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For each prime number $\ell$ and for each imaginary quadratic order of class number one or two, we determine all the possible $\ell$-adic Galois representations that occur for any elliptic curve with complex multiplication by such an order over its minimal field of definition, and then we determine all the isomorphism classes of elliptic curves that have a prescribed $\ell$-adic Galois representation., Comment: 38 pages; this version fixes some minor typos and includes a link in the introduction to a Github repository with code used in the paper

In a series of papers we classify the possible torsion structures of rational elliptic curves base-extended to number fields of a fixed degree. In this paper we turn our attention to the question of how the torsion of an elliptic curve with complex multiplication defined over the rationals grows over quadratic fields. We go further and we give an explicit characterization of the quadratic fields where the torsion grows in terms of some invariants attached to the curve., Comment: To appear in Glasnik Matemati\v{c}ki

This article is a contribution to the project of classifying the torsion growth of elliptic curve upon base-change. In this article we treat the case of elliptic curve defined over the rationals with complex multiplication. For this particular case, we give a description of the possible torsion growth over cubic fields and a completely explicit description of this growth in terms of some invariants attached to a given elliptic curve.

We present a fast algorithm that takes as input an elliptic curve defined over $\mathbb Q$ and an integer $d$ and returns all the number fields $K$ of degree $d'$ dividing $d$ such that $E(K)_{tors}$ contains $E(F)_{tors}$ as a proper subgroup, for all $F \varsubsetneq K$. We ran this algorithm on all elliptic curves of conductor less than 400.000 (a total of 2.483.649 curves) and all $d \leq 23$ and collected various interesting data. In particular, we find a degree 6 sporadic point on $X_1(4,12)$, which is so far the lowest known degree a sporadic point on $X_1(m,n)$, for $m\geq 2$., Comment: 15 pages, Added Supplementary material

Let $E$ be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer $A(E)$, that we call the {\it Serre's constant associated to $E$}, that gives necessary conditions to conclude that $\rho_{E,m}$, the mod m Galois representation associated to $E$, is non-surjective. In particular, if there exists a prime factor $p$ of $m$ satisfying ${\rm val}_p(m) > {\rm val}_p(A(E))>0$ then $\rho_{E,m}$ is non-surjective. {Conditionally under Serre's Uniformity Conjecture, w}e determine all the Serre's constants of elliptic curves without complex multiplication over the rationals that occur infinitely often. Moreover, we give all the possible combination of mod $p$ Galois representations that occur for infinitely many non-isomorphic classes of non-CM elliptic curves over $\mathbb{Q}$, and the known cases that appear only finitely. We obtain similar results for the possible combination of maximal non-surjective subgroups of ${\rm GL}_2(\mathbb{Z}_p)$. Finally, we conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre's constant., Comment: To appear in Exp. Math

Let $\mathcal{S}$ be a set of $n$ points in real four-dimensional space, no four coplanar and spanning the whole space. We prove that if the number of solids incident with exactly four points of $\mathcal{S}$ is less than $Kn^3$ for some $K=o(n^{\frac{1}{7}})$ then, for $n$ sufficiently large, all but at most $O(K)$ points of $\mathcal{S}$ are contained in the intersection of five linearly independent quadrics. Conversely, we prove that there are finite subgroups of size $n$ of an elliptic curve which span less than $\frac{1}{6}n^3$ solids containing exactly four points of $\mathcal{S}$.

Given an elliptic curve $E/\mathbb{Q}$ with torsion subgroup $G = E(\mathbb{Q})_{\rm tors}$ we study what groups (up to isomorphism) can occur as the torsion subgroup of $E$ base-extended to $K$, a degree 6 extension of $\mathbb{Q}$. We also determine which groups $H = E(K)_{\rm tors}$ can occur infinitely often and which ones occur for only finitely many curves. This article is a first step towards a complete classification of torsion growth of over sextic fields., Comment: To appear in Mathematics of Computation

We present the first exact calculation of the energy of the bound state of a one dimensional Dirac massive particle in weak short-range arbitrary potentials, using perturbation theory to fourth order (the analogous result for two dimensional systems with confinement along one direction and arbitrary mass is also calculated to second order). We show that the non--perturbative extension obtained using Pad\'e approximants can provide remarkably good approximations even for deep wells, in certain range of physical parameters. As an example, we discuss the case of two gaussian wells, comparing numerical and analytical results, predicted by our formulas., Comment: 11 pages, 4 figures

