关于有1模格的等价定义

进一步讨论有1模格的等价定义问题,得到并证明了一个(2,2,0)型代数成为有1模格的一个充分必要条件.这样大大简化了有1模格的等价定义.     

有1模格的另外两种等价定义

对具有单位元的模格,本文给出了两种更为简洁的等价条件,并根据模格所满足的幂等律、交换律、结合律、吸收律和模律,证明了有1模格的各种定义之间的等价性。     

Galois cohomology in degree 3 of function fields of curves over number fields

Let k be a field of characteristic not equal to 2. For n≥1, let denote the nth Galois Cohomology group. The classical Tate's lemma asserts that if k is a number field then given finitely many elements , there exist such that α_i=(a)∪(b_i), where for any λ∈k∗, (λ) denotes the image of k∗ in . In this paper we prove a higher dimensional analogue of the Tate's lemma.     

Witt equivalence of function fields of curves over local fields

Two fields are Witt equivalent if their Witt rings of symmetric bilinear forms are isomorphic. Witt equivalent fields can be understood to be fields having the same quadratic form theory. The behavior of finite fields, local fields, global fields, as well as function fields of curves defined over Archimedean local fields under Witt equivalence is well understood. Numbers of classes of Witt equivalent fields with finite numbers of square classes are also known in some cases. Witt equivalence of general function fields over global fields was studied in the earlier work [13 G?adki, P., Marshall, M. Witt equivalence of function fields over global fields. Trans. Am. Math. Soc., electronically published on April 11, 2017, doi: https://doi.org/10.1090/tran/6898 (to appear in print).[Crossref] , [Google Scholar]] by the authors and applied to study Witt equivalence of function fields of curves over global fields. In this paper, we extend these results to local case, i.e. we discuss Witt equivalence of function fields of curves over local fields. As an application, we show that, modulo some additional assumptions, Witt equivalence of two such function fields implies Witt equivalence of underlying local fields.     

Ihara's lemma for imaginary quadratic fields

An analogue over imaginary quadratic fields of a result in algebraic number theory known as Ihara's lemma is established. More precisely, we show that for a prime ideal p of the ring of integers of an imaginary quadratic field F, the kernel of the sum of the two standard p-degeneracy maps between the cuspidal sheaf cohomology is Eisenstein. Here Y₀ and Y₁ are analogues over F of the modular curves Y₀(N) and Y₀(Np), respectively. To prove our theorem we use the method of modular symbols and the congruence subgroup property for the group SL₂(Z[1/p]) which is due to Mennicke [J. Mennicke, On Ihara's modular group, Invent. Math. 4 (1967) 202-228] and Serre [J.-P. Serre, Le problème des groupes de congruence pour SL₂, Ann. of Math. (2) 92 (1970) 489-527].     

On Picard Modular Forms

We study moduli spaces of principally polarized abelian varieties with an automorphism of finite order. After some examples (e. g. hermitian modular forms) we compute the ring of Picard modular forms in the case considered by Picard.     

Class numbers of cyclotomic function fields

Let be a prime power and let be the finite field with elements. For each polynomial in , one could use the Carlitz module to construct an abelian extension of , called a Carlitz cyclotomic extension. Carlitz cyclotomic extensions play a fundamental role in the study of abelian extensions of , similar to the role played by cyclotomic number fields for abelian extensions of . We are interested in the tower of Carlitz cyclotomic extensions corresponding to the powers of a fixed irreducible polynomial in . Two types of properties are obtained for the -parts of the class numbers of the fields in this tower, for a fixed prime number . One gives congruence relations between the -parts of these class numbers. The other gives lower bound for the -parts of these class numbers.

     

Modular forms which behave like theta series

In this paper, we determine all modular forms of weights , , for the full modular group which behave like theta series, i.e., which have in their Fourier expansions, the constant term and all other Fourier coefficients are non-negative rational integers. In fact, we give convex regions in (resp. in ) for the cases (resp. for the cases ). Corresponding to each lattice point in these regions, we get a modular form with the above property. As an application, we determine the possible exceptions of quadratic forms in the respective dimensions.

     

Modular forms arising from divisor functions

For an infinite family of modular forms constructed from Klein forms we provide certain explicit formulas for their Fourier coefficients by using the theory of basic hypergeometric series (Theorem 2). By making use of these modular forms we investigate the bases of the vector spaces of modular forms of some levels (Theorem 5) and find its application.     

