Infinite symmetric power L-functions of the hyper-Kloosterman family
Bolun Wei, Liping Yang
TL;DR
This work extends p-adic cohomological methods to the hyper-Kloosterman family by constructing an infinite symmetric power cohomology and applying Frobenius to obtain a uniform lower bound for the q-adic Newton polygon of L( Sym^{κ,∞} Kl_n, T). It provides a cohomological description of the infinite symmetric power L-function, proves a Newton above Hodge property, and derives uniform lower bounds for finite symmetric powers L(Sym^k Kl_n, T) on slopes up to k. The results generalize Haessig’s 1-dimensional insights to a higher-dimensional setting and connect to Dwork’s conjecture and related geometric structures, with implications for understanding unit-root factors and potential Hodge-theoretic refinements. The paper also outlines the relation between finite and infinite symmetric power L-functions and discusses possible future improvements analogous to FSY’s work in the 1-dimensional case.
Abstract
The infinity symmetric power $L$-functions play a fundamental role in Wan's groundbreaking work on Dwork's conjecture[16]. Building upon this foundation, Haessig[8] established the $p$-adic estimates for these $L$-functions in the case of the one-dimensional Kloosterman family. In this paper, we extend Haessig's results by deriving a uniform lower bound for the $q$-adic Newton polygon of the infinite symmetric power $L$-functions associated with the hyper-Kloosterman family. For the $1$-dimensional Kloosterman family, Haessig[8] showed that there is a $p$-adic cohomology theory for the infinity symmetric power $L$-function. In this paper, we prove there is also a cohomological description of the infinity symmetric power $L$-function for the hyper-Kloosterman family. By applying the Frobenius endomorphism to this cohomology, we derive a uniform lower bound for the corresponding $L$-function.