We explore the phenomenology of a unique three-Higgs-doublet model based on the single CP symmetry of order 4 (CP4) without any accidental symmetries. The CP4 symmetry is imposed on the scalar potential and Yukawa interactions, strongly shaping both sectors of the model and leading to a very characteristic phenomenology. The scalar sector is analyzed in detail, and in the Yukawa sector we list all possible CP4-symmetric structures which do not run into immediate conflict with experiment, namely, do not lead to massless or mass-degenerate quarks nor to insufficient mixing or CP-violation in the CKM matrix. We show that the parameter space of the model, although very constrained by CP4, is large enough to comply with the electroweak precision data and the LHC results for the 125 GeV Higgs boson phenomenology, as well as to perfectly reproduce all fermion masses, mixing, and CP violation. Despite the presence of flavor changing neutral currents mediated by heavy Higgs scalars, we find through a parameter space scan many points which accurately reproduce the kaon CP-violating parameter $\epsilon_K$ as well as oscillation parameters in K and $B_{(s)}$ mesons. Thus, CP4 offers a novel minimalistic framework for building models with very few assumptions, sufficient predictive power, and rich phenomenology yet to be explored., Comment: 39 pages, 8 figures, 1 table; v2: expanded discussion, extra references, matches published version

We study how the torsion of elliptic curves over number fields grows upon base change, and in particular prove various necessary conditions for torsion growth. For a number field $F$, we show that for a large set of number fields $L$, whose Galois group of their normal closure over $F$ has certain properties, it will hold that $E(L)_{tors}=E(F)_{tors}$ for all elliptic curves $E$ defined over $F$. Our methods turn out to be particularly useful in studying the possible torsion groups $E(K)_{tors}$, where $K$ is a number field and $E$ is a base change of an elliptic curve defined over $\mathbb Q$. Suppose that $E$ is a base change of an elliptic curve over $\mathbb Q$ for the remainder of the abstract. We prove that $E(K)_{tors}=E(\mathbb Q)_{tors}$ for all elliptic curves $E$ defined over $\mathbb Q$ and all number fields $K$ of degree $d$, where $d$ is not divisible by a prime $\leq 7$. Using this fact, we determine all the possible torsion groups $E(K)_{tors}$ over number fields $K$ of prime degree $p\geq 7$. We determine all the possible degrees of $[\mathbb Q(P):\mathbb Q]$, where $P$ is a point of prime order $p$ for all $p$ such that $p\not\equiv 8 \pmod 9$ or $\left( \frac{-D}{p}\right)=1$ for any $D\in \{1,2,7,11,19,43,67,163\}$; this is true for a set of density $\frac{1535}{1536}$ of all primes and in particular for all $p<3167$. Using this result, we determine all the possible prime orders of a point $P\in E(K)_{tors}$, where $[K:\mathbb Q]=d$, for all $d\leq 3342296$. Finally, we determine all the possible groups $E(K)_{tors}$, where $K$ is a quartic number field and $E$ is an elliptic curve defined over $\mathbb Q$ and show that no quartic sporadic point on a modular curves $X_1(m,n)$ comes from an elliptic curve defined over $\mathbb Q$., Comment: 27 pages. The file contains text colored in blue; this text can be clicked on and is a link to the Magma code used to obtain that particular result

We explore the origin and Yukawa interactions of the scalars with peculiar CP-properties which were recently found in a multi-Higgs model based on an order-4 CP symmetry. We relate the existence of such scalars to the enhanced freedom of defining CP, even beyond the well-known generalized CP symmetries, which arises in models with several zero-charge scalar fields. We also show that despite possessing exotic CP quantum numbers, these scalars do not have to be inert: they can have CP-conserving Yukawa interactions provided the CP acts on fermions by also mixing generations. This paper focuses on formal aspects---exposed in a pedagogical manner---and includes a brief discussion of possible phenomenological consequences., Comment: 18 pages; v2: discussion expanded, references added; v3: more discussions and explanations provided, matches the published version

