Regulators

马勒测度、动机调节器和狄利克雷L-值

Mahler measure, motivic regulators and Dirichlet $L$-values

本文通过研究动机上同调和狄利克雷L函数导数的关系，建立了马勒测度与调节器之间的公式，并在特定假设下推导了单个L值的恒等式。

函数域算术中的调节器

Regulators in the Arithmetic of Function Fields

本文研究函数域算术中Anderson A-动机的调节器概念，证明了A-动机上同调的有限性，并在特定重量条件下显示调节器源和目标的维度相同，但发现调节器像可能不具有满秩，影响了Beilinson猜想的类比。

One-motives and a conjecture of Deligne

We introduce new motivic invariants of arbitrary varieties over a perfect field. These cohomological invariants take values in the category of one-motives (considered up to isogeny in positive characteristic). The algebraic definition of these invariants presented here proves a conjecture of Deligne. Applications include some cases of conjectures of Serre, Katz and Jannsen on the independence of $\ell$ of parts of the étale cohomology of arbitrary varieties over number fields and finite fields.

关于Picard模曲面的高阶调节子

On higher regulators of Picard modular surfaces

本文证明Picard模曲面的动机类位于内部动机上同调群中，并建立其与动机L函数非临界值的关系，从而验证Beilinson猜想。

Motivic periods and Grothendieck arithmetic invariants

We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective. Showing that a suitable Betti--de Rham realization of 1-motives is fully faithful we can verify this period conjecture in several cases. The divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical Grothendieck period conjecture for algebraic cycles on smooth and proper schemes. Th…

Non-Archimedean regulator maps and special values of L-functions

We define an analogue of the `Real' Deligne cohomology group at a prime of semi-stable or good reduction of a variety. We also define regulator maps to this group and formulate a conjecture about the image. This allows us to formulate a non-Archimedian version of Beilinson's Hodge-D-conjecture, S-integral and function field versions of Beilinson's global conjectures as well as a precise special value conjecture in the function field case. Finally we give a few examples where these conjectures are known to be true.

On p-adic regulators for GSp(4) x GL(2) and GSp(4) x GL(2) x GL(2)

We compute the images of motivic cohomology classes for GSp(4) x GL(2) and GSp(4) x GL(2) x GL(2) under the syntomic regulator, and relate them to non-critical values of p-adic L-functions.

Extensions of Motives and the Fundamental Group

In this paper we construct extensions of the Mixed Hodge structure on the fundamental group of a pointed algebraic curve. These extensions correspond to the regulator of certain explicit motivic cohomology cycles in the self product of the curve which were first constructed by Bloch and Beilinson. This leads to a new iterated integral expression for the regulator. Our result is a generalization of a result of Colombo's where she constructs the extension corresponding to a motivic cycle class in the Jacobian of a hyperelliptic curve constructed by Collino. This is to appear in the Mathematical …

黑塞三次曲线的调节子与超几何函数

Regulator of the Hesse cubic curves and hypergeometric functions

该研究构造黑塞三次曲线动机上同调的积分元素，用两类超几何函数表示其调节子，得到了Bloch-Beilinson猜想关于L函数特殊值的数值例子。

On higher regulators of Siegel threefolds II: the connection to the special value

We establish a connection between motivic cohomology classes over the Siegel threefold and special values of the degree four $L$-function of some cuspidal automorphic representations of $\mathrm{GSp}(4)$. Our computation relies on our previous work [Le15] and on an integral representation of the $L$-function due to Piatetski-Shapiro.