Math CS Colloquium, Anthony Várilly-Alvarado (Rice University):"Elliptic curves, torsion subgroups, and uniform bounds for Brauer groups of K3 surfaces"

Thursday, December 10, 2015
4:15 PM - 5:15 PM (ET)
Exley 121
Event Type
Seminar/Colloquium
Contact
Han Li
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Link
https://eaglet.wesleyan.edu/MasterCalendar/EventDetails.aspx?EventDetailId=65845

Abstract: Elliptic curves are smooth plane curves defined by a homogeneous equation of degree three that come with a marked point.  Results on elliptic integrals going back to Euler show that one can endow such a curve with an abelian group structure, making the marked point the origin of this group.  Mordell showed in 1922 that if E is an elliptic curve defined by an equation over the rational numbers Q, then the group of points E(Q) is finitely generated.  Surprisingly, there are only 15 possibilities for the torsion subgroup of E(Q).  This is a spectacular theorem of Mazur from 1977.  I will explore this circle of ideas for a higher dimensional analogue of elliptic curves: K3 surfaces.  Unlike ``abelian surfaces'', K3 surfaces have no group structure, so even understanding what the analogue of E(Q) should be is tricky.  I will explain how the Brauer group of  K3 surface comes to the rescue, argue for a conjecture along the lines of Mazur's theorem, and explain the impact this would have in our understanding of K3 surfaces.

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