Congruent modular forms and anticyclotomic Iwasawa theory

Abstract

Let $p$ be an odd prime. Consider normalized newforms $f_1,f_2$ that both satisfy the Heegner hypothesis for an imaginary quadratic field $K$ and suppose that they induce isomorphic residual Galois representations. In the work of Greenberg-Vatsal and Emerton-Pollack-Weston, the authors compare the cyclotomic Iwasawa $\mu$ and $\lambda$-invariants of $f_1$ and $f_2$. We extend this to the anticyclotomic indefinite setting by comparing the BDP $p$-adic $L$-functions attached to $f_1$ and $f_2$. Using this comparison, we obtain arithmetic implications for both generalized Heegner cycles and the Iwasawa main conjecture.