* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

Proceedings of the American Mathematical Society

Introduction to modular forms

Combinatory Analysis. By Major P. A. MacMohan. Vol. I. Pp. xx + 300. 15s. net. (Cam. Univ. Press)

Introduction To Elliptic Curves And Modular Forms

In this paper we define a new type of modular object and construct explicit examples of such functions. Our functions are closely related to cusp forms constructed by Zagier [37] which played an important role in the construction by Kohnen and Zagier [26] of a kernel function for the Shimura and Shintani lifts between half-integral and integral weight cusp forms. Although our functions share many properties in common with harmonic weak Maass forms, they also have some properties which strikingly contrast those exhibited by harmonic weak Maass forms. As a first application of the new theory developed in this paper, one obtains a new proof of the fact that the even periods of Zagier’s cusp forms are rational as an easy corollary.

http://www.mi.uni-koeln.de/Bringmann/qpoinc_IMRN_publish.pdf

Locally Harmonic Maass Forms and the Kernel of the Shintani Lift

In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous Ramanujan form x 2 + y 2 + 1Oz 2 ; equivalently the form 2x 2 + 5y 2 +4T z represents all integers greater than 1359, where T z denotes the triangular number z(z + 1)/2. Given positive integers a, b, c we employ modular forms and the theory of quadratic forms to determine completely when the general form ax 2 + by 2 + cT z represents sufficiently large integers and to establish similar results for the forms ax 2 + bT y + cT z and aT x + bT y + cT z . Here are some consequences of our main theorems: (i) All sufficiently large odd numbers have the form 2αx 2 + y 2 + z 2 if and only if all prime divisors of a are congruent to 1 modulo 4. (ii) The form αx 2 + y 2 + T z is almost universal (i.e., it represents sufficiently large integers) if and only if each odd prime divisor of a is congruent to 1 or 3 modulo 8. (iii) αx 2 +T y + T z is almost universal if and only if all odd prime divisors of a are congruent to 1 modulo 4. (iv) When υ 2 (α) ≠ 3, the form αT x + T y + Tz is almost universal if and only if all odd prime divisors of α are congruent to 1 modulo 4 and υ 2 (α) ≠ 5, 7, ..., where υ 2 (a) is the 2-adic order of α.

/pdf/on-almost-universal-mixed-sums-of-squares-and-triangular-34y1j5p2uq.pdf

On almost universal mixed sums of squares and triangular numbers

On divisors of modular forms

The first two authors and Kohnen have recently introduced a new class of modular objects called locally harmonic Maass forms, which are annihilated almost everywhere by the hyperbolic Laplacian operator. In this paper, we realize these locally harmonic Maass forms as theta lifts of harmonic weak Maass forms. Using the theory of theta lifts, we then construct examples of (non-harmonic) local Maass forms, which are instead eigenfunctions of the hyperbolic Laplacian almost everywhere.

/pdf/theta-lifts-and-local-maass-forms-3vm82izgjs.pdf

Theta lifts and local Maass forms

Andrews, Dyson, and Hickerson showed that 2 q-hypergeometric series, going back to Ramanujan, are related to real quadratic fields, which explains interesting properties of their Fourier coefficients. There is also an interesting relation of such series to automorphic forms. Here we construct more such examples arising from interesting combinatorial statistics.

/pdf/multiplicative-q-hypergeometric-series-arising-from-real-11nu38s6az.pdf

Ben Kane

Papers

Locally Harmonic Maass Forms and the Kernel of the Shintani Lift

On almost universal mixed sums of squares and triangular numbers

On divisors of modular forms

Theta lifts and local Maass forms

MULTIPLICATIVE q-HYPERGEOMETRIC SERIES ARISING FROM REAL QUADRATIC FIELDS