Ben Kane

On almost universal mixed sums of squares and triangular numbers

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.

Multiplicative q-hypergeometric series arising from real quadratic fields

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.

Duality and Differential Operators for Harmonic Maass Forms

Due to the graded ring nature of classical modular forms, there are many interesting relations between the coefficients of different modular forms. We discuss additional relations arising from Duality, Borcherds products, theta lifts. Using the explicit description of a lift for weakly holomorphic forms, we realize the differential operator \({D}^{k-1} := {( \frac{1} {2\pi \mathrm{i}} \frac{\partial } {\partial z})}^{k-1}\) acting on a harmonic Maass form for integers k > 2 in terms of \({\xi }_{2-k} := 2\mathrm{i}{y}^{2-k}\overline{ \frac{\partial } {\partial \overline{z}}}\) acting on a different form. Using this interpretation, we compute the image of D k − 1. We also answer a question arising in recent work on the p-adic properties of mock modular forms. Additionally, since such lifts are defined up to a weakly holomorphic form, we demonstrate how to construct a canonical lift from holomorphic modular forms to harmonic Maass forms.

REPRESENTATIONS OF INTEGERS BY TERNARY QUADRATIC FORMS

We investigate the representation of integers by quadratic forms whose theta series lie in Kohnens plus space , where p is a prime. Conditional upon certain GRH hypotheses, we show effectively that every sufficiently large discriminant with bounded divisibility by p is represented by the form, up to local conditions. We give an algorithm for explicitly calculating the bounds. For small p, we then use a computer to find the full list of all discriminants not represented by the form. Finally, conditional upon GRH for L-functions of weight 2 newforms, we give an algorithm for computing the implied constant of the Ramanujan–Petersson conjecture for weight 3/2 cusp forms of level 4N in Kohnens plus space with N odd and squarefree.

On a completed generating function of locally harmonic Maass forms

While investigating the Doi-Naganuma lift, Zagier defined integral weight cusp forms

Mock modular forms as

f_D

p

which are naturally defined in terms of binary quadratic forms of discriminant

-adic modular forms

D

A problem of Petersson about weight 0 meromorphic modular forms

. It was later determined by Kohnen and Zagier that the generating function for the

Equidistribution of Heegner points and ternary quadratic forms

f_D