We classify the possible torsion structures of rational elliptic curves over quintic number fields. In addition, let E be an elliptic curve defined over Q and let G = E(Q)_tors be the associated torsion subgroup. We study, for a given G, which possible groups G \subseteq H could appear such that H=E(K)_tors, for [K:Q]=5. In particular, we prove that at most there is a quintic number field K such that E(Q)_tors\neq E(K)_tors., Comment: The file contains text colored in blue; this text can be clicked on and is a link to the Magma code used to obtain that particular result. To appear in Journal of Algebra

Let E be an elliptic curve defined over Q and let G = E(Q)_tors be the associated torsion subgroup. We study, for a given G, which possible groups G <= H could appear such that H=E(K)_tors, for [K:Q]=4 and H is one of the possible torsion structures that occur infinitely often as torsion structures of elliptic curves defined over quartic number fields., Comment: In this new version we fix some errors in Table 5

Let E be an elliptic curve over Q, and let n=>1. The central object of study of this article is the division field Q(E[n]) that results by adjoining to Q the coordinates of all n-torsion points on E(Q). In particular, we classify all curves E/Q such that Q(E[n]) is as small as possible, that is, when Q(E[n])=Q(zeta_n), and we prove that this is only possible for n=2,3,4, or 5. More generally, we classify all curves such that Q(E[n]) is contained in a cyclotomic extension of Q or, equivalently (by the Kronecker-Weber theorem), when Q(E[n])/Q is an abelian extension. In particular, we prove that this only happens for n=2,3,4,5,6, or 8, and we classify the possible Galois groups that occur for each value of n., Comment: In this version we fix an error in the proof of Proposition 3.7. We thank Tyler Genao for pointing out this error to us. In Table 4 the elliptic curve 46800cw4 has been replaced by the elliptic curve 486720dr3. The elliptic curve 46800cw4 does not have 2-adic image equal to X58f

In this article, we study the minimal degree [K(T):K] of a p-subgroup T <= E(\overline{K})_tors for an elliptic curve E/K defined over a number field K. Our results depend on the shape of the image of the p-adic Galois representation \rho_{E,p^infty}:Gal_K-->GL(2,Z_p). However, we are able to show that there are certain uniform bounds for the minimal degree of definition of T. When the results are applied to K=Q and p=2, we obtain a divisibility condition on the minimal degree of definition of any subgroup of E[2^n] that is best possible., Comment: Math. Res. Lett., to appear

A variant for the superspin one-half massive superparticle in $ 4D $, $ \mathcal{N}=1 $, based on Dirac superfields, is offered. As opposed to the current known models that use spinor chiral superfields, the propagating fields of the supermultiplet are those of the lowest mass dimensions possible: scalar, Dirac and vector fields. Besides the supersymmetric chiral condition, the Dirac superfields are not further constrained, allowing a very straightforward implementation of the path-integral method. The corresponding superpropagators are presented. In addition, an interaction super Yukawa potential, formed by Dirac and scalar chiral superfields, is given in terms of their component fields. The model is first presented for the case of two superspin one-half superparticles related by the charged conjugation operator, but in order to treat the case of neutral superparticles, the Majorana condition on the Dirac superfields is also studied. We compare our proposal with the known models of spinor superfields for the one-half superparticle and show that it is equivalent to them., Comment: 22 pages. Matches published version

Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)_tors and the torsion subgroup E(K)_tors, where K is a cubic number field. In particular, We study the number of cubic number fields K such that E(Q)_tors\neq E(K)_tors., Comment: Rocky Mountain Journal of Mathematica, to appear

Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)_tors and the torsion subgroup E(K)_tors, where K is a quadratic number field.

Let E be an elliptic curve defined over Q and let G=E(Q)_tors be the associated torsion group. In a previous paper, the authors studied, for a given G, which possible groups G\leq H could appear such that H=E(K)_tors, for [K:Q]=2. In the present paper, we go further in this study and compute, under this assumption and for every such G, all the possible situations where G\neq H. The result is optimal, as we also display examples for every situation we state as possible. As a consequence, the maximum number of quadratic number fields K such that E(Q)_tors\neq E(K)_tors is easily obtained., Comment: To appear in Revista de la Real Academia de Ciencias Exactas, F\'isicas y Naturales. Serie A. Matem\'atica RACSAM