The fundamental theorem of Weil II (for curves) with ultra
product coefficients 
-- Anna Cadoret, February 9, 2018

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l-adic cohomology was built to provide an etale cohomology
with coefficients in a field of characteristic 0. This, via the
Grothendieck trace formula, gives  a cohomological interpretation of
L-functions - a fundamental tool in Deligne's theory of weights
developed in Weil II. Instead of l-adic coefficients one can consider
coefficients in ultra products of finite fields. I will state the
fundamental theorem of Weil II for curves in this setting and explain
briefly what are the difficulties to overcome to adjust Deligne's proof.
I will then discuss how this ultra product variant of Weil II allows to
extend to arbitrary coefficients  previous results of Gabber and Hui,
Tamagawa and myself for constant Z_l-coefficients.