Period Rings with Big Coefficients and Applications I

On algebra of big zeta values

The algebra of big zeta values we introduce in this paper is an intermediate object between multiple zeta values and periods of the multiple zeta motive. It consists of number series generalizing multiple zeta values, the simplest examples, which are not multiple zeta series, are Tornheim sums. We show that convergent big zeta values are periods of the moduli space of stable curves of genus zero on one hand and multiple zeta values on the other hand. It gives an alternative way to prove that any such period may be expressed as a rational linear combination of multiple zeta values and a simple …

On some local properties of sequences of big Galois representations

In this article, we prove that for a convergent sequence of residually absolutely irreducible representations of the absolute Galois group of a number field $F$ with coefficients in a domain finite over a power series ring over a $p$-adic integer ring, the set of places of $F$ where some of the representations ramifies has density zero. Using this, we extend a result of Das--Rajan to such convergent sequences. We also establish a strong multiplicity one theorem for big Galois representations.

Iwasawa theory of de Rham (ϕ,Γ)-modules over the Robba rings

The aim of this article is to study Bloch-Kato's exponential map and Perrin-Riou's "big" exponential map purely in terms of (ϕ,Γ)-modules over the Robba ring. We first generalize the definition of Bloch-Kato's exponential map for all the (ϕ,Γ)-modules without using Fontaine's rings B_{cris}, B_{dR} of p-adic periods and then we generalize the construction of Perrin-Riou's "big" exponential map for all the de Rham (ϕ,Γ)-modules and prove that this map interpolates our Bloch-Kato's exponential map and the dual exponential map. Finally, we prove a theorem concerning to the determinant of our "big…

扭曲的 Lubin-Tate 大 Witt 向量与 Fleck-Sun-Wan 同余式

Twisted Lubin-Tate Big Witt Vectors and Fleck-Sun-Wan Congruences

本文构造了与扭曲 Lubin-Tate 形式群相关的大 Witt 理论，引入 Fleck 算子并证明了 Fleck-Sun-Wan 同余式的 Lubin-Tate 类比，推广了现有 Witt 理论。

Period sets of linear recurrences over finite fields and related commutative rings

After giving an overview of the existing theory regarding the periods of sequences defined by linear recurrences over finite fields, we give explicit descriptions of the sets of periods that arise if one considers all sequences over $\mathbb{F}_q$ generated by linear recurrences for a fixed choice of the degree $k$ in the range $1 \leq k \leq 4$. We also investigate the periods of sequences generated by linear recurrences over rings of the form $\mathbb{F}_{q_1} \oplus \ldots \oplus \mathbb{F}_{q_r}$.

多元周期环

Multivariate Period Rings

提出一种更符合经典理论的多元周期环新研究方法，并证明B-容许表示构成Tannakian子范畴。

Graded rings of integral Jacobi forms

We determine the structure of the bigraded ring of weak Jacobi forms with integral Fourier coefficients. This ring is the target ring of a map generalising the Witten and elliptic genera and a partition function of $(0,2)$-model in string theory. We also determine the structure of the graded ring of all weakly holomorphic Jacobi forms of weight zero and integral index with integral Fourier coefficients. These forms are the data for Borcherds products for the Siegel paramodular groups.

An introduction to p-adic period rings

This paper is the augmented notes of a course I gave jointly with Laurent Berger in Rennes in 2014. Its aim was to introduce the periods rings B crys and B dR and state several comparison theorems between{é}tale and crystalline or de Rham cohomologies for p-adic varieties.

New estimates of double trigonometric sums with exponential functions

We establish a new bound for the exponential sum \begin{eqnarray*} \sum_{x\in\mathcal{X}}\Big|\sum_{y\in \mathcal{Y}}γ(y)\exp(2πi a λ^{xy}/p)\Big|, \end{eqnarray*} where $λ$ is an element of the residue ring modulo a large prime number $p,$ $\mathcal{X}$ and $\mathcal{Y}$ are arbitrary subsets of the residue ring modulo $p-1$ and $γ(n)$ are any complex numbers with $|γ(n)| \le 1.$ In particular, we improve several previously known bounds.

二维伪表示的大像

Big images of two-dimensional pseudorepresentations

本文扩大了Bellaïche关于二维伪表示像的子环B，用共轭自扭解释其概念并证明其最优性，关联伴随迹环，还恢复并扩展了多种GL₂伽罗瓦表示的算术大像结果。