The Tamagawa number conjecture and Kolyvagin's conjecture for motives of modular forms

TL;DR

This paper proves cases of the $p$-part of the Tamagawa number conjecture for motives of modular forms of even weight ≥4, using Kolyvagin's conjecture and $p$-adic methods, extending known results beyond weight 2.

Contribution

It establishes new cases of the $p$-TNC for higher weight modular motives and proves a higher weight analogue of Kolyvagin's conjecture via $p$-adic variation methods.

Findings

Proved $p$-TNC for a large class of modular motives in rank 1 and 0.

Established a higher weight analogue of Kolyvagin's conjecture.

Derived consequences for Selmer groups and parity results.

Abstract

Assuming specific instances of two general conjectures in arithmetic algebraic geometry (bijectivity of $p$-adic regulator maps, injectivity of $p$-adic Abel-Jacobi maps), we prove several cases of the $p$-part of the Tamagawa number conjecture ($p$-TNC) of Bloch-Kato and Fontaine-Perrin-Riou for (homological) motives of modular forms of even weight $\geq4$ in analytic rank $1$. More precisely, we prove our results for a large class of newforms $f$ and prime numbers $p$ that are ordinary for $f$ and such that the weight of $f$ is congruent to $2$ modulo $2(p-1)$. Inspired by work of W. Zhang in weight $2$, the key ingredient in our strategy is an analogue for $p$-adic Galois representations attached to higher (even) weight newforms of Kolyvagin's conjecture on the $p$-indivisibility of derived Heegner points on elliptic curves, which we prove via a $p$-adic variation method exploiting the arithmetic of Hida families. Along the way, we also prove (under similar assumptions) the $p$-TNC for modular motives in analytic rank $0$ and the rationality conjecture of Beilinson and Deligne on the existence of zeta elements on the fundamental line in analytic ranks $0$ and $1$. Prior to this work, the only known results on (questions related to) the $p$-TNC for modular motives were in weight $2$ and analytic rank $\leq1$ and in even weight and analytic rank $0$. As further applications of our result on Kolyvagin's conjecture in higher weight, we deduce a structure theorem for Selmer groups, $p$-parity results, converse theorems and higher rank results for modular forms and modular motives.