Xavier Xarles

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.

Rational periodic sequences for the Lyness recurrence

Consider the celebrated Lyness recurrence

On a conjecture of Rudin on squares in arithmetic progressions

x_{n+2}=(a+x_{n+1})/x_{n}

Five squares in arithmetic progression over quadratic fields

with

On p-adic intermediate Jacobians

a\in\mathbb{Q}

Additive reduction of algebraic tori

. First we prove that there exist initial conditions and values of

The scheme of connected components of the Néron model of an algebraic torus.

a

Component groups of Néron models via rigid uniformization.

for which it generates periodic sequences of rational numbers with prime periods

Local Bounds for Torsion Points on Abelian Varieties

1,2,3,5,6,7,8,9,10

Squares in arithmetic progression over number fields

or