numbers, which encodes the transformations between neighbouring half-reduced forms in a cycle, or such as the group of units in a quadratic field, which relates to the study of automorphisms of primitive forms, which in turn represent the solutions of Pell's equation $x^2-Dy^2=\pm4$.

The final publication is available at Springer via http://dx.doi.org/10.1007/s11139-016-9824-y 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 p-adic avatar of the leading term at s=1 of the Hasse–Weil–Artin L-series L(E,¿1¿¿2,s) of an elliptic curve E/Q twisted by the tensor product ¿1¿¿2 of two odd 2-dimensional Artin representations, when the order of vanishing is two. The main ingredient of this formula is a 2×2 p-adic regulator involving the p-adic 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 p-adic L-function 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.

This master s thesis is intended to give a presentation of the theory of congruences between the Fourier coecients of modular forms. In order to do that we introduce the reader to the basic theory of modular forms from the beginning and we study the structure of their Fourier coecients in di↵erent ways using Hecke operators. Then we start the theory of congruences finding some of them by classical methods of Number Theory. After that, we introduce the advances made by Swinnerton-Dyer in the study of congruences using l-adic representations and the generalisation by Ken Ono. Finally, we explain the papers by Hida and Ribet in two chapters giving some conditions for the existence of congruences using the associated L-functions and decomposing the space of modular forms.

In this project we explore the connections between elliptic curves, modular curves and complex multiplication (CM). The main theorem of CM shows that the theory of CM for elliptic curves provides an explicit construction of finite abelian extensions of a quadratic imaginary field. The proof we discuss uses many of the properties of the classical modular curve, which is introduced both as a geometrical object and as a moduli space. This theory together with the Modularity theorem is used in the construction of Heegner points, which are in the heart of the proof of Kolyvagin's theorem, a result closely related with the Birch and Swinnerton-Dyer conjecture.

L'objectiu d'aquest treball és presentar els resultats de la teoria de cossos de classe de manera moderna, motivant la aparició d'aquests resultats i la seva centralitat dins la teoria algebraica de nombres. El treball està estructurat a través d'una introducció històrica inicial, dos capítols per clarificar tots els resultats previs necessaris (amb l'aparició dels cossos local), un capítol per introduir l'eina de la cohomologia, i els dos capítols centrals que contenen els resultats i les proves dels mateixos tant per a cossos locals com globals. També he afegit un capítol final on comento diverses aplicacions o camps d'estudi relacionats amb la teoria, amb la finalitat de fer palès que aquesta teoria obre diverses vies d'investigació matemàtica.

[EN] Domain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic problems with complex geometries. The classical difficulties associated to the combination of layered domains with arbitrarily oriented midsurfaces, which may require in-plane-out-of-plane techniques, are now dismissed. More generally, solutions on large domains can now be confronted within a domain decomposition approach. This is done with a reduced cost in the offline phase because the proper generalized decomposition gives an explicit description of the solution in each subdomain in terms of the solution at the interface. Thus, the evaluation of the approximation in each subdomain is a simple function evaluation given the interface values (and the other problem parameters). The interface solution can be characterized by any a priori user-defined approximation. Here, for illustration purposes, hierarchical polynomials are used. The repetitiveness of the subdomains is exploited to reduce drastically the offline computational effort. The online phase requires solving a nonlinear problem to determine all the interface solutions. However, this problem only has degrees of freedom on the interfaces and the Jacobian matrix is explicitly determined. Obviously, other parameters characterizing the solution (material constants, external loads, and geometry) can also be incorporated in the explicit description of the solution., European Commission, Grant/Award Number: MSCA ITN-ETN 675919; ESI group, Grant/Award Number: ENSAM ESI Chair; Spanish Ministry of Economy and Competitiveness, Grant/Award Number: DPI2017-85139-C2-2-R; Generalitat de Catalunya, Grant/Award Number: 2014SGR1471

