Tag Archives: automorphic forms

Notes from a Talk at Building Bridges 4

On 18 July 2018 I gave a talk at the 4th Building Bridges Automorphic Forms Workshop, which is hosted at the Renyi Institute in Budapest, Hungary this year. In this talk, I spoke about counting points on hyperboloids, with a certain focus on counting points on the three dimensional hyperboloid

$$\begin{equation} X^2 + Y^2 = Z^2 + h \end{equation}$$

for any fixed integer $h$.

I gave a similar talk at the 32nd Automorphic Forms Workshop in Tufts in March. I don’t say this during my talk, but a big reason for giving these talks is to continue to inspire me to finish the corresponding paper. (There are still a couple of rough edges that need some attention).

The methodology for the result relies on the spectral expansion of half-integral weight modular forms. This is unfriendly to those unfamiliar with the subject, and particularly mysterious to students. But there is a nice connection to a topic discussed by Arpad Toth during the previous week’s associated summer school.

Arpad sketched a proof of the spectral decomposition of holomorphic modular cusp forms on $\Gamma = \mathrm{SL}(2, \mathbb{Z})$. He showed that
$$\begin{equation} L^2(\Gamma \backslash \mathcal{H}) = \textrm{cuspidal} \oplus \textrm{Eisenstein}, \tag{1}
where the cuspidal contribution comes from Maass forms and the Eisenstein contribution comes from line integrals against Eisenstein series.

The typical Eisenstein series $$\begin{equation} E(z, s) = \sum_{\gamma \in \Gamma_\infty \backslash \Gamma} \textrm{Im}(\gamma z)^s \end{equation}$$ only converges for $\mathrm{Re}(s) > 1$, and the initial decomposition in $(1)$ implicitly has $s$ in this range.

To write down the integrals appearing in the Eisenstein spectrum explicitly, one normally shifts the line of integration to $1/2$. As Arpad explained, classically this produces a pole at $s = 1$ (which is the constant function).

In half-integral weight, the Eisenstein series has a pole at $s = 3/4$, with the standard theta function

$$\begin{equation} \theta(z) = \sum_{n \in \mathbb{Z}} e^{2 \pi i n^2 z} \end{equation}$$

as the residue. (More precisely, it’s a constant times $y^{1/4} \theta(z)$, or a related theta function for $\Gamma_0(N)$). I refer to this portion of the spectrum as the residual spectrum, since it comes from often-forgotten residues of Eisenstein series. Thus the spectral decomposition for half-integral weight objects is a bit more complicated than the normal case.

When giving talks involving half-integral weight spectral expansions to audiences including non-experts, I usually omit description of this. But for those who attended the summer school, it’s possible to at least recognize where these additional terms come from.

The slides for this talk are available here.

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Paper: The Second Moments of Sums of Fourier Coefficients of Cusp Forms

This is joint work with Thomas Hulse, Chan Ieong Kuan, and Alex Walker.

We have just uploaded a paper to the arXiv on the second moment of sums of Fourier coefficients of cusp forms. This is the first in a trio of papers that we will be uploading and submitting in the near future.

Suppose $latex {f(z)}$ and $latex {g(z)}$ are weight $latex {k}$ holomorphic cusp forms on $latex {\text{GL}_2}$ with Fourier expansions

$$\begin{align} f(z) &= \sum_{n \geq 1} a(n) e(nz) \\
g(z) &= \sum_{n \geq 1} b(n) e(nz). \end{align}$$

Denote the sum of the first $latex {n}$ coefficients of a cusp form $latex {f}$ by $$ S_f(n) := \sum_{m \leq n} a(m). \tag{1}$$

We consider upper bounds for the second moment of $latex {S_f(n)}$.

The famous Ramanujan-Petersson conjecture gives us that $latex {a(n)\ll n^{\frac{k-1}{2} + \epsilon}}$. So one might assume $latex {S_f(X) \ll X^{\frac{k-1}{2} + 1 + \epsilon}}$. However, we expect the better bound $$ S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{4} + \epsilon}, \tag{2}$$

which we refer to as the “Classical Conjecture,” echoing Hafner and Ivić [HI].

