Let $\Gamma_n(\mathcal{\scriptstyle{O}}_\mathbb{K})$ denote the Hermitian modular group of degree $n$ over an imaginary-quadratic number field $\mathbb{K}$. In this paper we determine its maximal discrete extension in $SU(n,n;\mathbb{C})$, which coincides with normalizer $\Gamma_n(\mathcal{\scriptstyle{O}}_{\mathbb{K}})$. The description involves $n$-torsion subgroup ideal class This is defined...

We study the automorphisms of modular curves associated to Cartan subgroups $\mathrm{GL}_2(\mathbb Z/n\mathbb Z)$ and certain their normalizers. prove that if $n$ is large enough, all are induced by ramified covering complex upper half-plane. get new results for non-split prime level $p\ge 13$: curve $X_{\text{ns}}^+(p)$ has no non-trivial automorphisms, whereas $X_{\text{ns}}(p)$ exactly one a...

We use the method of quadratic Chabauty on quotients $X_0^+(N)$ modular curves $X_0(N)$ by their Fricke involutions to provably compute all rational points these for prime levels $N$ genus four, five, and six. find that only such with exceptional are $137$ $311$. In particular there no those five More precisely, we determine $N=137,173,199,251,311,157,181,227,263,163,197,211,223,269,271,359$.

It is shown that Ramanujan’s cubic transformation of the Gauss hypergeometric function 2F1 arises from a relation between modular curves, namely the covering of X0(3) by X0(9). In general, when 2 N 7, the N-fold cover of X0(N) by X0(N) gives rise to an algebraic hypergeometric transformation. The N = 2, 3, 4 transformations are arithmetic–geometric mean iterations, but the N = 5, 6, 7 transform...

Certain identities of Ramanujan may be succinctly expressed in terms of the rational function ğχ = f̆χ − 1 f̆χ on the modular curve X0(N), where f̆χ = wNfχ and fχ is a certain modular unit on the Nebentypus cover Xχ(N) introduced by Ogg and Ligozat for prime N ≡ 1 (mod 4) and wN is the Fricke involution. These correspond to levels N = 5, 13, where the genus gN of X0(N) is zero. In this paper we st...

A theory of local old– and newforms for representations of GSp(4) over a p–adic field with Iwahori– invariant vectors is developed. The results are applied to Siegel modular forms of degree 2 with square-free level with respect to various congruence subgroups. Introduction For representations of GL(2) over a p–adic field F there is a well-known theory of local newforms due to Casselman, see [Ca...

Diffusion Theory in the New Media Environment: Toward an Integrated Technology Adoption Model David J. Atkin, Daniel S. Hunt & Carolyn A. Lin To cite this article: David J. Atkin, Daniel S. Hunt & Carolyn A. Lin (2015) Diffusion Theory in the New Media Environment: Toward an Integrated Technology Adoption Model, Mass Communication and Society, 18:5, 623-650, DOI: 10.1080/15205436.2015.1066014 T...

For a given order R in an imaginary quadratic field K, we study the specialization of the set CM(R) of Heegner points on the Shimura curve X =X0(D,N) at primes p | DN . As we show, if p does not divide the conductor of R, a point P ∈ CM(R) specializes to a singular point (resp. a connected component) of the special fiber X̃ of X at p if p ramifies (resp. does not ramify) in K. Exploiting the mod...

Abstract In [20], Rohrlich proved a modular analog of Jensen’s formula. Under certain conditions, the Rohrlich–Jensen formula expresses an integral log-norm $\log \Vert f $ ${\mathrm {PSL}}(2,{\mathbb {Z}})$ form in terms Dedekind Delta function evaluated at divisor . [2], authors re-interpreted as evaluating regularized inner product and extended result to compute with what amounts powers Haup...