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

Source abstract

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 $$\ge 4$$ ≥ 4 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 $$2(p-1)$$ 2 ( p - 1 ) . Inspired by work of W. Zhang in weight 2, which builds on the congruence method originally developed by Bertolini–Darmon, 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 $$\le 1$$ ≤ 1 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.