# Asymptotic result for the equation $x^2 + dy^6 = z^p$

Posted in
Speaker:
Franco Golfieri
Affiliation:
University of Aveiro (CIDMA)
Date:
Wed, 13/09/2023 - 12:05 - 12:25
Location:
MPIM Lecture Hall
In order to solve some Diophantine equations, the general approach is to start associating a geometric object to a putative solution. In our cases, these objects are Q-curves over an imaginary quadratic extension. A result of Ribet implies that these particular curves have (after a twist by a Hecke character) a Galois representation defined over the rationals, and hence, by Serre's conjectures and some Taylor-Wiles' results on modularity, it has associated a rational modular form f. Using a Ribet result this modular form will be congruent module a prime to a modular form g in a smaller space.

The idea is to try to discard, via some methods, these latter modular forms. By Eichler-Shimura, they have associated an abelian variety and a field of coefficients. However, the newforms g of this smaller space whose coefficient field matches the one of f in general pass this elimination procedure. There is a plausible situation that might appear (because the curve is defined over an imaginary quadratic field) which is that the building block of the abelian variety associated to g might have dimension two. On the other hand, the abelian variety associated to f has a 1-dimensional building block, the Q-curve we started with. Then a reasonable question might be: Is it true that the newform g also has a building block of dimension 1? If so, what is the minimum field of definition of the elliptic curve? One of the main contributions of the work done in  is to provide a positive answer to this latter question, as well as to answer some other relations between the algebra of endomorphisms of their abelian varieties.
A non-trivial application of our results is solving the Diophantine equation $x^2+dy^6=z^p$, that the authors studied at  and , but this time asymptotically for the parameter d.

Keywords: Endomorphisms of $GL_2$-type abelian varieties, Diophantine equations, Elliptic curves, Modular forms
References:
 Franco Golfieri, Ariel Pacetti, Lucas Villagra Torcomian. “Asymptotic results for the equations $x^4+dy^2=z^p$ and $x^2+dy^6=z^p$”, https://arxiv.org/abs/2211.16334
 Ariel Pacetti and Lucas Villagra Torcomian. “Q-curves, Hecke characters and some Diophantine equations” In Math. Comp., 91(338):1233-1257, 2022
 Franco Golfieri, Ariel Pacetti, and Lucas Villagra Torcomian. “On the equation $x^2+dy^6=z^p$ for square-free $1\leq d \leq 20$”. In International Journal of Number Theory https://doi.org/10.1142/S1793042123500562

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