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Neelam Saikia

Publications and source records attributed to Neelam Saikia.

18 recordsLinked to original sources
arXiv2026

Weighted averages of $p$-adic hypergeometric functions and traces of Frobenius of elliptic curves

In this paper, we aim to study traces of Frobenius of certain one parameter families of elliptic curves and their relationships with $p$-adic hypergeometric functions. For example, we consider a DIK family of curves and establish the trace of Frobenius as weighted averages of special values of certain families of $p$-adic hypegeometric functions, where the average is taken over the arrays of parameters. Moreover, we consider Jacobi curves and express the trace of Frobenius as a special values of $p$-adic hypergeomtric functions. As a consequence of these results we obtain four summation identities for the $p$-adic hypegeometric functions that arise from the DIK family. Furthermore, we obtain $p$-adic analogous of Euler and Pfaff transformations for certain $p$-adic hypergemetric functions.

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arXiv2025

Fourth power moment of twisted Kloosterman sum and Hurwitz class numbers

In this paper, we investigate the fourth power moment of twisted Kloosterman sum and its relationship with Hurwitz class number. We derive an explicit formula expressing this moment in terms of weighted sums involving Hurwitz class numbers. Our approach involves analyzing point counting formulas associated with the resolution of certain Calabi-Yau threefold. Furthermore, we study the asymptotic behaviour of weighted sums of Hurwitz class numbers that appear in the moment formula. To derive these asymptotic formulas, we employ the theory of harmonic Maass forms, mock modular forms and holomorphic projections. As an application of these asymptotic results, we obtain the asymptotic formula for the fourth power moment of twisted Kloosterman sums.

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arXiv2025

Distribution of the Hessian values of Gaussian hypergeometric functions

We consider a special family of Gaussian hypergeometric functions whose entries are cubic and trivial characters over finite fields. The special values of these functions are known to give the Frobenius traces of families of Hessian elliptic curves. Using the theory of harmonic Maass forms and mock modular forms, we prove that the limiting distribution of these values is semi-circular (i.e. $SU(2)$), confirming the usual Sato-Tate distribution in this setting.

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arXiv2024

Distribution of rational points of an algebraic surface over finite fields

The number of points on a certain one parameter family of algebraic surface over a finite field $\F_p$ can be expressed as $p^2+A_p(λ),$ where $A_p(λ)$ is a character sum and $λ$ is an element of the finite field $\F_p.$ In this paper, we study the distribution of the term $A_p(λ)$ as the surface varies over a large family of algebraic surfaces of fixed genus and growing $p.$ The power moments of $A_p$'s are weighted sums of Catalan numbers. As a consequence of these results, we obtain limiting distributions of certain families of hypergeometric functions over large finite fields.

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arXiv2023

Values of $p$-adic hypergeometric functions, and $p$-adic analogue of Kummer's linear identity

Let $p$ be an odd prime and $\mathbb{F}_p$ be the finite field with $p$ elements. This paper focuses on the study of values of a generic family of hypergeometric functions in the $p$-adic setting which we denote by ${_{3n-1}G_{3n-1}}(p, t),$ where $n\geq1$ and $t\in\mathbb{F}_p$. These values are expressed in terms of numbers of zeros of certain polynomials over $\mathbb{F}_p$. These results lead to certain $p$-adic analogues of classical hypergeometric identities. Namely, we obtain $p$-adic analogues of particular cases of a Gauss' theorem and a Kummer's theorem. Moreover, we examine the zeros of these functions. For instance, if $n$ is odd then we obtain zeros of ${_{3n-1}G_{3n-1}}(p, t)=0$ under certain condition on $t$. In contrast we show that if $n$ is even then the function ${_{3n-1}G_{3n-1}}(p, t)$ has no non-trivial zeros for any prime $p$.

