We study the group of automorphisms of Shimura curves X0(D,N) attached to an Eichler order of square-free level N in an indefinite rational quaternion algebra of discriminant D > 1. We prove that, when the genus g of the curve is greater than or equal to 2, Aut(X0(D,N)) is a 2-elementary abelian group which contains the group of Atkin-Lehner involutions W0(D,N) as a subgroup of index 1 or 2. It...

Theory, Communications and Coding Theory Concepts By Guillermo Atkin (Associate Professor, Department of Electrical & Computer Engineering, Illinois Institute of Technology, Chicago, IL 60616, United States) Wei Zhang (Assistant Professor, Department of Biological, Chemical, Physical Sciences, College of Science and Letters; Joint Assistant Professor, National Center for Food Safety and Technol...

WN ′ = W (N) N ′ denotes the corresponding Atkin–Lehner involution defined for Γ0(N). (If N ′ = 1, W1 means the identity operator.) Then we define the modular group Γ ∗ 0 (N) to be Γ ∗ 0 (N) = 〈Γ0(N) ∪ {WN ′}N ′ ‖N 〉, i.e., Γ ∗ 0 (N) is generated by Γ0(N) and {WN ′}N ′ ‖N . Then Γ ∗ 0 (N) is a normalizer of Γ0(N) in GL + 2 (Q) = {A ∈M2(Q) | detA > 0}. The factor group Γ ∗ 0 (N)/Γ0(N) is abelian...

The purpose of this note is to introduce a method for proving the existence of no rational points on a coarse moduli space X of abelian varieties over a given number field K, in cases where the moduli problem is not fine and points in X(K) may not be represented by an abelian variety (with additional structure) admitting a model over the field K. This is typically the case when the abelian vari...

Let K be a quadratic imaginary field and π an irreducible regular algebraic cuspidal automorphic representation of GL(2,AK). Under the assumption that the central character χπ is isomorphic to its complex conjugate, Taylor et al. associated p-adic Galois representations ρπ,p : GK → GL(2,Qp) which are unramified except at finitely many places, and such that the truncated L-function of the Galois...

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the p-adic analogue of the threeterm recursion satisfied ...

In the previous two papers with the same title ([LLY05] by W.C. Li, L. Long, Z. Yang and [ALL05] by A.O.L. Atkin, W.C. Li, L. Long), the authors have studied special families of cuspforms for noncongruence arithmetic subgroups. It was found that the Fourier coefficients of these modular forms at infinity satisfy three-term Atkin and Swinnerton-Dyer congruence relations which are the p-adic anal...

Jin Ho Kwak (POSTECH) Title: Enumeration and genus distribution of maps on surfaces Abstract: Two 2-cell embeddings ı : X → S and  : X → S of a connected graph X into a closed orientable surface S are congruent if there are an orientation-preserving surface homeomorphism h on S and a graph automorphism γ of X such that ıh = γ. When we restrict γ as the identity, we say two embeddings are equi...