Arithmetic of Eisenstein quotients

TL;DR

This paper investigates the arithmetic properties of Eisenstein quotients of modular Jacobians, providing theoretical criteria and concrete examples, especially focusing on levels that are prime or squares of primes.

Contribution

It introduces a general framework for Eisenstein quotients of modular Jacobians and establishes criteria for the non-triviality of Heegner points on these quotients.

Findings

Criteria for non-triviality of Heegner points

Analysis of Eisenstein quotients at prime levels

Concrete examples at prime and prime square levels

Abstract

In this paper, we will study the arithmetic of the Eisenstein part of the modular Jacobians. In the first section, we introduce some general preliminaries of the arithmetic theory of modular curves that we will need later. In the second section, we give an example of modular abelian varieties due to Gross and study its properties in some details. In the third section, we define Eisenstein quotients of the modular Jacobians in general and give a criterion of the non-triviality of Heegner points on such Eisenstein quotients. The last two sections return to the concrete examples when the level of the modular Jacobian ia a prime or a square of a prime.