Sansuc’s formula and Tate global duality (d’après Rosengarten).

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• Brian Conrad (Stanford University)
• Tuesday 25 October 2016, 16:00-17:00
• MR13.

If you have a question about this talk, please contact Jack Thorne.

Tamagawa numbers are canonical (finite) volumes attached to smooth connected affine groups G over global fields k; they arise in mass formulas and local-global formulas for adelic integrals. A conjecture of Weil (proved long ago for number fields, and recently by Lurie and Gaitsgory for function fields) asserts that the Tamagawa number of a simply connected semisimple group is equal to 1; for special orthogonal groups this expresses the Siegel Mass Formula. Sansuc pushed this further (using a lot of class field theory) to give a formula for the Tamagawa number of any connected reductive G in terms of two finite arithmetic invariants: its Picard group and degree-1 Tate-Shafarevich group.

Over number fields it is elementary to remove the reductivity hypothesis from Sansuc’s formula, but over function fields that is a much harder problem; e.g., the Picard group can be infinite. Work in progress by my PhD student Zev Rosengarten is likely to completely solve this problem. He has formulated an alternative version, proved it is always finite, and established the formula in many new cases. We will discuss some aspects of this result, including one of its key ingredients: a generalization of Tate local and global duality to the case of coefficients in any positive-dimensional (possibly non-smooth) commutative affine algebraic k-group scheme and its (typically non-representable) GL_1-dual.

This talk is part of the Number Theory Seminar series.

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