Construction of Anti-Cyclotomic Euler Systems of Abelian Varieties Associated to $X_1(N)$
Let $K$ be an imaginary quadratic field, $N$ be a positive integer, $f(z)$ be a newform of level $Γ_1(N)$, and $A_f$ be the abelian variety associated to $f$. For each $τ\in K$ ($\operatorname{Im} τ>0$), we construct a certain point $P_τ$ 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 $Γ_1(N)$ and nontrivial character. Then, we show that $P_τ$'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 $P_τ$ and the rank of $A_f(K)$.
math.NT↗