On the Tamagawa number conjecture for modular forms twisted by anticyclotomic Hecke characters
Takamichi Sano
TL;DR
The paper proves that the Tamagawa number conjecture for the critical motive $M_f(r)\otimes\chi$ attached to a modular form twisted by an anticyclotomic Hecke character follows from the Iwasawa main conjecture for the Bertolini–Darmon–Prasanna $p$-adic L-function, provided $L(f,χ^{-1},r)\neq 0$. It introduces a $\Lambda$-adic element $\mathfrak{z}_S$ built from the local epsilon element and the BDP $p$-adic L-function, showing it is a $\Lambda$-basis under the IMC and interpolates the critical $L$-value via a Deligne-period comparison of CM periods. The work also provides an elliptic-curve corollary in analytic rank zero with extra hypotheses, discusses potential rank-one extensions via $p$-adic Gross–Zagier formulas, and situates the construction within a broader Euler-system framework for higher-rank motives. Overall, it bridges the Iwasawa theory of $p$-adic L-functions with the Bloch–Kato Tamagawa framework in a mixed setting of modular forms and anticyclotomic characters, and outlines a path toward generalizing these ideas beyond class number one."
Abstract
Let $f \in S_{2r}(Γ_0(N))$ be a normalized newform of weight $2r$ which is good at $p$. Let $K$ be an imaginary quadratic field of class number one in which every prime divisor of $pN$ splits. Let $χ$ be an anticyclotomic Hecke character of $K$ which is crystalline at the primes above $p$ and such that $L(f,χ,r)\neq 0$. We prove that the Tamagawa number conjecture for the critical value $L(f,χ,r)$ follows from the Iwasawa main conjecture for the Bertolini-Darmon-Prasanna $p$-adic $L$-function.