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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the $ L$-function of multiplicative character sums

Author: John Dollarhide
Journal: Trans. Amer. Math. Soc. 365 (2013), 1637-1668
MSC (2010): Primary 11L40, 14F30
Published electronically: July 25, 2012
MathSciNet review: 3003277
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Abstract: Let $ \mathbb{F}_{q}$ be a finite field and let $ \chi _{1}, \dots , \chi _{r}$ be multiplicative characters on $ \mathbb{F}_{q}$. Suppose there are homogeneous polynomials $ f_{1},\dots , f_{r}$ of degrees $ d_{1}, \dots , d_{r}$ in $ \mathbb{F}_{q} [x_{1}, \dots , x_{n}]$ and suppose that $ f_{1}, \dots , f_{r}$ define smooth hypersurfaces in $ \mathbb{P}^{n-1}$ that have normal crossings. When the character sum $ S = \sum _{x \in \mathbb{P}^{n-1}(\mathbb{F}_{q})} \chi _{1}(f_{1}(x)) \dots \chi _{r}(f_{r}(x))$ is well defined, we compute the $ p$-adic Dwork cohomology of the $ L$-function associated to $ S$. In particular, we give a lower bound for the $ p$-adic Newton polygon of the $ L$-function and give a formula for the Hilbert series of the non-vanishing cohomology in terms of $ n,r,q$ and the $ d_{j}$'s.

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John Dollarhide
Affiliation: La Paz, Bolivia

Keywords: Character sums, $L$-function, $p$-adic cohomology, Newton polygon
Received by editor(s): January 23, 2009
Received by editor(s) in revised form: August 28, 2011
Published electronically: July 25, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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