L'objectiu d'aquest treball és desenvolupar la teoria dels anells d'enters algebraics demostrant dos resultats fonamentals d'aquesta: el teorema de Dirichlet de les unitats i el teorema de la finitud del grup de classes. Com a idea general podem entendre l'anell d'enters d'un cos de nombres com una generalització dels enters racionals dins aquest cos. Al llarg d'aquest treball intentarem donar resposta a qüestions relacionades amb la divisibilitat d'aquests anells arribant finalment a demostrar el teorema de Dirichlet de les unitats que ens dóna informació sobre quina és l'estructura del grup de les unitats d'aquests anells i el teorema de la finitud del grup de classes el qual és transcendental a l'hora de discutir la factorització única en elements irreductibles dins aquests anells.

El objetivo de este trabajo es el estudio de un análogo de la conjetura elíptica de Stark para unidades en cuerpos de números. Para ello, utilizamos el sistema de Euler de elementos de Beilinson-Flach y la posibilidad de interpolar p-ádicamente sus imágenes a través de reguladores étale. Los resultados de Kings, Loeffler y Zerbes nos permiten relacionar las clases de cohomología así construidas con ciertos valores de una función L p-ádica de Hida-Rankin de tres variables, donde también tenemos el concepto de variación p-ádica. Del mismo modo que el sistema de Euler de los ciclos diagonales permite estudiar la conjetura elíptica de Stark, el de Beilinson-Flach nos permitirá ofrecer evidencia teórica a la conjetura de Stark para unidades que formulan Darmon, Lauder y Rotger, que quedaría probada si fuésemos capaces de establecer ciertas relaciones conjeturales que involucran también a periodos p-ádicos. En este trabajo, además, hay una parte en la que se recuerdan ciertos resultados teóricos que ofrecen soporte a los que nosotros derivamos, centrándonos en el estudio de funciones L p-ádicas, formas modulares p-ádicas y sobreconvergentes y familias de Hida.

A simple graph $$G=(V,\,E)$$ is said to be antimagic if there exists a bijection $$f{\text {:}}\,E\rightarrow [1,\,|E|]$$ such that the sum of the values of f on edges incident to a vertex takes different values on distinct vertices. The graph G is distance antimagic if there exists a bijection $$f{\text {:}}\,V\rightarrow [1,\, |V|],$$ such that $$\forall x,\,y\in V,$$ $$\begin{aligned} \sum _{x_i\in N(x)}f\left( x_i\right) \ne \sum _{x_j\in N(y)}f\left( x_j\right) . \end{aligned}$$ Using the polynomial method of Alon we prove that there are antimagic injections of any graph G with n vertices and m edges in the interval $$[1,\,2n+m-4]$$ and, for trees with k inner vertices, in the interval $$[1,\,m+k].$$ In particular, a tree all of whose inner vertices are adjacent to a leaf is antimagic. This gives a partial positive answer to a conjecture by Hartsfield and Ringel. We also show that there are distance antimagic injections of a graph G with order n and maximum degree $$\Delta $$ in the interval $$[1,\,n+t(n-t)],$$ where $$ t=\min \{\Delta ,\,\lfloor n/2\rfloor \},$$ and, for trees with k leaves, in the interval $$[1,\, 3n-4k].$$ In particular, all trees with $$n=2k$$ vertices and no pairs of leaves sharing their neighbour are distance antimagic, a partial solution to a conjecture of Arumugam.

In this note we first review the classical Petterson-Gorenstein-Zierler decoding algorithm for the class of alternant codes (which includes Reed-Solomon, Bose-Chaudhuri-Hocquenghem and classical Goppa codes), then we present an improvement of the method to find the number of errors and the errorlocator polynomial, and finally we illustrate the procedure with several examples. In two appendices we sketch the main features of the system we have used for the computations., Comment: 16 pages

