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Register a new userOn Pisots $d$-th root conjecture for function fields and related GCD estimates
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We propose a function-field analog of Pisots $d$-th root conjecture on linear recurrences, and prove it under some non-triviality assumption. Besides a recent result of Pasten-Wang on B{u}chis $d$-th power problem, our main tool, which is also developed in this paper, is a function-field analog of an GCD estimate in a recent work of Levin and Levin-Wang. As an easy corollary of such GCD estimate, we also obtain an asymptotic result.
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Let $K$ be a local function field of characteristic $l$, $mathbb{F}$ be a finite field over $mathbb{F}_p$ where $l e p$, and $overline{rho}: G_K rightarrow text{GL}_n (mathbb{F})$ be a continuous representation. We apply the Taylor-Wiles-Kisin method over certain global function fields to construct a mod $p$ cycle map $overline{text{cyc}}$, from mod $p$ representations of $text{GL}_n (mathcal{O}_K)$ to the mod $p$ fibers of the framed universal deformation ring $R_{overline{rho}}^square$. This allows us to obtain a function field analog of the Breuil--Mezard conjecture. Then we use the technique of close fields to show that our result is compatible with the Breuil-Mezard conjecture for local number fields in the case of $l e p$, obtained by Shotton.
For a global function field K of positive characteristic p, we show that Artin conjecture for L-functions of geometric p-adic Galois representations of K is true in a non-trivial p-adic disk but is false in the full p-adic plane. In particular, we prove the non-rationality of the geometric unit root L-functions.
Using the following $_4F_3$ transformation formula $$ sum_{k=0}^{n}{-x-1choose k}^2{xchoose n-k}^2=sum_{k=0}^{n}{n+kchoose 2k}{2kchoose k}^2{x+kchoose 2k}, $$ which can be proved by Zeilbergers algorithm, we confirm some special cases of a recent conjecture of Z.-W. Sun on integer-valued polynomials.
We establish cancellation in short sums of certain special trace functions over $mathbb{F}_q[u]$ below the P{o}lya-Vinogradov range, with savings approaching square-root cancellation as $q$ grows. This is used to resolve the $mathbb{F}_q[u]$-analog of Chowlas conjecture on cancellation in M{o}bius sums over polynomial sequences, and of the Bateman-Horn conjecture in degree $2$, for some values of $q$. A final application is to sums of trace functions over primes in $mathbb{F}_q[u]$.
We explore whether a root lattice may be similar to the lattice $mathscr O$ of integers of a number field $K$ endowed with the inner product $(x, y):={rm Trace}_{K/mathbb Q}(xcdottheta(y))$, where $theta$ is an involution of $K$. We classify all pairs $K$, $theta$ such that $mathscr O$ is similar to either an even root lattice or the root lattice $mathbb Z^{[K:mathbb Q]}$. We also classify all pairs $K$, $theta$ such that $mathscr O$ is a root lattice. In addition to this, we show that $mathscr O$ is never similar to a positive-definite even unimodular lattice of rank $leqslant 48$, in particular, $mathscr O$ is not similar to the Leech lattice. In appendix, we give a general cyclicity criterion for the primary components of the discriminant group of $mathscr O$.
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