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Let $X$ a smooth projective geometrically connected curve over a finite field $k$. Let $J$ a smooth commutative group scheme over $X$ and $F$ the function field of $X$.

Do we have a formula to compute $H^{1}(k,J(\mathbb{A})/J(F))$ where $\mathbb{A}$ is the ring of adeles?

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I think you might be able to deal with such a cohomology group using global tate duality. Try for example the book "Cohomology of number fields" by Neukirch, Schmidt and Winberg. Note that despite the title, they do indeed study all global fields – Daniel Loughran Jan 31 '13 at 18:14
This question looks like it may be incorrectly formulated: the Galois group of $k$ has no evident nontrivial action on $J(\mathbf{A})/J(F)$, so what could be interesting about this cohomology and more specifically: how would one find oneself confronted with wanting to know about it? – user30180 Feb 3 '13 at 7:01

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