Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on the hypergeometric datum $\big((\frac{1}{2},\frac{1}{6},\frac{5}{6}),(1,1)\big)$ and symmetric squares of elliptic curves

Pengcheng Zhang

TL;DR

The paper provides a self-contained proof of a mod $p$ congruence linking the truncated hypergeometric sums associated to $((\frac{1}{2},\frac{1}{6},\frac{5}{6}),(1,1))$ with the symmetric squares of elliptic curves. It achieves this by reinterpreting $a_{\mathfrak{p}}(E)^2$ via a pair of $j$-equivalent curves $E_0$ and $E_1$, and then connecting the resulting traces to truncated hypergeometric sums through Clausen-type identities and precise $p$-adic truncation lemmas. The argument splits into two residue cases depending on $(\tfrac{z_0}{\mathfrak{p}})$ with $z_0=1-1728/j_0$, giving explicit ${}_{3}F_{2}$-valued congruences at $1728/j_0$ and $1-z_0$, thereby clarifying the role of the Legendre symbol $(\tfrac{1-1728/j_0}{\mathfrak{p}})$. Overall, the note offers a thorough, self-contained exposition of how hypergeometric truncations encode symmetric-square information of elliptic curves in mod $p$ arithmetic.

Abstract

This is an expository note on a mod $p$ congruence relating the truncated hypergeometric sums associated to $\big((\frac{1}{2},\frac{1}{6},\frac{5}{6}),(1,1)\big)$ to symmetric squares of elliptic curves.

A note on the hypergeometric datum $\big((\frac{1}{2},\frac{1}{6},\frac{5}{6}),(1,1)\big)$ and symmetric squares of elliptic curves

TL;DR

The paper provides a self-contained proof of a mod congruence linking the truncated hypergeometric sums associated to with the symmetric squares of elliptic curves. It achieves this by reinterpreting via a pair of -equivalent curves and , and then connecting the resulting traces to truncated hypergeometric sums through Clausen-type identities and precise -adic truncation lemmas. The argument splits into two residue cases depending on with , giving explicit -valued congruences at and , thereby clarifying the role of the Legendre symbol . Overall, the note offers a thorough, self-contained exposition of how hypergeometric truncations encode symmetric-square information of elliptic curves in mod arithmetic.

Abstract

This is an expository note on a mod congruence relating the truncated hypergeometric sums associated to to symmetric squares of elliptic curves.
Paper Structure (5 sections, 16 theorems, 76 equations)

This paper contains 5 sections, 16 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on elliptic curves
  3. Reinterpretation of $a_\mathfrak{p}(E)^2$
  4. The case when $(\frac{1-1728/j_0}{\mathfrak{p}})=1$
  5. The case when $(\frac{1-1728/j_0}{\mathfrak{p}})=-1$

Key Result

Theorem 1.1

Let $L$ be a number field, $E/L$ be an elliptic curve with $j(E)\notin\{0,1728\}$, and $\mathfrak{p}$ be a prime of $L$ that is a good prime of $E$. Suppose that $\mathfrak{p}\nmid 6$ and $v_\mathfrak{p}(j(E))=0=v_\mathfrak{p}((j(E)-1728))$. Then,

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 22 more