No description available.

Let Mk(∞) (Gamma, nu) denote the space of weight k weakly holomorphic weight modular forms with poles only at the cusp (∞), and let widehat Mk(∞) (Gamma, nu) subseteq Mk(∞) (Gamma, nu) denote the space of weight k weakly holomorphic modular forms in Mk(∞) (Gamma, nu) which vanish at every cusp other than (∞). We construct canonical bases for these spaces in terms of Maass--Poincaré series, and show that the coefficients of these bases satisfy Zagier duality.

The primary arithmetic information attached to a Maass cusp form is its Laplace eigenvalue. However, in the case of cuspidal Maass forms, the range that these eigenvalues can take is not well-understood. In particular it is unknown if, given a real number r, one can prove that there exists a primitive Maass cusp form with Laplace eigenvalue 1/4 + r2. Conversely, given the Fourier coefficients of a primitive Maass cusp form f on Γ0(D), it is not clear whether or not one can determine its Laplace eigenvalue. In this paper we show that given only a finite number of Fourier coefficients one can first determine the level D, and then compute the Laplace eigenvalue to arbitrarily high precision. The key to our results will be understanding the resonance and rapid decay properties of Maass cusp forms. Let f be a primitive Maass cusp form with Fourier coefficients λf (n). The resonance sum for f is given by SX(f;α;β) = Εn≥1λf(n)‑Φ(n/X) e(αnβ) where φ ∈ Cc∞((1, 2)) is a Schwartz function and α ∈ R and β, X > 0 are real numbers. Sums of this form have been studied for many different classes of functions f, including holomorphic modular forms for SL(2, Z), and Maass cusp forms for SL(n,Z). In this paper we take f to be a primitive Maass cusp form for a congruence subgroup Γ0(D) ⊂ SL(2, Z). Thus our result extends the family of automorphic forms for which their resonance properties are understood. Similar analysis and algorithms can be easily implemented for holomorphic cusp forms for Γ0(D). Our techniques include Voronoi summation, weighted exponential sums, and asymptotics expansions of Bessel functions. We then use these estimates in a new application of resonance sums. In particular we show that given only limited information about a Maass cusp form f (in particular a finite list of high Fourier coefficients), one can determine its level and estimate its spectral parameter, and thus its Laplace eigenvalue. This is done using a large parallel computing cluster running MATLAB and Mathematica

Ramanujan introduced his now celebrated mock theta functions in 1920, grouping them into families parameterized by an integer called the order. In 2010 Bringmann and Ono discovered generalizations of Ramanujan's mock theta functions for any order relatively prime to 6; this result was later strengthened by Garvan in 2016. It was also shown that by adding suitable nonholomorphic completion terms to the mock theta functions the family of mock theta functions corresponding to a given order constitute a complex vector space which is closed under the action of the modular group. We strengthen the Bringmann, Ono, and Garvan result by constructing a vector-valued modular form of weight 1/2 transforming according the Weil representation for orders greater than 3 by introducing an algorithm which simultaneously numerically constructs the form and proves its transformation laws. We also explicitly construct the 7th order form and prove analytically that it has the proper modular transformations. It is conjectured the same method will apply for other orders.