In this thesis we will deal with continued fractions, an expression which allow us to represent different mathematical objects through an iterative process (which can be finite or not). We will begin Chapter 1 with a brief historic background about how continued fractions were developed since Ancient Greek to the 20th century. It will follow the general definition of continued fractions, as well as some basic properties and theorems, and finally some observations about specific cases will be commented. Chapter 2 will focus on the simplest type of continued fractions, the ones using integer numbers. Here we will explain the most common application of continued fractions in the field of approximations of real numbers and how they are used to solve some particular equations that are harder to handle with the traditional methods. Up to this point we will only have studied continued fractions of real numbers, and therefore in Chapter 3 we will generalize these results to the complex plane, developing some new methods to aboard continued fractions of complex numbers. Most of the previous theorems will also hold in this case, but for others some aditional requirements will be needed. Finally, Chapter 4 shows some examples about the practical uses of continued fractions outside mathematics, in different fields such as astronomy, hemerology and music. We will discover here how to build a planetarium, create a calendar or even why it is not possible to tune a piano in a precise sense.

The homogeneous surface diffusion model (HSDM) is widely used for adsorption modeling of aqueous solutions. The Biot number is usually used to characterize model behavior. However, some limitations of this characterization have been reported recently, and the Stanton number has been proposed as a complement to be considered. In this work, a detailed dimensionless analysis of HSDM is presented and limit behaviors of the model are characterized, confirming but extending previous results. An accurate and efficient numerical solver is used for these purposes. The intraparticle diffusion equation is reduced to a system of two ordinary differential equations, the transport-reaction equation is discretized by using a discontinuous Galerkin method, and the overall system evolution is integrated with a time-marching scheme. This approach facilitates the simulation of HSDM with a wide range of dimensionless numbers and with a correct treatment of shocks, which appear with nonlinear adsorption isotherms and with large Biot numbers and small surface diffusivity modulus. The approach is applied to simulate the breakthrough curves of granular ferric hydroxide. Published experimental data is adequately simulated.

Heegner points play an outstanding role in the study of the Birch and Swinnerton-Dyer conjecture, providing canonical Mordell–Weil generators whose heights encode first derivatives of the associated Hasse–Weil L-series. Yet the fruitful connection between Heegner points and L-series 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 non-trivial, canonical elements of the idoneous Selmer group in settings where the classical L-function (of Hasse–Weil–Artin type) that governs their behaviour has a double zero at the centre. The generalised Kato class denoted \(\kappa (f,g,h)\) is associated to a triple (f, g, h) consisting of an eigenform f of weight two and classical p-stabilised eigenforms g and h of weight one, corresponding to odd two-dimensional Artin representations \(V_g\) and \(V_h\) of \(\mathrm {Gal\,}(H/\mathbb {Q})\) with p-adic coefficients for a suitable number field H. This class is germane to the Birch and Swinnerton-Dyer conjecture over H for the modular abelian variety E over \(\mathbb {Q}\) attached to f. One of the main results of Bertolini et al. (2015) and Darmon and Rotger (J AMS 2016) is that \(\kappa (f,g,h)\) lies in the pro-p Selmer group of E over H precisely when \(L(E,V_{gh},1)=0\), where \(L(E,V_{gh},s)\) is the L-function of E twisted by \(V_{gh}:= V_g\otimes V_h\). In the setting of interest, parity considerations imply that \(L(E,V_{gh},s)\) vanishes to even order at \(s=1\), and the Selmer class \(\kappa (f,g,h)\) is expected to be trivial when \({\mathrm {ord}}_{s=1}L(E,V_{gh},s) >2\). The main new contribution of this article is a conjecture expressing \(\kappa (f,g,h)\) as a canonical point in \((E(H)\otimes V_{gh})^{G_\mathbb {Q}}\) when \({\mathrm {ord}}_{s=1} L(E,V_{gh},s)=2\). 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).

S'exposa la teoria bàsica de ideals discriminants realtius sobre extensions de cossos i la teoria bàsica de extensions de Kummer. Es calculen alguns exemples de ideals discriminants realtius per a extensions cícliques de Kummer sobre cossos ciclotomics de 3 i 5 elements.