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October 13, 2025Open Access

Congruent modular forms and anticyclotomic Iwasawa theory

Key Points

Comparative analysis reveals new links between anticyclotomic L-functions and modular forms, enhancing Iwasawa theory.
Results suggest that the BDP p-adic L-functions of forms inducing isomorphic Galois representations exhibit significant arithmetic properties.
The study extends previous work on cyclotomic invariants to an indefinite context, offering broader implications for Heegner cycles.
Findings provide support for the Iwasawa main conjecture in the context of modular forms and their associated cycles.

Abstract

Let p be an odd prime. Consider normalized newforms f₁, f₂ 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 and -invariants of f₁ and f₂. We extend this to the anticyclotomic indefinite setting by comparing the BDP p-adic L-functions attached to f₁ and f₂. Using this comparison, we obtain arithmetic implications for both generalized Heegner cycles and the Iwasawa main conjecture.

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Cite This Study

Dac-Nhan-Tam Nguyen (Fri,) studied this question.

synapsesocial.com/papers/68ecc715d1cc7436f7d189cd https://doi.org/https://doi.org/10.48550/arxiv.2503.00247

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