Matija Kazalicki

There Are Infinitely Many Rational Diophantine Sextuples

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple. In this paper, we prove that there exist infinitely many rational Diophantine sextuples.

2-adic properties of modular functions associated to Fermat curves

For an odd integer N, we study the action of Atkins U(2)-operator on the modular function x(tau) associated to the Fermat curve: X^N +Y^N = 1. The function x(tau) is modular for the Fermat group Phi(N), generically a noncongruence subgroup. If x(tau) = q^(-1) + sum a(iN-1)q^(iN-1), we essentially prove that lim n->0 a(n) = 0 in the 2-adic topology.

More on Diophantine Sextuples

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple, and Dujella, Kazalicki, Mikic and Szikszai recently proved that there exist infinitely many rational Diophantine sextuples.

LINEAR RELATIONS FOR COEFFICIENTS OF DRINFELD MODULAR FORMS

Choie, Kohnen and Ono have recently classified the linear relations among the initial Fourier coefficients of weight k modular forms on SL2(ℤ), and they employed these results to obtain particular p-divisibility properties of some p-power Fourier coefficients that are common to all modular forms of certain weights. Using this, they reproduced some famous results of Hida on non-ordinary primes. Here we generalize these results to Drinfeld modular forms.