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Register a new userA Class of Incomplete Character Sums
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Using $ell$-adic cohomology of tensor inductions of lisse $overline{mathbb Q}_ell$-sheaves, we study a class of incomplete character sums.
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We deduce Katzs theorems for $(A,B)$-exponential sums over finite fields using $ell$-adic cohomology and a theorem of Denef-Loeser, removing the hypothesis that $A+B$ is relatively prime to the characteristic $p$. In some degenerate cases, the Betti number estimate is improved using toric decomposition and Adolphson-Sperbers bound for the degree of $L$-functions. Applying the facial decomposition theorem in cite{W1}, we prove that the universal family of $(A,B)$-polynomials is generically ordinary for its $L$-function when $p$ is in certain arithmetic progression.
We calculate the E-polynomials of certain twisted GL(n,C)-character varieties M_n of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n,F_q) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,C)-character variety. The calculation also leads to several conjectures about the cohomology of M_n: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious Hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n = 2.
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