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We construct analogs of the classical Δ-function for quotients of the upper half plane 𝓗 by certain arithmetic triangle groups Γ coming from quaternion division algebras B. We also establish a relative integrality result concerning modular functions of the form Δ(αz)/Δ(z) for α in B⁺. We give two explicit examples at the end.

Analyse spectrale et prolongement analytique : séries d’Eisenstein, fonctions $z e t a$ et nombre de solutions d’équations diophantiennes

Gilles Lachaud (1982)

Annales scientifiques de l'Université de Clermont. Mathématiques

Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan.

Bruce C. Berndt (1978)

Journal für die reine und angewandte Mathematik

Arithmetic differential equations in several variables

Alexandru Buium, Santiago R. Simanca (2009)

Annales de l’institut Fourier

We survey recent work on arithmetic analogues of ordinary and partial differential equations.

Arithmetic of the modular function $j_{1, 4}$

Chang Heon Kim, Ja Kyung Koo (1998)

Acta Arithmetica

We find a generator $j_{1, 4}$ of the function field on the modular curve X₁(4) by means of classical theta functions θ₂ and θ₃, and estimate the normalized generator $N (j_{1, 4})$ which becomes the Thompson series of type 4C. With these modular functions we investigate some number theoretic properties.

Arithmétique de courbes modulaires

A.P. OGG (1974/1975)

Seminaire de Théorie des Nombres de Bordeaux

Arithmetisch ganze Differentiale der Modulfunktionenkörper 6. und 7. Stufe

Rolf Berndt, Klaus Schramm (1977)

Acta Arithmetica

Asymptotic formulae for partition ranks

Jehanne Dousse, Michael H. Mertens (2015)

Acta Arithmetica

Using an extension of Wright's version of the circle method, we obtain asymptotic formulae for partition ranks similar to formulae for partition cranks which where conjectured by F. Dyson and recently proved by the first author and K. Bringmann.

Asymptotic Formulae for the Coefficients of a Class of Modular Forms.

Gunnar Dirdal (1972)

Mathematica Scandinavica

Asymptotic formulas for the coefficients of certain automorphic functions

Jaban Meher, Karam Deo Shankhadhar (2015)

Acta Arithmetica

We derive asymptotic formulas for the coefficients of certain classes of weakly holomorphic Jacobi forms and weakly holomorphic modular forms (not necessarily of integral weight) without using the circle method. Then two applications of these formulas are given. Namely, we estimate the growth of the Fourier coefficients of two important weak Jacobi forms of index 1 and non-positive weights and obtain an asymptotic formula for the Fourier coefficients of the modular functions $θ^{k} / η^{l}$ for all integers k,l...

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