Chandrasekharan and Narasimhan [CN] proved that the Classical Conjecture is true on average by showing that $$ \sum_{n \leq X} \lvert S_f(n) \rvert^2 = CX^{k- 1 + \frac{3}{2}} + B(X), \tag{3}$$

where $latex {B(x)}$ is an error term, $$ B(X) = \begin{cases} O(X^{k}\log^2(X)) \ \Omega\left(X^{k – \frac{1}{4}}\frac{(\log \log \log X)^3}{\log X}\right), \end{cases} \tag{4}$$

and $latex {C}$ is the constant, $$ C = \frac{1}{(4k + 2)\pi^2} \sum_{n \geq 1}\frac{\lvert a(n) \rvert^2}{n^{k + \frac{1}{2}}}. \tag{5}$$

A application of the Cauchy-Schwarz inequality to~(3) leads to the on-average statement that $$ \frac{1}{X} \sum_{n \leq X} |S_f(n)| \ll X^{\frac{k-1}{2} + \frac{1}{4}}. \tag{6}$$

From this, [HI] were able to show in some cases that $$ S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{3}}. \tag{7}$$

Better lower bounds are known for $latex {B(X)}$. In the same work [HI] improved the lower bound of [CN] for full-integral weight forms of level one and showed that $$ B(X) = \Omega\left(X^{k – \frac{1}{4}}\exp\left(D \tfrac{(\log \log x )^{1/4}}{(\log \log \log x)^{3/4}}\right)\right), \tag{8}$$

for a particular constant $latex {D}$.

The question of better understanding $latex {B(X)}$ is analogous to understanding the error term in the circle problem or divisor problem. In our paper, we introduce the Dirichlet series $$D(s, S_f \times S_g) := \sum_{n \geq 1} \frac{S_f(n) \overline{S_g(n)}}{n^{s + k – 1}}$$

D(s, S_f \times \overline{S_g}) &:= \sum_{n \geq 1} \frac{S_f(n)S_g(n)}{n^{s + k – 1}} and provide their meromorphic continuations. From our review of the literature, these Dirichlet series and their meromorphic continuations are new and provide new approaches to the classical problems related to $latex {S_f(n)}$.

Our primary result is the meromorphic continuation of $latex {D(s, S_f \times S_g)}$. As a first application, we prove a smoothed generalization to~(3).

Theorem 1 Suppose either that $latex {f = g}$ is a Hecke eigenform or that $latex {f}$ and $latex {g}$ have real coefficients. \begin{equation*} \frac{1}{X} \sum_{n \geq 1}\frac{S_f(n)\overline{S_g(n)}}{n^{k – 1}}e^{-n/X} = CX^{\frac{1}{2}} + O_{f,g,\epsilon}(X^{-\frac{1}{2} + \theta + \epsilon}) \end{equation*} where \begin{equation*} C = \frac{\Gamma(\tfrac{3}{2}) }{4\pi^2} \frac{L(\frac{3}{2}, f\times g)}{\zeta(3)}= \frac{\Gamma(\tfrac{3}{2})}{4\pi ^2} \sum_{n \geq 1} \frac{a(n)\overline{b(n)}}{n^{k + \frac{1}{2}}}, \end{equation*} and $latex {\theta}$ denotes progress towards Selberg’s Eigenvalue Conjecture. Similarly, \begin{equation*} \frac{1}{X} \sum_{n \geq 1}\frac{S_f(n)S_g(n)}{n^{k – 1}}e^{-n/X} = C’X^{\frac{1}{2}} + O_{f,g,\epsilon}(X^{-\frac{1}{2} + \theta + \epsilon}), \end{equation*} where \begin{equation*} C’ = \frac{\Gamma(\tfrac{3}{2})}{4\pi^2} \frac{L(\frac{3}{2}, f\times \overline{g})}{\zeta(3)} = \frac{\Gamma(\tfrac{3}{2})}{4\pi ^2} \sum_{n \geq 1} \frac{a(n)b(n)}{n^{k + \frac{1}{2}}}.\end{equation*}

We have a complete meromorphic continuation, and it would not be hard to give additional terms in the asymptotic. But the next terms come from zeroes of the zeta function and are complicated to nail down exactly.

Choosing $latex {f = g}$, we recover a proof of the Classical Conjecture on Average. More interestingly, we show that the secondary growth terms do not arise from a pole, nor are there prescribed polar reasons for growth. The secondary growth in the classical result comes from choosing a sharp cutoff instead of the nicely behaving and natural smooth cutoffs.

We prove analogous results for sums of normalized Fourier coefficients $$ S_f^\alpha(n) := \sum_{m \leq n} \frac{a(m)}{m^\alpha} \tag{9}$$

for $latex {0 \leq \alpha < k}$.

In the path to proving these results, we explicitly demonstrate remarkable cancellation between Rankin-Selberg convolution $latex {L}$-functions $latex {L(s, f\times g)}$ and shifted convolution sums $$ Z(s, 0; f,g) := \sum_{n, h} \frac{a(n)\overline{b(n-h)}}{n^{s + k – 1}}. \tag{10}$$

Comparing our results and methodologies with the main results of [CN] guarantees similar cancellation for general level and general weight, including half-integral weight forms.

We provide additional applications of the meromorphic continuation of $latex {D(s, S_f \times S_g)}$ in forthcoming works, which will be uploaded to the arXiv and described briefly here soon.

For exact references, see the paper.

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