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On the zeta function of a projective complete intersection. (English) Zbl 1232.11092

Summary: We compute a basis for the \(p\)-adic Dwork cohomology of a smooth complete intersection in projective space over a finite field and use it to give \(p\)-adic estimates for the action of Frobenius on this cohomology. In particular, we prove that the Newton polygon of the characteristic polynomial of Frobenius lies on or above the associated Hodge polygon. This result was first proved by B. Mazur using crystalline cohomology [Bull. Am. Math. Soc. 78, 653–667 (1972; Zbl 0258.14006), Ann. Math. (2) 98, 58–95 (1973; Zbl 0261.14005)].

MSC:

11M38 Zeta and \(L\)-functions in characteristic \(p\)
14F30 \(p\)-adic cohomology, crystalline cohomology
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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Full Text: arXiv Euclid

References:

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