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Register a new userOn p-adic approximation of sums of binomial coefficients
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We propose higher-order generalizations of Jacobsthals $p$-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients $binom{ip}{p}$ ($i=1,2,dots$) that are divisible by arbitrarily large powers of prime $p$.
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Recently, Ni and Pan proved a $q$-congruence on certain sums involving central $q$-binomial coefficients, which was conjectured by Guo. In this paper, we give a generalization of this $q$-congruence and confirm another $q$-congruence, also conjectured by Guo. Our proof uses Ni and Pans technique and a simple $q$-congruence observed by Guo and Schlosser.
We investigate the $p$-adic valuation of Weil sums of the form $W_{F,d}(a)=sum_{x in F} psi(x^d -a x)$, where $F$ is a finite field of characteristic $p$, $psi$ is the canonical additive character of $F$, the exponent $d$ is relatively prime to $|F^times|$, and $a$ is an element of $F$. Such sums often arise in arithmetical calculations and also have applications in information theory. For each $F$ and $d$ one would like to know $V_{F,d}$, the minimum $p$-adic valuation of $W_{F,d}(a)$ as $a$ runs through the elements of $F$. We exclude exponents $d$ that are congruent to a power of $p$ modulo $|F^times|$ (degenerate $d$), which yield trivial Weil sums. We prove that $V_{F,d} leq (2/3)[Fcolon{mathbb F}_p]$ for any $F$ and any nondegenerate $d$, and prove that this bound is actually reached in infinitely many fields $F$. We also prove some stronger bounds that apply when $[Fcolon{mathbb F}_p]$ is a power of $2$ or when $d$ is not congruent to $1$ modulo $p-1$, and show that each of these bounds is reached for infinitely many $F$.
In this paper, we focus on a family of generalized Kloosterman sums over the torus. With a few changes to Haessig and Sperbers construction, we derive some relative $p$-adic cohomologies corresponding to the $L$-functions. We present explicit forms of bases of top dimensional cohomology spaces, so to obtain a concrete method to compute lower bounds of Newton polygons of the $L$-functions. Using the theory of GKZ system, we derive the Dworks deformation equation for our family. Furthermore, with the help of Dworks dual theory and deformation theory, the strong Frobenius structure of this equation is established. Our work adds some new evidences for Dworks conjecture.
We generalise Dworks theory of $p$-adic formal congruences from the univariate to a multi-variate setting. We apply our results to prove integrality assertions on the Taylor coefficients of (multi-variable) mirror maps. More precisely, with $mathbf z=(z_1,z_2,...,z_d)$, we show that the Taylor coefficients of the multi-variable series $q(mathbf z)=z_iexp(G(mathbf z)/F(mathbf z))$ are integers, where $F(mathbf z)$ and $G(mathbf z)+log(z_i) F(mathbf z)$, $i=1,2,...,d$, are specific solutions of certain GKZ systems. This result implies the integrality of the Taylor coefficients of numerous families of multi-variable mirror maps of Calabi-Yau complete intersections in weighted projective spaces, as well as of many one-variable mirror maps in the Tables of Calabi-Yau equations [arXiv:math/0507430] of Almkvist, van Enckevort, van Straten and Zudilin. In particular, our results prove a conjecture of Batyrev and van Straten in [Comm. Math. Phys. 168 (1995), 493-533] on the integrality of the Taylor coefficients of canonical coordinates for a large family of such coordinates in several variables.
Francis Castro, et al [2] computed the exact divisibility of families of exponential sums associated to binomials $F(X) = aX^{d_1} + bX^{d_2}$ over $mathbb{F}_p$, and a conjecture is presented for related work. Here we study this question.
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