Binary Hermitian forms over a cyclotomic field

Let ζ be a primitive fifth root of unity and let F be the cyclotomic field . Let be the ring of integers. We compute the Voronoï polyhedron of binary Hermitian forms over F and classify -conjugacy classes of perfect forms. The combinatorial data of this polyhedron can be used to compute the cohomology of the arithmetic group and Hecke eigenforms.     

A note on inverses of cyclotomic mapping permutation polynomials over finite fields

In this note, we give a shorter proof of the result of Zheng, Yu, and Pei on the explicit formula of inverses of generalized cyclotomic permutation polynomials over finite fields. Moreover, we characterize all these cyclotomic permutation polynomials that are involutions. Our results provide a fast algorithm (only modular operations are involved) to generate many classes of generalized cyclotomic permutation polynomials, their inverses, and involutions.     

Modular independence and generator matrices for codes over $${\mathbb {Z}_m}$$

We introduce a new notion of modular independence to define bases and the generator matrices for the codes over the ring of integers of general modulus m. We define standard forms for such generator matrices, and discuss how to find such forms and the parity check matrices.      

All double coverings of cyclotomic fields

A double covering of a Galois extension K/F in the sense of [3] is an extension /K of degree ≤2 such that /F is Galois. In this paper we determine explicitly all double coverings of any cyclotomic extension over the rational number field in the complex number field. We get the results mainly by Galois theory and by using and modifying the results and the methods in [2] and [3]. Project 10571097 supported by NSFC     

Modular forms and differential operators

In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for eachn ≥ 0 a bilinear operation which assigns to two modular formsf andg of weightk andl a modular form [f, g]_n of weightk +l + 2n. In the present paper we study these “Rankin-Cohen brackets” from two points of view. On the one hand we give various explanations of their modularity and various algebraic relations among them by relating the modular form theory to the theories of theta series, of Jacobi forms, and of pseudodifferential operators. In a different direction, we study the abstract algebraic structure (“RC algebra”) consisting of a graded vector space together with a collection of bilinear operations [,]_n of degree + 2n satisfying all of the axioms of the Rankin-Cohen brackets. Under certain hypotheses, these turn out to be equivalent to commutative graded algebras together with a derivationS of degree 2 and an element Φ of degree 4, up to the equivalence relation (∂,Φ) ~ (∂ - ϕE, Φ - ϕ² + ∂(ϕ)) where ϕ is an element of degree 2 andE is the Fuler operator (= multiplication by the degree). Dedicated to the memory of Professor K G Ramanathan     

Universal octonary diagonal forms over some real quadratic fields

In this paper, we will prove there are infinitely many integers n such that n ²— 1 is square-free and admits universal octonary diagonal quadratic forms. Received: November 2, 1998.     

Paley-like graphs over finite fields from vector spaces

Motivated by the well-known Paley graphs over finite fields and their generalizations, in this paper we explore a natural multiplicative-additive analogue of such graphs arising from vector spaces over finite fields. Namely, if $n\ge 2$ and $U⊊{\mathbb{F}}_{q^n}$ is an ${\mathbb{F}}_q$-vector space, $G_U$ is the (undirected) graph with vertex set $V(G_U)={\mathbb{F}}_{q^n}$ and edge set $E(G_U)=\{(a,b)\in {\mathbb{F}}_{q^n}^2|a\ne b,ab\in U\}$. We describe the structure of an arbitrary maximal clique in $G_U$ and provide bounds on the clique number $\omega (G_U)$ of $G_U$. In particular, we compute the largest possible value of $\omega (G_U)$ for arbitrary q and n. Moreover, we obtain the exact value of $\omega (G_U)$ when $U⊊{\mathbb{F}}_{q^n}$ is any ${\mathbb{F}}_q$-vector space of dimension $d_U\in \{1,2,n-1\}$.     

Fluctuations in the number of points on smooth plane curves over finite fields

In this note, we study the fluctuations in the number of points on smooth projective plane curves over a finite field Fq as q is fixed and the genus varies. More precisely, we show that these fluctuations are predicted by a natural probabilistic model, in which the points of the projective plane impose independent conditions on the curve. The main tool we use is a geometric sieving process introduced by Poonen (2004) [8].