Symmetric power L-functions of a weighted hyper-Kloosterman family

Abstract

As a natural generalization of the classical hyper-Kloosterman family studied by D. Haessig and S. Sperber, we study the $k$-th symmetric power $L$-functions attached to a weighted hyper-Kloosterman family $$K{l}_{n,m}(t;x_1,\cdot \cdot \cdot ,x_n)=x_1^m+x_2\cdot \cdot \cdot +x_n+\frac{t}{x_1{x}_2\cdot \cdot \cdot x_n}.$$ Under suitable conditions, we determine the bounds of degrees of these $L$-functions and prove that their $q$-adic Newton polygons admit uniform lower bounds.

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