UCSD Number Theory Seminar (Math 209)

Thursday 2-3pm, AP&M 7421

Winter Quarter 2015

For previous quarters' schedule, click here.

January 8

Elisa Lorenzo-Garcia (University of Leiden)
Bad reduction of genus 3 curves with complex multiplication

Let C be a smooth, absolutely irreducible genus 3 curve over a number field M. Suppose that the Jacobian of C has complex multiplication by a sextic CM-field K. Suppose further that K contains no imaginary quadratic subfield. We give a bound on the primes p of M such that the stable reduction of C at p contains three irreducible components of genus 1.

Joint work with Bouw, Cooley, Lauter, Manes, Newton, Ozman.

January 15

Djordjo Milovic (Leiden)

January 22

Peter Stevenhagen (Leiden)

January 29

Grzegorz Banaszak (Adam Mickiewicz U. and UCSD)

February 5

Danny Neftin (University of Michigan)

March 19

Mona Merling (Johns Hopkins University)
Equivariant algebraic K-theory

The algebraic K theory space K(R) is defined as a topological group completion, which on \pi_0 is just the usual algebraic group completion of a monoid which yields K_0(R). Amazingly, it turns out that this space not only has a multiplication on it which is associative and commutative up to homotopy, but it is an infinite loop space. This means that it represents a spectrum (the stable analogue of a space), and therefore a cohomology theory. We construct equivariant algebraic K-theory for G-rings. However, spectra with G-action (called naive G-spectra) are not robust enough for stable homotopy theory, and the objects of study in equivariant stable homotopy theory are genuine G-spectra, which correspond to cohomology theories graded on representations.

Our construction of "genuine" equivariant algebraic K-theory recovers as its fixed points the K-theory of twisted group rings, and as particular cases equivariant topological real and complex K-theory, Atiyah's Real K-theory and statements previously formulated in terms of naive G-spectra for Galois extensions. For example, we can reinterpret the map from the Quillen-Lichtenbaum conjecture and the assembly map from Carlsson's conjecture in terms of genuine G-spectra and their fixed points.

We will not assume background in topology and will explain all the concepts from homotopy theory that arise in the talk.