On the structure of Artin $L$-functions

S. A. Stepanov

Izvestiya. Mathematics , Tome 78 (2014) no. 1, pp. 154-168

Résumé

We consider the generating Artin $L$-function $$ L(z)=L(z,f)=\exp\Bigl(\,\sum_{\nu=1}^{\infty}\frac{T_\nu}{\nu} z^\nu\Bigr) $$ for the character sums $$ T_\nu=\sum_{x_1,\dots,x_n\in\mathbb F_{q^\nu}}\psi_\nu(f(x_1,\dots,x_n)), $$ where $\mathbb F_q$ is a finite field, $\mathbb F_{q^\nu}$ is a finite extension of $\mathbb F_q$, $\psi_\nu(\alpha)$ is a non-trivial additive character of $\mathbb F_{q^\nu}$, and $f\in\mathbb F_q[x_1,\dots,x_n]$ is a polynomial of degree $d\geqslant 2$, and give an elementary proof of Bombieri's conjecture on the algebraic structure of $L(z)$ in the case $n=2$.

Détail

Keywords: finite fields, sums of characters for polynomials in many variables, Artin $L$-function, polarized symmetric polynomials in many variables, Waring's theorem on symmetric polynomials.
Mots-clés : Bombieri's conjecture