It is a long-standing problem to produce curves whose Jacobians have
endomorphism algebras containing specified algebras. For instance,
Tautz, Top, Verberkmoes, Mestre, and Brumer have produced families of
genus (p-1)/2 whose Jacobians have real multiplication by the real
subfield of Q(zeta_p). Serre and Ekedahl have produced various
families of curves whose Jacobians are isogenous to products of
elliptic curves. In this talk we discuss a viewpoint, inspired by the
work of Brumer, that places all these results within one framework.
The main idea is to consider curves which have interesting structures
as branched covers of the projective line. We use this viewpoint to
produce the first known examples of curves of genus (p-1)/n (n=4, 6)
whose Jacobians have real multiplication by the index-n subfield of
Q(zeta_p).
M.I.T. Algebraic Geometry Seminar - Jordan Ellenberg
Friday May 7, 4:00pm-5:30pm
Room 2-255
Endomorphism algebras of Jacobians
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