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arXiv2022

AGM and jellyfish swarms of elliptic curves

The classical $\mathrm{AGM}$ produces wonderful interdependent infinite sequences of arithmetic and geometric means with common limit. For finite fields $\mathbb{F}_q,$ with $q\equiv 3\pmod 4,$ we introduce a finite field analogue $\mathrm{AGM}_{\mathbb{F}_q}$ that spawns directed finite graphs instead of infinite sequences. The compilation of these graphs reminds one of a $\mathit{jellyfish~swarm},$ as the 3D renderings of the connected components resemble $\mathit{jellyfish}$ (i.e. tentacles connected to a bell head). These swarms turn out to be more than the stuff of child's play; they are taxonomical devices in number theory. Each jellyfish is an isogeny graph of elliptic curves with isomorphic groups of $\mathbb{F}_q$-points, which can be used to prove that each swarm has at least $(1/2-\varepsilon)\sqrt{q}$ jellyfish. Additionally, this interpretation gives a description of the $\mathit{class~numbers}$ of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish.

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arXiv2022

Distribution of values of Gaussian hypergeometric functions

In the 1980's, Greene defined {\it hypergeometric functions over finite fields} using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Apéry-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. We study the value distribution (over large finite fields) of natural families of these functions. For the $_2F_1$ functions, the limiting distribution is semicircular (i.e. $SU(2)$), whereas the distribution for the $_3F_2$ functions is the {\it Batman} distribution for the traces of the real orthogonal group $O_3$.

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arXiv2022

Sato-Tate Distribution of $p$-adic hypergeometric functions

Recently Ono, Saad and the second author \cite{KHN} initiated a study of value distribution of certain families of Gaussian hypergeometric functions over large finite fields. They investigated two families of Gaussian hypergeometric functions and showed that they satisfy semicircular and Batman distributions. Motivated by their results we aim to study distributions of certain families of hypergeometric functions in the $p$-adic setting over large finite fields. In particular, we consider two and six parameters families of hypergeometric functions in the $p$-adic setting and obtain that their limiting distributions are semicircular over large finite fields. In the process of doing this we also express the traces of $p$th Hecke operators acting on the spaces of cusp forms of even weight $k\geq4$ and levels 4 and 8 in terms of $p$-adic hypergeometric function which is of independent interest. These results can be viewed as $p$-adic analogous of some trace formulas of \cite{ah, ah-ono, fop}.

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arXiv2021

Zeros of hypergeometric functions in the $p$-adic setting

Let $p$ be an odd prime and $\mathbb{F}_p$ be the finite field with $p$ elements. McCarthy \cite{mccarthy-pacific} initiated a study of hypergeometric functions in the $p$-adic setting. This function can be understood as $p$-adic analogue of Gauss' hypergeometric function, and also some kind of extension of Greene's hypergeometric function over $\mathbb{F}_p$. In this paper we investigate values of two generic families of McCarthy's hypergeometric functions denoted by ${_nG_n}(t)$, and ${_n\widetilde{G}_n}(t)$ for $n\geq3$, and $t\in\mathbb{F}_p$. The values of the function ${_nG_n}(t)$ certainly depend on whether $t$ is $n$-th power residue modulo $p$ or not. Similarly, the values of the function ${_n\widetilde{G}_n}(t)$ rely on the incongruent modulo $p$ solutions of $y^n-y^{n-1}+\frac{(n-1)^{n-1}t}{n^n}\equiv0\pmod{p}$. These results generalize special cases of $p$-adic analogues of Whipple's theorem and Dixon's theorem of classical hypergeometric series. We examine zeros of the functions ${_nG_n}(t)$, and ${_n\widetilde{G}_n}(t)$ over $\mathbb{F}_p$. Moreover, we look into the values of $t$ for which ${_nG_n}(t)=0$ for infinitely many primes. For example, we show that there are infinitely many primes for which ${_{2k}G_{2k}}(-1)=0$. In contrast, for $t\neq0$ there is no prime for which ${_{2k}\widetilde{G}_{2k}}(t)=0$.

