Delta symbols and the subconvexity problem

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1. Problems

Obtain cancellations in additive twists of GL(3) Fourier coefficients

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Problem 1.02.
[Matthew Young] Let $\pi$ be a GL(3) Hecke–Maass cusp form with the n-th Fourier coefficient $\lambda_\pi(1,n)$ . For $\alpha\in\mathbb{R}$ , prove a bound of the form $\sum_{n\ll N} \lambda_\pi(1,n) e(n\alpha) \ll_\pi N^{1-\delta},$ for some $\delta>0$ using a delta method. Aim to improve upon Miller’s bound.
Shifted convolution sum via various delta methods

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Problem 1.04.
[Roman Holowinsky]
For a holomorphic or a Hecke–Maass cusp form $f$ with the nth Fourier coefficient $\lambda_f(n)$ , obtain cancellations in the summation $\sum_{n\leq N} \lambda_f(n)\lambda_f(n+1),$ via
Duke-Friedlander-Iwaniec’s delta method
Trivial delta method
GL(2) Petersson trace formula as a delta method
GL(3) Kuznetsov trace formula as a delta method.

Note: We do not yet have an explicit GL(3) Kuznetsov trace formula with arbitrary level and nebentypus.
Level aspect subconvexity for GL(2)

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Problem 1.06.
[Roman Holowinsky] Use a delta method to obtain level aspect subconvexity for GL(2) L-functions.
Relationship between Trivial delta method and Nelson’s Kernel method

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Problem 1.08.
[Paul Nelson]
Let $f$ be a holomorphic or Hecke–Maass cusp form of level $1$ and $\chi$ be a Dirichlet character of level $p$ . For fixed $f$ , a level aspect Burgess-type bound for $L(f\times\chi, 1/2)$ can be obtained by (among many others) these two methods:
Trivial delta method,
Using the Kernel $K(z,w) = \sum_{\gamma\in\Gamma_0(p)} \omega(z^{-1}\gamma w)$ for $z, x\in \mathbb{H}$ and $\omega$ a smooth bump function, defined appropriately.
Find a precise relationship between the two methods.
Character sums appearing in the works of Nelson, and Sharma

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Problem 1.1.
[Paul Nelson] Relate or compare the exponential sums appearing in the work of Paul Nelson [cite], and the work of Prahlad Sharma [cite] in the proof for level aspect subconvexity bound for $\rm GL(1)$ twists of $\rm GL(3)\times GL(2)$ $L$ -functions.
Subconvexity bound for trace function twists

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Problem 1.12.
[Philippe Michel] Subconvexity bounds for $\rm GL(1)$ twists of a fixed $\rm GL(2)\times GL(2)$ $L$ -function has been proved using a delta symbol by C. Raju [cite] in the archimedean aspect, and by A. Ghosh [cite] in the level aspect. Can we generalize these techniques to replace the $\rm GL(1)$ twist by a trace function, or an appropriate archimedean analog?
Subconvexity for $\rm GL(3)\times GL(1)$ via averaging over a family

Let $\pi$ be a fixed Hecke-Maass cusp form for $\rm SL_3(\mathbb{Z})$ , and $\chi$ be a character of level $M$ . Munshi [cite] used Petersson trace formula as a delta method to obtain a subconvexity bound for $L(1/2, \pi\times\chi)$ as $M$ varies.

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Problem 1.14.
[Rizwan Khan] Can we obtain a subconvexity bound by averaging over a family, e.g. by bounding $\sum_{\chi\bmod M} |L(1/2, \chi)|^2.$
Application to Quantum Unique Ergodicity

Holowinsky and Soundararajan proved the Quantum Unique Ergodicity conjecture in the case of holomorphic Hecke eigenforms by combining two different approaches: By proving a non-trivial bound on certain shifted convolution sum, and by proving (weak) subconvexity bound for certain $L$ -functions.

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Problem 1.16.
[Roman Holowinsky] Obtaining a bound of the form $\sum_{f,g \in \mathcal{B}_k} L(1/2, sym^2f) L(1/2, sym^2g)\big|\sum_{n\leq k} \lambda_f(n)\lambda_g(n+h)\big|^2 \ll k^{4-\delta},$ for $|h|\ll 1$ and some $\delta>0$ would imply the result of Holowinsky and Soundararajan (conditionally on non-negativity of the central values $L(1/2, sym^2f)$ ).

