A multi-Frey approach to Fermat equations of signature (r,r,p)(r,r,p)

In this paper, we give a resolution of the generalized Fermat equations $$x^5 + y^5 = 3 z^n \text{ and } x^{13} + y^{13} = 3 z^n,$$ for all integers $n \ge 2$, and all integers $n \ge 2$ which are not a multiple of $7$, respectively, using the modular method with Frey elliptic curves over totally real fields. The results require a refined application of the multi-Frey technique, which we show to be effective in new ways to reduce the bounds on the exponents $n$. We also give a number of results for the equations $x^5 + y^5 = d z^n$, where $d = 1, 2$, under additional local conditions on the solutions. This includes a result which is reminiscent of the second case of Fermat's Last Theorem, and which uses a new application of level raising at $p$ modulo $p$.Comment: Includes more details regarding the connection of this paper with its sequel 'Some extensions of the modular method and Fermat-equations of signature (13,13,n)'. More precisely: extended Remark 7.4; added details on the computational parts of the proofs of Proposition 9 and Theorem 2; included new comments and polished the auxiliary Magma files for Proposition 9 and Theorem

Hecke and sturm bounds for Hilbert modular forms over real quadratic fields

Let K be a real quadratic field and OK its ring of integers. Let Γ be a congruence subgroup of SL2(OK) and M(k1,k2)(Γ) be the finite dimensional space of Hilbert modular forms of weight (k1, k2) for Γ. Given a form f(z) ∈ M(k1,k2)(Γ), how many Fourier coefficients determine it uniquely in such space? This problem was solved by Hecke for classical forms, and Sturm proved its analogue for congruences modulo a prime ideal. The present article solves the same problem for Hilbert modular forms over K. We construct a finite set of indices (which depends on the cusps desingularization of the modular surface attached to Γ) such that the Fourier coefficients of any form in such set determines it uniquely.Fil: Burgos Gil, Jose Ignacio. Instituto de Ciencias Matemáticas; EspañaFil: Pacetti, Ariel Martín. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin