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Bloch's exact sequence for surfaces over local fields

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Pierre Matsumi
U of Nottingham
Mit, 31/03/2010 - 14:15 - 15:15
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Let $k$ be a local field such that $[k:Q_p] < \infty$ and $X$ be a proper smooth variety over $k$ with good reduction. Define $SK_1(X):=Coker(\partial: \bigoplus_{All C on X} K_2^M(k(C)) \to \bigoplus_{All points x on X} k(x)^*)$, where $\partial$ is the tame-symbol map. There is a reciprocity homomorphism   $\rho_S:SK_1(S) \to \pi_1^{ab}(S)$   to the abelianized fundamental group of S. During my last stay in MPIM, I proved class field theory for S=elliptic fibration, by which I mean $\rho_S/m$ is bijective for any $m > 1$. Inspired by Faltings, I made much progress on my research thereafter and recently established that for any $m>1 \rho_S/m$ is bijective with $S$ an arbitrary surface, assuming good reduction. We mimic the original approach by S. Bloch who proved a certain Bloch's exact sequence to capture the size of  $\pi_1^{ab}(S)$. The difference between Bloch's and ours is that we combine two semi-global theories successfully to deduce the global result. In the talk, I try to speak only high-lights and self-contained manner so that all audience enjoy what is the fascinating point in the result.

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