Use a delta method in order to prove the above bound.
$GL(3)$ Kuznetsov trace formula in the level aspect

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Problem 1.18.
[Jack Buttcane] Develop a Kuznetsov trace formula for automorphic forms for the group $\Gamma_0(N) = \left\lbrace \begin{pmatrix} \star & \star & \star \\ \star & \star & \star \\ 0 & 0 & \star \end{pmatrix} \bmod N \right\rbrace$ for any integer $N\geq1$ .

For the purpose of applications, develop a Kuznetsov trace formula for automorphic forms for the group $\Gamma_B(N) = \left\lbrace \begin{pmatrix} \star & \star & \star \\ 0 & \star & \star \\ 0 & 0 & \star \end{pmatrix} \bmod N \right\rbrace$ .
Subconvexity bound for Rankin-Selberg $L$ -functions

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Problem 1.2.
[Paul Nelson] Obtain a subconvexity bound for a $\rm GL_n\times GL_m$ Rankin–Selberg $L$ -function when $n$ and $m$ are not coprime.

The simplest first cases would be obtaining a subconvexity bound (in any aspect) for a $\rm GL_4\times GL_2$ $L$ -function or a $\rm GL_3\times GL_3$ $L$ -function.
Cancellations in additive twists on average

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Problem 1.22.
[Xiaoqing Li] Let $\alpha\in\mathbb{R}$ and $f_j$ be a Maass form for $\rm SL(4, \mathbb{Z})$ with eigenvalue $t_j$ and Fourier coefficients $\{\lambda_j(1,1,n)\}_{n\geq1}$ . For parameters $T$ and $N$ , prove a non-trivial bound for the averaged sum $\sum_{t_j\sim T} \bigg|\sum_{n\leq N}\lambda_j(1,1,n)e(n\alpha) \bigg|^2.$

If $\{f_j\}$ are Maass forms for $\rm SL(3, \mathbb{Z})$ improve Miller’s bound on average over the spectral parameter $t_j$ .
Asymptotic for second moment of $\rm GL_2\times GL_2$ in level aspect

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Problem 1.24.
[Roman Holowinsky] Let $\mathcal{B}_k(p)$ be a basis for holomorphic or Hecke-Maass forms for $\Gamma_0(p)\subset SL_2(\mathbb{Z})$ of weight $k$ . Obtain an asymptotic with a power saving error term for the second moment $\sum_{f\in\mathcal{B}_k(p)}\sum_{g\in\mathcal{B}_\ell(q)} |L(1/2, f\times g)|^2,$ as the level $p$ and $q$ become larger, the weights $k$ and $\ell$ are fixed.
Subconvexity bound for $\rm GL_3\times GL_2$ in level aspect

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Problem 1.26.
[Ritabrata Munshi] Let $\pi$ be a fixed Hecke-Maass cusp form for $SL_3(\mathbb{Z})$ and $f$ be a holomorphic for Hecke Maass cusp form for $\Gamma_0(M)$ . Obtain a subconvexity bound estimate for $L(1/2, \pi\times f)$ as $M$ grows.
Shifted convolution sum for $\rm GL_3\times GL_2$ in level aspect

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Problem 1.28.
[Roman Holowinsky] Let $\pi$ be a fixed Hecke-Maass cusp form for $SL_3(\mathbb{Z})$ with Fourier coefficients $\{\lambda_\pi(r,n)\}$ and $f$ be a holomorphic for Hecke Maass cusp form for $\Gamma_0(M)$ with Fourier coefficients $\{\lambda_f(n)\}$ . Obtain a non-trivial bound for the shifted sum, $\begin{equation*} \sum_{n<N}\lambda_\pi(r,n)\lambda_f(n+1), \end{equation*}$ in the convexity range $N\sim M^{3/2}$ .
$\rm GL_3$ Kuznetsov trace formula as a $\delta$ -method

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Problem 1.3.
[Ritabrata Munshi] Obtain $t$ -aspect subconvexity for a $\rm GL_2$ $L$ -function by using $\rm GL_3$ Kuznetsov trace formula as a $\delta$ -method.
Delta method over Number fields

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Problem 1.32.
[Matthew Young] Design a delta symbol method over number fields in order to generalize the classical techniques to the number fields setting.
Strong hybrid bounds for $L(1/2, f\times \chi)$

Let $f$ be a holomorphic or Hecke-Maass cusp form of level $p$ , and $\chi$ be a primitive Dirchlet character of level $q$ . Conrey–Iwaniec obtained a Weyl type subconvexity bound for $L(1/2, f\times\chi)$ when $p=q$ , while Khan has obtained a Burgess-type bound when $p\asymp q$ , and Yang has proved a Burgess-type bound over number fields in a similar range.

