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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the $L$-function of multiplicative character sums
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by John Dollarhide PDF
Trans. Amer. Math. Soc. 365 (2013), 1637-1668 Request permission

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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Additional Information
  • John Dollarhide
  • Affiliation: La Paz, Bolivia
  • Email: dollarhi@hotmail.com
  • Received by editor(s): January 23, 2009
  • Received by editor(s) in revised form: August 28, 2011
  • Published electronically: July 25, 2012
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 1637-1668
  • MSC (2010): Primary 11L40, 14F30
  • DOI: https://doi.org/10.1090/S0002-9947-2012-05279-2
  • MathSciNet review: 3003277