Construction of Anti-Cyclotomic Euler Systems of Abelian Varieties Associated to $X_1(N)$

TL;DR

This paper constructs special points on abelian varieties linked to modular forms over imaginary quadratic fields, generalizing Heegner points, and demonstrates they satisfy Euler system properties to relate their non-torsionness to the rank of the variety.

Contribution

The paper introduces a new construction of points on abelian varieties associated to modular forms, extending Birch's Heegner points to a broader setting with Euler system properties.

Findings

Constructed points $P_\tau$ over extended ring class fields.

Proved $P_\tau$ satisfy distribution and congruence relations.

Potential to relate non-torsion points to the rank of $A_f(K)$.

Abstract

Let $K$ be an imaginary quadratic field, $N$ be a positive integer, $f(z)$ be a newform of level $\Gamma_1(N)$, and $A_f$ be the abelian variety associated to $f$. For each $\tau \in K$ ($\operatorname{Im} \tau >0$), we construct a certain point $P_\tau$ on $A_f$ defined over an extended ring class field of $K$ of level $N$. Our construction generalizes Birch's construction of the Heegner points to the abelian varieties associated to modular forms of level $\Gamma_1(N)$ and nontrivial character. Then, we show that $P_\tau$'s satisfy the distribution and congruence relations of an Euler system, which implies that it should be possible to apply the Euler system techniques to them to show a relation between the non-torsionness of $P_\tau$ and the rank of $A_f(K)$.