  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.
