$x^2+y^3=z^7$

Abstract: The generalized Fermat equation $ax^p + by^q = cz^r$ admits a fascinating trichotomy in the theory of its primitive integer solutions: if the invariant $\chi=1/p+1/q+1/r-1$ is positive, there are either zero or infinitely many solutions (Beukers), if $\chi=0$ it reduces to solving certain elliptic curves, and if $\chi<0$ there are only finitely many solutions (Darmon-Granville). The striking similarity between this result and the finiteness of rational points on algebraic curves over $\Q$ of genus $g\ge2$ (Faltings' theorem) is not a coincidence. In fact, primitive integer solutions to the generalized Fermat equations correspond to rational points on a "stacky curve" whose Euler characteristic is $\chi$. The Riemann existence theorem guarantees us a finite étale covering of this stacky curve by an ordinary curve (which is in our case a branched covering of $\mathbb{P}^1$ with prescribed ramification). In this talk, I will explain this general theory in the case $\chi<0$ and focus on the explicit computations due to Poonen-Schaefer-Stoll, using twists of the triply branched covering $\pi\colon X(7)\to\mathbb{P}^1$ and their pullbacks to the punctured affine surface $S=\text{Spec }\Z[x,y,z]/(x^2+y^3-z^7)\setminus0$ to determine the primitive integer solutions to $x^2+y^3=z^7$. This is an example of the descent obstruction applied to $G$-torsors over $S$, where $G=\text{Aut }\pi=\text{PSL}_2(\mathbb{F}_7)$!

*Note different time and location.

Reference: Poonen-Schaefer­-Stoll, Twists of $X(7)$ and primitive solutions to $x^2+y^3=z^7$; Darmon, Faltings plus epsilon, Wiles plus epsilon, and the Generalized Fermat Equation; Darmon-Granville, On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$.

algebraic geometrynumber theory

Audience: advanced learners

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STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Spring 2026 topic: The descent obstruction.

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.

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