## 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.## References

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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