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Problem 1.34.
[Matthew Young] Prove strong hybrid bounds for other ranges of $p$ and $q$ .
Subconvexity bound $\rm GL_2$ in the ‘conductor dropping’ case

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Problem 1.36.
[Matthew Young]
Let $f$ be a holomorphic or Hecke-Maass cusp form for $\rm GL_2$ . Use a $\delta$ -method to prove the following subconvexity bounds for the following $L$ -functions:

Level aspect subconvexity bound for $L(1/2, f)$ where $f$ has level $p$ prime (and may have non-trivial nebentypus).

Subconvexity for $L(1/2+it_f, f)$ , where $f$ is a Maass form of level $1$ and spectral parameter $t_f$ .
Bound for short second moment of $\rm GL_4$ $L$ -functions

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Problem 1.38.
[Keshav Aggarwal] Let $f$ be a fixed Hecke-Maass cusp form for $\rm GL_4(\mathbb{Z})$ , $T$ and $\Delta$ be parameters. Prove a non-trivial bound for $\int_T^{T+\Delta} |L(\frac12+it, f)|^2 dt,$ for $\Delta = T^\delta$ with some $\delta<1$ .
Applications of trace formula towards point counting

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Problem 1.4.
[Anurag Sahay] Various circle methods and $\delta$ -methods have been successfully used to count points on varieties. Could one use Petersson or Kuznetsov trace formula as a $\delta$ -method to count point on varieties?
Hardy-type theorem for $\rm GL_3$ $L$ -functions

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Problem 1.42.
Use a $\delta$ -method to prove that there are infinitely many critical zeros for a $\rm GL_3(\mathbb{Z})$ $L$ -functions.

Cite this as: AimPL: Delta symbols and the subconvexity problem , available at http://aimpl.org/deltasubconvex2.

Delta symbols and the subconvexity problem

1. Problems

Obtain cancellations in additive twists of GL(3) Fourier coefficients

Problem 1.02.

Shifted convolution sum via various delta methods

Problem 1.04.

Level aspect subconvexity for GL(2)

Problem 1.06.

Relationship between Trivial delta method and Nelson’s Kernel method

Problem 1.08.

Character sums appearing in the works of Nelson, and Sharma

Problem 1.1.

Subconvexity bound for trace function twists

Problem 1.12.

Subconvexity for \rm GL(3)\times GL(1)\rm GL(3)\times GL(1) via averaging over a family

Problem 1.14.

Application to Quantum Unique Ergodicity

Problem 1.16.

GL(3)GL(3) Kuznetsov trace formula in the level aspect

Problem 1.18.

Subconvexity bound for Rankin-Selberg LL-functions

Problem 1.2.

Cancellations in additive twists on average

Problem 1.22.

Asymptotic for second moment of \rm GL_2\times GL_2\rm GL_2\times GL_2 in level aspect

Problem 1.24.

Subconvexity bound for \rm GL_3\times GL_2\rm GL_3\times GL_2 in level aspect

Problem 1.26.

Shifted convolution sum for \rm GL_3\times GL_2\rm GL_3\times GL_2 in level aspect

Problem 1.28.

\rm GL_3\rm GL_3 Kuznetsov trace formula as a \delta\delta-method

Problem 1.3.

Delta method over Number fields

Problem 1.32.

Strong hybrid bounds for L(1/2, f\times \chi)L(1/2, f\times \chi)

Problem 1.34.

Subconvexity bound \rm GL_2\rm GL_2 in the ‘conductor dropping’ case

Problem 1.36.

Bound for short second moment of \rm GL_4\rm GL_4 LL-functions

Problem 1.38.

Applications of trace formula towards point counting

Problem 1.4.

Hardy-type theorem for \rm GL_3\rm GL_3 LL-functions

Problem 1.42.

Subconvexity for $\rm GL(3)\times GL(1)$ via averaging over a family

$GL(3)$ Kuznetsov trace formula in the level aspect

Subconvexity bound for Rankin-Selberg $L$ -functions

Asymptotic for second moment of $\rm GL_2\times GL_2$ in level aspect

Subconvexity bound for $\rm GL_3\times GL_2$ in level aspect

Shifted convolution sum for $\rm GL_3\times GL_2$ in level aspect

$\rm GL_3$ Kuznetsov trace formula as a $\delta$ -method

Strong hybrid bounds for $L(1/2, f\times \chi)$

Subconvexity bound $\rm GL_2$ in the ‘conductor dropping’ case

Bound for short second moment of $\rm GL_4$ $L$ -functions

Hardy-type theorem for $\rm GL_3$ $L$ -functions