Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-2012-05279-2
Published electronically: July 25, 2012
MathSciNet review: 3003277
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11L40, 14F30

Retrieve articles in all journals with MSC (2010): 11L40, 14F30


Additional Information

John Dollarhide
Affiliation: La Paz, Bolivia
Email: dollarhi@hotmail.com

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.