Zhi-Wei Sun
Let {a_s(mod n_s)}_{s=1}^k (k>1) be a finite system of residue classes with the moduli n_1,...,n_k distinct. By means of algebraic integers we show that the range of the covering function w(x)=|{1\le s\le k: x=a_s (mod n_s)}| is not contained in any residue class with modulus greater one. In particular, the values of w(x) cannot have the same parity.
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Bob Hough recently disproved a long-standing conjecture of Paul Erdős regarding covering systems. Inspired by his seminal paper, we describe analogs of covering systems to Boolean functions, and more generally, the problem of covering discrete hyper-boxes by non-parallel lower dimensional hyper-subboxes. We point out that very often primes are red herrings. This is definitely the case for covering system, and who knows, perhaps also for the Riemann Hypothesis.
Results on $L$-functions of Artin-Schreier coverings by Dwork, Bombieri and Adolphson-Sperber are generalized to $L$-functions of Witt coverings.
Initially motivated by the relations between Anabelian Geometry and Artin's L-functions of the associated Galois-representations, here we study the list of zeta-functions of genus two abelian coverings of elliptic curves over finite fields. Our goal is to provide a complete description of such a list.
For a Brylinski-Deligne covering group of a general linear group, we calculate some values of unramified Whittaker functions for certain representations that are analogous to the theta representations.
Campana conjecture for coverings of toric varieties over function fields
本文在函数域上证明了沿边界具有高重数的射影环面簇上Campana点的Vojta abc猜想，并以此推导出函数域上射影环面簇有限覆叠的Campana猜想版本。
This is a research announcement concerning a series of constructions obtained by applying the "doubling method" from the theory of automorphic forms to covering groups. Using these constructions, we obtain partial tensor product L-functions attached to generalized Shimura lifts, which may be defined in a natural way since at almost all places the representations are unramified principal series.
Quadratic Forms, Exact Covering Systems, and Product Identities for Theta Functions
本文建立整二次型与精确覆盖系统的联系，提出结构框架生成无穷多涉及拉马努金theta函数的乘积恒等式，统一解释二十二个Rogers-Ramanujan恒等式，并推导三元二次型相关新恒等式。
In this work we develop an integral representation for the partial $L$-function of a pair $π\timesτ$ of genuine irreducible cuspidal automorphic representations, $π$ of the $m$-fold covering of Matsumoto of the symplectic group $Sp_{2n}$, and $τ$ of a certain covering group of $GL_k$, with arbitrary $m$, $n$ and $k$. Our construction is based on the recent extension by Cai, Friedberg, Ginzburg and the author, of the classical doubling method of Piatetski-Shapiro and Rallis, from rank-$1$ twists to arbitrary rank twists. We prove a basic global identity for the integral and compute the local in…
In this paper, we prove that there is a natural correspondence between product identities for theta functions and integer matrix exact covering systems. We show that since $\mathbb{Z}^n$ can be taken as the disjoint union of a lattice generated by $n$ linearly independent vectors in $\mathbb{Z}^n$ and a finite number of its translates, certain products of theta functions can be written as linear combinations of other products of theta functions. We firstly give a general theorem to write a product of $n$ theta functions as a linear combination of other products of theta functions. Many known i…
In this note we compute some local unramified integrals defined on metaplectic covering groups of $GL$. These local integrals which were introduced by Suzuki, represent the standard tensor product $L$ function $L(π^{(n)}\times τ^{(n)},s)$ and extend the well known local integrals which represent $L(π\times τ,s)$. The computation is done using a certain "generating function" which extends a similar function introduced by the author in a previous paper. In the last section we discuss the Conjectures of Suzuki and introduce a global integral which unfolds to the above local integrals. This last p…
Bounds for Erdős covering systems in global function fields
本文研究了全球函数域中覆盖系统的界，证明了在特定条件下不存在具有乘法次数s的覆盖系统，并改进了之前与李、王和伊合作的研究结果。
We develop the local theory of the generalized doubling method for the $m$-fold central extension $Sp_{2n}^{(m)}$ of Matsumoto of the symplectic group. We define local $γ$-, $L$- and $ε$-factors for pairs of genuine representations of $Sp_{2n}^{(m)}\times\widetilde{GL}_k$ and prove their fundamental properties, in the sense of Shahidi. Here $\widetilde{GL}_k$ is the central extension of $GL_k$ arising in the context of the Langlands--Shahidi method for covering groups of $Sp_{2n}\times GL_k$. We then construct the complete $L$-function for cuspidal representations and prove its global function…
On Erdős covering systems in global function fields
本文基于 distortion 方法，证明了在特定条件下，全球函数域中不存在具有特定乘数的覆盖系统，尤其指出当 q≥759 时，F_q[x] 上不存在具有不同模数的覆盖系统。
On Covering Radii in Function Fields
本文研究了正特征设定下数的几何学中的覆盖半径问题，找到了凸体覆盖半径的闭式，证明了Woods和Minkowski猜想的正特征类似物，并得出正特征下Gruber-Mordell谱在所有维度均平凡的结论。
This paper studies a covering map phi from the upper half plane to the complex plane with a triangular lattice excised. This map is interesting as it factorises Klein's J invariant. Its derivative has properties which are a slight generalisation of modular functions, and (phi')^6 is a modular function of weight 12. There is a homomorphism from the modular group Gamma to the affine transformations of the complex plane which preserve the excised lattice. With respect to this action phi is a map of Gamma-sets. Identification of the excised lattice with the root lattice of sl_3(C) allows functions…
Regulators on some abelian coverings of $\mathbb{P}^1$ minus $n+2$ points
本文研究了超椭圆曲线的动机上同调中的调节器构造，并通过超几何函数验证了其非平凡性及整数性，同时探讨了 $L$-函数特殊值的贝利松猜想。
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