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for questions on one dimensional algebraic varieties over any field, including questions of moduli, and questions about specific curves.

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Geometric (or at least non-cohomological) proof of Lefschetz trace formula for curves

This was proved by Weil using his intersection theory. For a modern exposition, see 11.2 of Milne, J. S. Jacobian varieties. Arithmetic geometry (Storrs, Conn., 1984), 167--212, Springer, New York, 19 …
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