On unit root formulas for toric exponential sums
The main result generalises the unit root results of Dwork on Kloosterman sums.NEWLINENEWLINELet \(A\subset {\mathbb Z}^n\) be a finite subset that spans \({\mathbb R}^n\) and set \(f_\Lambda(X)=\sum_{a\;\text{in}\;A}\Lambda_aX^a\) be a Laurent polynomial in \(X=(X_1,\ldots,x_n)\). Let \({\mathbb F}_q\) be the finite field of \(q=p^\epsilon\) elements. Fix a non-trivial additive character \(\Theta:{\mathbb F}_q\rightarrow{\mathbb Q}_p(\zeta_p)\) and let \(\Theta_{\overline\lambda}=\Theta\cdot\text{Tr}_{{\mathbb F}_q(\overline\lambda)/{\mathbb F}_q}\) be the induced additive character of \({\mathbb F}_q(\overline\lambda)\). For each positive integer \(l\), let \({\mathbb F}_q(\overline\lambda,l)\) denote the extension of degree \(l\) of \({\mathbb F}_q(\overline\lambda)\) and define the exponential sum NEWLINE\[CARRIAGE_RETURNNEWLINES_l=S_l(f_{\overline\lambda},\Theta_{\overline\lambda},{\mathbb T}^n)=\sum_{x\;\text{in}\;{\mathbb T}^n({\mathbb F}_q(\overline\lambda,l))}\Theta_{\overline\lambda}\cdot\text{Tr}_{{\mathbb F}_q(\overline\lambda,l)/{\mathbb F}_q(\overline\lambda)}(f_{\overline\lambda}(x))CARRIAGE_RETURNNEWLINE\]NEWLINE where the sum runs over the \(n\)-torus \({\mathbb T}^n\) over \({\mathbb F}_q(\overline\lambda)\). The associated \(L\)-function, \(L(f_{\overline\lambda}; T)\) is in \({\mathbb Q}(\zeta_p)(T)\) and its reciprocal zees and poles are algebraic integers. By the Dwork trace formula, just one of the reciprocal zeros and poles is a \(p\)-adic unit and it is one of the reciprocal zeros of \(L(f_{\overline\lambda}; T)^{(-1)^{n+1}}\). Denote this unit root by \(u(\overline\lambda)\). The goal is an explicit \(p\)-adic analytic formula for \(u(\overline\lambda)\).NEWLINENEWLINEWrite \(\exp f_\Lambda(x)=\prod_{a\;\text{in} A}\exp(\Lambda_aX^a)=\sum_{i\;\text{in}\;{\mathbb Z}^n}F_i(\Lambda)X^i\) so that NEWLINE\[CARRIAGE_RETURNNEWLINEF_i(\Lambda)=\sum_{_{\substack{ u=(u_a)\\ \sum u_aa=i}}}{\Lambda^u\over\prod_au_a!.}CARRIAGE_RETURNNEWLINE\]NEWLINE Fix \(\pi\) such that \(\pi^{p-1}=-p\) and \(\Theta(1)\equiv\pi\bmod\pi^2\). The \(F_i(\pi\Lambda)\) converge \(p\)-adically for \(\Lambda\) with \(|\Lambda_a|<1\) for all \(a\). Set \({\mathcal F}(\Lambda)=F_0(\pi\Lambda)/F_0(\pi\Lambda^p)\). The main result of the paper is that \({\mathcal F}(\Lambda))\) converges \(p\)-adically for \(|\Lambda_a|\leq1\) for all \(a\) and the unit root is given by NEWLINE\[CARRIAGE_RETURNNEWLINEu(\overline\lambda)={\mathcal F}(\lambda){\mathcal F}(\lambda^p){\mathcal F}(\lambda^{p^2})\cdots{\mathcal F}(\lambda^{p^{\epsilon d(\overline\lambda)-1}})CARRIAGE_RETURNNEWLINE\]NEWLINE where \(\lambda\) denotes the Teichmüller lifting of \(\overline\lambda\) and \(d(\overline\lambda)=[{\mathbb F}_q(\overline\lambda):{\mathbb F}_q]\).
scientific article; zbMATH DE number 6064722