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What is the form of zeta function of an elliptic curve over $\mathbb{F}_p(t)$? Does it satisfy a Riemann hypothesis?

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It is a polynomial in $p^{-s}$ and yes, it satisfies the Riemann hypothesis. See for example Theorem 2.2.1 of Lecture 4 of Douglas Ulmer's Elliptic curves over function fields available on the arXiv: 1101.1939v1.

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