On mod p representations which are defined over F_p: II

Arithmetic and Differential Swan Conductors of rank one representations with finite local monodromy

We consider a complete discrete valuation field of characteristic p, with possibly non perfect residue field. Let V be a rank one continuous representation with finite local monodromy of its absolute Galois group. We will prove that the Arithmetic Swan conductor of V (defined after Kato in [Kat89] which fits in the more general theory of [AS02] and [AS06]) coincides with the Differential Swan conductor of the associated differential module $D^†(V)$ defined by Kedlaya in [Ked]. This construction is a generalization to the non perfect residue case of the Fontaine's formalism as presented in [Tsu…

Representations of degenerate poly-Bernoulli polynomials

As is well-known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate version of such functions and polynomials, degenerate polylogarithm functions were introduced and degenertae poly-Bernoulli polynomials were defined by means of the degenerate polylogarithm functions, and some their properties were investigated. The aim of this paper is to furthur study some properties of the degenertae poly-Bernoulli polynomials by using three formulas coming from the recently developed lambda-umbral calculus. In more detail, among other things, we represent t…

Morita's Theory for the Symplectic Groups

We construct and study the holomorphic discrete series representation and the principal series representation of the symplectic group $\mathrm{Sp}(2n,F)$ over a $p$-adic field $F$ as well as a duality between some sub-representations of these two representations. The constructions of these two representations generalize those defined in Morita and Murase's works. Moreover, Morita built a duality for $\mathrm{SL}(2, F)$ defined by residues. We view the duality we defined as an algebraic interpretation of Morita's duality in some extent and its generalization to the symplectic groups.

On weights of $l$-adic representation

Let $J$ be an abelian variety over a number field such that the center of its endomorphism ring is equal to the ring of integers. If the endomorphism ring splits at a prime number $l$, then the $l$-adic representation is defined by the minuscule weights (microweights) of simple classical Lie algebras.

Galois action on the $\bar\mathbb{Q}$-isogeny classes of abelian surfaces with quaternionic multiplication

An abelian variety over a number field is called L-abelian variety if, for any element of the absolute Galois group of a number field L, the conjugated abelian variety is isogenous to the given one by means of an isogeny that preserves the Galois action on the endomorphism rings. We can think of them as generalizations of abelian varieties defined over L with endomorphisms also defined over L. In the one dimensional case, an elliptic curve defined over L gives rise to a Galois representation provided by the Galois action on its Tate module. This classical Galois representation has been a centr…

On L-factors attached to generic representations of unramified U(2,1)

Let G be the unramified unitary group in three variables defined over a p-adic field of odd residual characteristic. In this paper, we establish a theory of newforms for the Rankin-Selberg integral for G introduced by Gelbart and Piatetski-Shapiro. We describe L and epsilon-factors defined through zeta integrals in terms of newforms. We show that zeta integrals of newforms for generic representations attain L-factors. As a corollary, we get an explicit formula for epsilon-factors of generic representations.

On epsilon factors attached to supercuspidal representations of unramified U(2,1)

Let G be the unramified unitary group in three variables defined over a p-adic field F of odd resudual characteristic. Gelbart, Piatetski-Shapiro and Baruch attached zeta integrals of Rankin-Selberg type to irreducible generic representations of G. In this paper, we formulate a conjecture on L- and epsilon-factors defined through zeta integrals in terms of local newforms for G, which is an analogue of the result by Casselman and Deligne for GL(2). We prove our conjecture for the generic supercuspidal representations of G.

由冗余字母系统中的数表示定义的分形函数

Fractal functions defined in terms of number representations in systems with a redundant alphabet

基于冗余字母系统的数表示定义的分形函数，通过数的展开算法研究其几何性质与连续性，揭示了函数在不同参数下的自相似结构与奇点特征。

Comparison of types for inner forms of $\mathrm{GL}_{N}$

Let $F$ be a non-archimedean local field, $A$ be a central simple $F$-algebra, and $G$ be the multiplicative group of $A$. To construct types for supercuspidal representations of $G$, simple types by Sécherre--Stevens and Yu's construction are already known. In this paper, we compare these constructions. In particular, we show essentially tame supercuspidal representations of $G$ defined by Bushnell--Henniart are nothing but tame supercuspidal representations defined by Yu.

Theta operators on Siegel modular forms and Galois representations

We describe the effect of the differential operators defined by Boecherer-Nagaoka, Flander-Ghitza and Yamauchi on the Galois representations (conjecturally) attached to Siegel modular eigenforms.

利用微积分的$\text{GL}_2(\text{${\bf F}_q$})$模表示

Modular representations of $\mathrm{GL}_2({\mathbb F}_q)$ using calculus

该研究证明$\text{GL}_2(\text{${\bf F}_q$})$的某些模诱导表示可表为对称幂表示上算子的余核，通过微分算子定义同构并改进theta滤过中商的周期性结果。

关于具有扩展伽罗华表示的阿贝尔簇的下降性质

Descent properties for an abelian variety with extended Galois representation

本文研究了具有扩展伽罗华表示的阿贝尔簇的下降性质，证明了在特定条件下，类型I的阿贝尔簇的逆向成立。

On the monotonicity of additive representation functions

We study the monotonicity behavior of three slightly differently defined additive representation functions (as initiated by Erdős, Sárközy and Sós), answering one open question and another one partially, and give a slightly simpler proof for a result of Chen and Tang.

Deformation of Curves with Automorphisms and representations on Riemann-Roch spaces

We study the deformation theory of nonsigular projective curves defined over algebraic closed fields of positive characteristic. We show that under some assumptions the local deformation problem for automorphisms of powerseries can be reduced to a deformation problem for matrix representations. We study both equicharacteristic and mixed deformations in the case of two dimensional representations.

A Prime Number Theorem for Rankin-Selberg L-functions over Number fields

We prove a prime number theorem first for the classical Rankin-Selberg L-function $L(s,π\timesπ')$ over any Galois extension with $π$ and $π'$ unitary automorphic cuspidal representations of $GL_n$ and $GL_m$ respectively with at least one of the representations subject to a self-contragredient assumption. We then extend these results to two representations $π$ defined on $GL_n/E$ and $π'$ defined on $GL_m/F$ with $E$ and $F$ cylic algebraic number fields of coprime degree where $π$ and $π'$ admit a base change lift from $\mathbb{Q}$ again given a self-contragredient assumption on at least one…

On the Rankin-Selberg $L$-factors for ${\rm SO}_{5}\times{\rm GL}_2$

Let $π$ and $τ$ be a smooth generic representation of ${\rm SO}_5$ and ${\rm GL}_2$ respectively over a non-archimedean local field. Assume that $π$ is irreducible and $τ$ is irreducible or induced of Langlands' type. We show that the $L$- and $ε$-factors attached to $π\timesτ$ defined by the Rankin-Selberg integrals and the associated Weil-Deligne representation coincide. Similar compatibility results are also obtained for the local factors defined by the Novodvorsky's local zeta integrals attached to generic representations of ${\rm GSp}_4\times{\rm GL}_2$.

Hurt-Sada阵列与Zeckendorf表示

The Hurt-Sada Array and Zeckendorf Representations

本文研究Hurt-Sada无限阵列及相关Sada序列，揭示其与黄金比例φ、Zeckendorf表示的紧密关联，并探讨了若干相关序列，为数论相关研究提供了新的联系视角。