Markoff-Rosenberger triples in geometric progression

Solutions of the Markoff-Rosenberger equation ax^2+by^2+cz^2 = dxyz such that their coordinates belong to the ring of integers of a number field and form a geometric progression are studied.Comment: To appear in Acta Mathematica Hungaric

An algorithm for determining torsion growth of elliptic curves

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 materia

Markoff-Rosenberger triples in arithmetic progression

We study the solutions of the Rosenberg--Markoff equation ax^2+by^2+cz^2 = dxyz (a generalization of the well--known Markoff equation). We specifically focus on looking for solutions in arithmetic progression that lie in the ring of integers of a number field. With the help of previous work by Alvanos and Poulakis, we give a complete decision algorithm, which allows us to prove finiteness results concerning these particular solutions. Finally, some extensive computations are presented regarding two particular cases: the generalized Markoff equation x^2+y^2+z^2 = dxyz over quadratic fields and the classic Markoff equation x^2+y^2+z^2 = 3xyz over an arbitrary number field.Comment: To appear in Journal of Symbolic Computatio

Torsion growth over cubic fields of rational elliptic curves with complex multiplication

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

Explicit characterization of the torsion growth of rational elliptic curves with complex multiplication over quadratic fields

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

Complete classification of the torsion structures of rational elliptic curves over quintic number fields

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⊆H could appear such that H=E(K)tors, for [K:Q]=5. In particular, we prove that at most there is one quintic number field K such that the torsion grows in the extension K/Q, i.e., E(Q)tors⊊E(K)torsThe author was partially supported by the grant MTM2015-68524-

On the modularity level of modular abelian varieties over number fields

Let f be a weight two newform for Γ1(N) without complex multiplication. In this article we study the conductor of the absolutely simple factors B of the variety Af over certain number fields L. The strategy we follow is to compute the restriction of scalars ResL/Q(B), and then to apply Milne's formula for the conductor of the restriction of scalars. In this way we obtain an expression for the local exponents of the conductor NL(B). Under some hypothesis it is possible to give global formulas relating this conductor with N. For instance, if N is squarefree, we find that NL(B) belongs to Z and NL(B)fLdimB=NdimB, where fL is the conductor of LThis work was supported in part by grants MTM 2009-07291 and CCG08-UAM/ESP-3906. This work was supported in part by grants 2009 SGR 1220 and MTM2009-13060-C02-0

Five squares in arithmetic progression over quadratic fields

We give several criteria to show over which quadratic number fields Q(sqrt{D}) there should exists a non-constant arithmetic progressions of five squares. This is done by translating the problem to determining when some genus five curves C_D defined over Q have rational points, and then using a Mordell-Weil sieve argument among others. Using a elliptic Chabauty-like method, we prove that the only non-constant arithmetic progressions of five squares over Q(sqrt{409}), up to equivalence, is 7^2, 13^2, 17^2, 409, 23^2. Furthermore, we give an algorithm that allow to construct all the non-constant arithmetic progressions of five squares over all quadratic fields. Finally, we state several problems and conjectures related to this problem.Comment: To appear in Revista Matem\'atica Iberoamerican

Arithmetic progressions of four squares over quadratic fields

Let d be a squarefree integer. Does there exist four squares in arithmeti progression over Q( √ d)? We shall give a partial answer to this question, depending on the value of d. In the a rmative ase, we onstru t expli it arithmeti progressions onsisting of four squares over Q( √ d)Research of the first author was supported in part by grant MTM 2009-07291 (Ministerio de Educación y Ciencia, Spain) and CCG08- UAM/ESP–3906 (Universidad Autónoma de Madrid – Comunidad de Madrid, Spain). Research of the second author was supported in part by grant MTM 2006-01859 (Ministerio de Educación y Ciencia, Spain