Enrique González-Jiménez

On symmetric square values of quadratic polynomials

We prove that there does not exist a non-square quadratic polynomial with integer coefficients and an axis of symmetry which takes square values for N consecutive integers for N=7 or N >= 9. At the opposite, if N <= 6 or N=8 there are infinitely many.

On fields of definition of torsion points of elliptic curves with complex multiplication

For any elliptic curve E defined over the rationals with complex multiplication (CM) and for every prime p, we describe the image of the mod p Galois representation attached to E. We deduce information about the field of definition of torsion points of these curves; in particular, we classify all cases where there are torsion points over Galois number fields not containing the field of definition of the CM.

On arithmetic progressions on Edwards curves

Let be m ∈ Z>0 and a,q ∈ Q. Denote by APm(a,q) the set of rational numbers d such that a,a + q,...,a + (m − 1)q form an arithmetic progression in the Edwards curve Ed : x 2 +y 2 = 1+dx 2 y 2 . We study the set APm(a, q) and we parametrize it by the rational points of an algebraic curve.

On a conjecture of Rudin on squares in arithmetic progressions

Let Q(N;q,a) denotes the number of squares in the arithmetic progression qn+a, for n=0, 1,...,N-1, and let Q(N) be the maximum of Q(N;q,a) over all non-trivial arithmetic progressions qn + a. Rudins conjecture asserts that Q(N)=O(Sqrt(N)), and in its stronger form that Q(N)=Q(N;24,1) if N=> 6. We prove the conjecture above for 6 8 for some integer k, where GP_k is the k-th generalized pentagonal number, then Q(N)=Q(N;q,a) with gcd(q,a) squarefree and q> 0 if and only if (q,a)=(24,1).

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.

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.

Galois theory, discriminants and torsion subgroup of elliptic curves

Abstract We find a tight relationship between the torsion subgroup and the image of the mod 2 Galois representation associated to an elliptic curve defined over the rationals. This is shown using some characterizations for the squareness of the discriminant of the elliptic curve.

Three cubes in arithmetic progression over quadratic fields

We study the problem of the existence of arithmetic progressions of three cubes over quadratic number fields

Growth of torsion groups of elliptic curves upon base change

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