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arXiv2020

Zeros of $p$-adic hypergeometric functions, $p$-adic analogues of Kummer's and Pfaff's identities

We classify all the zeros and non-zero values of a family of hypergeometric series in the $p$-adic setting. These values of hypergeometric series in the $p$-adic setting lead to transformations of hypergeometric series in the $p$-adic setting which can be described as $p$-adic analogues of Kummer's and Pfaff's linear transformations on classical hypergeometric series. We also evaluate certain summation identities for hypergeometric series in the $p$-adic setting as well as Gaussian hypergeometric series.

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arXiv2020

$p$-Adic hypergeometric functions in the connections with certain twisted Kloosterman sheaf sum and modular forms

In this paper we establish certain identities connecting $p$-adic hypergeometric functions with 4-th twisted Kloosterman sheaf sum. To prove these identities we express certain character sum over finite field in terms of special values of $p$-adic hypergeometric functions. One conjecture of Evans behaves as a bridge to connect $p$-adic hypergeometric functions with Kloosterman sheaf sum. We also connect $p$-adic hypergeometric functions with Fourier coefficients of certain modular forms.

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arXiv2018

Supercongruences for truncated hypergeometric series and p-adic gamma function

We prove three more general supercongruences between truncated hypergeometric series and $p$-adic Gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez-Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for ${_7}F_6$ truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.

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arXiv2018

Certain character sums and hypergeometric series

We prove two transformations for the $p$-adic hypergeometric series which can be described as $p$-adic analogues of a Kummer's linear transformation and a transformation of Clausen. We first evaluate two character sums, and then relate them to the $p$-adic hypergeometric series to deduce the transformations. We also find another transformation for the $p$-adic hypergeometric series from which many special values of the $p$-adic hypergeometric series as well as finite field hypergeometric functions are obtained.

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arXiv2016

Summation identities and transformations for hypergeometric series

We find summation identities and transformations for the McCarthy's $p$-adic hypergeometric series by evaluating certain Gauss sums which appear while counting points on the family $$Z_λ: x_1^d+x_2^d=dλx_1x_2^{d-1}$$ over a finite field $\mathbb{F}_p$. A. Salerno expresses the number of points over a finite field $\mathbb{F}_p$ on the family $Z_λ$ in terms of quotients of $p$-adic gamma function under the condition that $d|p-1$. In this paper, we first express the number of points over a finite field $\mathbb{F}_p$ on the family $Z_λ$ in terms of McCarthy's $p$-adic hypergeometric series for any odd prime $p$ not dividing $d(d-1)$, and then deduce two summation identities for the $p$-adic hypergeometric series. We also find certain transformations and special values of the $p$-adic hypergeometric series. We finally find a summation identity for the Greene's finite field hypergeometric series.

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arXiv2014

Certain Transformations for Hypergeometric series in $p$-adic setting

In \cite{mccarthy2}, McCarthy defined a function $_{n}G_{n}[\cdots]$ using the Teichmüller character of finite fields and quotients of the $p$-adic gamma function. This function extends hypergeometric functions over finite fields to the $p$-adic setting. In this paper, we give certain transformation formulas for the function $_{n}G_{n}[\cdots]$ which are not implied from the analogous hypergeometric functions over finite fields.

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arXiv2013

p-adic Gamma function and traces of Frobenius of elliptic curves

In \cite{mccarthy2}, McCarthy defined a function $_{n}G_{n}[\cdots]$ using Teichmüller character of finite fields and quotients of $p$-adic gamma function, and expressed the trace of Frobenius of elliptic curves in terms of special values of $_{2}G_{2}[\cdots]$. We establish two different expressions for the traces of Frobenius of elliptic curves in terms of the function $_{2}G_{2}[\cdots]$. As a result, we obtain two relations between special values of the function $_{2}G_{2}[\cdots]$ with different parameters.

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