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Rank of elliptic curves and class groups of real quadratic fields

Kalyan Banerjee

TL;DR

The paper addresses the problem of connecting the positive Mordell-Weil rank of an elliptic curve $E$ given by $y^2=x^3+ax+b$ to nontrivial elements in the class groups of real quadratic fields $K=\mathbb{Q}(\sqrt{p^3+ap+b})$ via specialization from a rank generator. It develops the spreading technique of a rank point to a Weil divisor on the integral model $E_{\mathbb{Z}}$ and leverages Chow/Hilbert schemes to produce nonprincipal class elements in $\mathrm{Cl}(K)$ for infinitely many primes $p$, with Siegel's theorem ensuring infinitude. The work highlights a novel use of Weil divisors rather than Cartier divisors, enabling a functorial, motivic link from the elliptic curve to class groups and extending to singular varieties. Overall, it provides a constructive bridge between elliptic-curve arithmetic and class groups of real quadratic fields, offering a framework to realize large class groups from elliptic data and potential generalizations to higher dimensions.

Abstract

In this paper, we are going to prove the relation between rank of elliptic curves and the non-triviality of class groups of infinitely many real quadratic fields.

Rank of elliptic curves and class groups of real quadratic fields

TL;DR

The paper addresses the problem of connecting the positive Mordell-Weil rank of an elliptic curve given by to nontrivial elements in the class groups of real quadratic fields via specialization from a rank generator. It develops the spreading technique of a rank point to a Weil divisor on the integral model and leverages Chow/Hilbert schemes to produce nonprincipal class elements in for infinitely many primes , with Siegel's theorem ensuring infinitude. The work highlights a novel use of Weil divisors rather than Cartier divisors, enabling a functorial, motivic link from the elliptic curve to class groups and extending to singular varieties. Overall, it provides a constructive bridge between elliptic-curve arithmetic and class groups of real quadratic fields, offering a framework to realize large class groups from elliptic data and potential generalizations to higher dimensions.

Abstract

In this paper, we are going to prove the relation between rank of elliptic curves and the non-triviality of class groups of infinitely many real quadratic fields.
Paper Structure (2 sections, 2 theorems, 38 equations)

This paper contains 2 sections, 2 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Proof of the theorem \ref{['thm1']}

Key Result

Theorem 1.1

Let $E$ be an elliptic curve given by equation where $a,b$ are rational integers and $4a^3+27b^2\neq 0$. Let the Mordell-Weil group $E({\mathbb Q})$ has positive rank and there exists a rational point on $E$, except the rank point. Then for an infinite number of prime numbers $p$, $p^3+ap+b$ is a square free positive integer and the class group

Theorems & Definitions (3)

  • Theorem 1.1
  • Theorem 2.1
  • proof