A $p$-adic analogue of the Borel regulator and the Bloch-Kato exponential map

On the non-vanishing of generalized Kato classes for elliptic curves of rank $2$

We prove the first cases of a conjecture by Darmon--Rotger on the non-vanishing of generalized Kato classes attached to elliptic curves $E$ over $\mathbf{Q}$ of rank $2$. Our method also shows that the non-vanishing of generalized Kato classes implies that the $p$-adic Selmer group of $E$ is $2$-dimensional. The main novelty in the proof is a formula for the leading term at the trivial character of an anticyclotomic $p$-adic $L$-function attached to $E$ in terms of the derived $p$-adic height of generalized Kato classes and an enhanced $p$-adic regulator.

2-部分的Bloch-Kato猜想及某些椭圆曲线的K₂的不可除结果

The 2-part of the Bloch-Kato conjecture, and indivisibility results, for $K_2$ of some elliptic curves

本文研究特定整数u的2-部分Bloch-Kato猜想，分析Selmer群的2- torsion，并构造K₂的不可除元素，通过Merkur'ev-Suslin同构证明其不可除性，同时探讨2-可除性界及数值验证。

Wach modules, regulator maps, and epsilon-isomorphisms in families

We prove the local epsilon-isomorphism conjecture of Fukaya and Kato [FK06] for certain crystalline families of G_Qp-representations. This conjecture can be regarded as a local analogue of the Iwasawa main conjecture for families. Our work extends earlier work of Kato for rank-1 modules (cf. [Ven13]), of Benois and Berger for crystalline G_Qp-representations with respect to the cyclotomic extension (cf. [BB08]), as well as of Loeffler, Venjakob, and Zerbes (cf. [LVZ13]) for crystalline G_Qp- representations with respect to abelian p-adic Lie extensions of Qp. Nakamura [Nak13, Nak14] has also f…

Iwasawa theory of overconvergent modular forms, I: Critical $p$-adic $L$-functions

We construct an Euler system of $p$-adic zeta elements over the eigencurve which interpolates Kato's zeta elements over all classical points. Applying a big regulator map gives rise to a purely algebraic construction of a two-variable $p$-adic $L$-function over the eigencurve. As a first application of these ideas, we prove the equality of the $p$-adic $L$-functions associated with a critical-slope refinement of a modular form by the works of Bellaïche/Pollack-Stevens and Kato/Perrin-Riou.

A p-adic regulator map and finiteness results for arithmetic schemes

A main theme of the paper is a conjecture of Bloch-Kato on the image of $p$-adic regulator maps for a proper smooth variety $X$ over an algebraic number field $k$. The conjecture for a regulator map of particular degree and weight is related to finiteness of two arithmetic objects: One is the $p$-primary torsion part of the Chow group in codimension 2 of $X$. Another is an unramified cohomology group of $X$. As an application, for a regular model ${\mathscr X}$ of $X$ over the integer ring of $k$, we show an injectivity result on torsion of a cycle class map from the Chow group in codimension …

On the p-adic Beilinson conjecture for number fields

We formulate a conjectural p-adic analogue of Borel's theorem relating regulators for higher K-groups of number fields to special values of the corresponding zeta-functions, using syntomic regulators and p-adic L-functions. We also formulate a corresponding conjecture for Artin motives, and state a conjecture about the precise relation between the p-adic and classical situations. Parts of he conjectures are proved when the number field (or Artin motive) is Abelian over the rationals, and all conjectures are verified numerically in some other cases.

Coleman maps and the p-adic regulator

This paper is a sequel to our earlier paper "Wach modules and Iwasawa theory for modular forms" (arXiv: 0912.1263), where we defined a family of Coleman maps for a crystalline representation of the Galois group of Qp with nonnegative Hodge-Tate weights. In this paper, we study these Coleman maps using Perrin-Riou's p-adic regulator L_V. Denote by H(Γ) the algebra of Qp-valued distributions on Γ= Gal(Qp(μ(p^\infty) / Qp). Our first result determines the H(Γ)-elementary divisors of the quotient of D_{cris}(V) \otimes H(Γ) by the H(Γ)-submodule generated by (ϕ* N(V))^{ψ= 0}, where N(V) is the Wac…

Polylogarithms, regulators and Arakelov motivic complexes

We construct an explicit regulator map from the weigh n Bloch Higher Chow group complexto the weight n Deligne complex of a regular complex projective algebraic variety X. We define the Arakelovweight n motivic complex as the cone of this map shifted by one. Its last cohomology group is (a version of) the Arakelov Chow group defined by H. Gillet. and C.Soule. We relate the Grassmannian n-logarithms (defined as in [G5]) to geometry of the symmetric space for GL_n(C). For n=2 we recover Lobachevsky's formula for the volume of an ideal geodesic tetrahedron via the dilogarithm. Using the relations…

Local theta-regulators of an algebraic number -- p-adic Conjectures

Let K/Q be Galois and let eta in K* be such that the multiplicative Z[G]-module generated by eta is of Z-rank n.We define the local theta-regulators Delta\_p^theta(eta) in F\_p for the Q\_p-irreducible characters theta of G=Gal(K/Q). Let V\_theta be the theta-irreducible representation. A linear representation L^theta=delta.V\_theta is associated withDelta\_p^theta(eta) whose nullity is equivalent to delta$\ge$1 (Theorem 3.9). Each Delta\_p^theta(eta) yields Reg\_p^theta(eta) modulo p in the factorization $\prod$\_theta (Reg\_p^theta(eta))^phi(1) of Reg\_p^G(eta) := Reg\_p(eta)/p^[K : Q] (norm…

动机关联子、簇簇与扎吉尔关于ζ(F,4)的猜想

Motivic correlators, cluster varieties and Zagier's conjecture on zeta(F,4)

研究证明数域F的戴德金ζ函数在s = 4处取值的扎吉尔猜想，通过定义特定映射等工具，还给出椭圆曲线E的L函数在s = 4处值的公式，为自由性猜想中F的动机李余代数权重4部分提供证据。