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

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Franco Golfieri
University of Aveiro (CIDMA)
Mit, 13/09/2023 - 12:05 - 12:25
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 [1] 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 [2] and [3], but this time asymptotically for the parameter d.
Keywords: Endomorphisms of $GL_2$-type abelian varieties, Diophantine equations, Elliptic curves, Modular forms
[1] Franco Golfieri, Ariel Pacetti, Lucas Villagra Torcomian. “Asymptotic results for the equations $x^4+dy^2=z^p$ and $x^2+dy^6=z^p$”,
[2] Ariel Pacetti and Lucas Villagra Torcomian. “Q-curves, Hecke characters and some Diophantine equations” In Math. Comp., 91(338):1233-1257, 2022
[3] 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


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