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SUMMARY:Arav Karighattam (MIT)
DTSTART:20260319T200000Z
DTEND:20260319T213000Z
DTSTAMP:20260422T220800Z
UID:STAGE/153
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/STAGE/153/">
 $x^2+y^3=z^7$</a>\nby Arav Karighattam (MIT) as part of STAGE\n\nLecture h
 eld in Room 2-143 in the MIT Simons Building.\n\nAbstract\nThe generalized
  Fermat equation $ax^p + by^q = cz^r$ admits a fascinating trichotomy in t
 he 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 solution
 s (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 rati
 onal points on algebraic curves over $\\Q$ of genus $g\\ge2$ (Faltings' th
 eorem) is not a coincidence.  In fact\, primitive integer solutions to the
  generalized Fermat equations correspond to rational points on a "stacky c
 urve" whose Euler characteristic is $\\chi$.  The Riemann existence theore
 m guarantees us a finite étale covering of this stacky curve by an ordina
 ry curve (which is in our case a branched covering of $\\mathbb{P}^1$ with
  prescribed ramification).  In this talk\, I will explain this general the
 ory in the case $\\chi<0$ and focus on the explicit computations due to Po
 onen-Schaefer-Stoll\, using twists of the triply branched covering $\\pi\\
 colon X(7)\\to\\mathbb{P}^1$ and their pullbacks to the punctured affine s
 urface $S=\\text{Spec }\\Z[x\,y\,z]/(x^2+y^3-z^7)\\setminus0$ 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 $G=\\text{
 Aut }\\pi=\\text{PSL}_2(\\mathbb{F}_7)$!\n\n<b>*Note different time and lo
 cation.</b>\n\nReference: <a href="https://arxiv.org/abs/math/0508174">Poo
 nen-Schaefer­-Stoll</a>\, Twists of $X(7)$ and primitive solutions to $x^
 2+y^3=z^7$\; <a href="https://www.math.mcgill.ca/darmon/pub/Articles/Expos
 itory/04.Aisenstadt-prize/paper.pdf">Darmon</a>\, Faltings plus epsilon\, 
 Wiles plus epsilon\, and the Generalized Fermat Equation\; <a href="https:
 //www.math.mcgill.ca/darmon/pub/Articles/Research/12.Granville/pub12.pdf">
 Darmon-Granville</a>\, On the equations $z^m=F(x\,y)$ and $Ax^p+By^q=Cz^r$
 .\n
LOCATION:https://researchseminars.org/talk/STAGE/153/
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