Partial zeta functions, partial exponential sums, and p-adic estimates

Noah Bertram; Xiantao Deng; C. Douglas Haessig; Yan Li

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arXiv · 2106.09755

Partial zeta functions, partial exponential sums, and p-adic estimates

Noah Bertram·Xiantao Deng·C. Douglas Haessig·Yan Li

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Abstract

Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation their rationality is surprising, and even simple examples are delicate to compute. For instance, we give a detailed description of the partial zeta function of an affine curve where the number of unit poles varies, a property different from classical zeta functions. On the other hand, they do retain some properties similar to the classical case. To this end, we give Chevalley-Warning type bounds for partial zeta functions and L-functions associated to partial exponential sums.

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Noah Bertram, Xiantao Deng, C. Douglas Haessig, Yan Li. 2021-06-17. Partial zeta functions, partial exponential sums, and p-adic estimates. https://arxiv.org/abs/2106.09755

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