论文标题
Tamagawa数字猜想和Kolyvagin的猜想是模块化形式的动机
The Tamagawa number conjecture and Kolyvagin's conjecture for motives of modular forms
论文作者
论文摘要
假设在算术代数几何形状（$ p $ -Adic调节器地图，注射率的三个概念性，$ p $ - p $ -Adic Abel-Jacobi地图）的特定实例，我们证明了$ p $的几个案例，用于$ p $ - tamagawa number number number number（$ p $ -tnc-tnc-per）的部分（分析等级中的模块化形式的动机均为$ \ geq4 $ $ 1 $。更准确地说，我们证明了我们的大量新形式$ f $和Prime数字$ P $普通的$ F $的结果，因此$ f $的重量是一致的，$ 2 $ MODULO $ 2 $ 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.一路上，我们还证明（在类似的假设下）$ p $ -TNC的分析等级模块化动机$ 0 $以及贝林森和迪格尼的合理性猜想是关于分析等级基本线上Zeta Elements在分析等级中存在$ 0 $ 0 $和$ 1 $的。在这项工作之前，模块化动机的$ P $ -TNC的唯一已知结果是重量$ 2 $，分析等级$ \ leq1 $，甚至重量和分析等级$ 0 $。随着我们对Kolyvagin的猜想的进一步应用，我们推断出Selmer组的结构定理，$ P $ - PARITY结果，Converse定理和模块化形式和模块化动机的较高等级结果。
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.
