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1. Problems

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


      Add remark

      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
          1. Duke-Friedlander-Iwaniec’s delta method
          2. Trivial delta method
          3. GL(2) Petersson trace formula as a delta method
          4. 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:
                  1. Trivial delta method,
                  2. 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


                                                          Add remark

                                                          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


                                                              Add remark

                                                              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.

                                                                      Add remark

                                                                      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:

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

                                                                          2. 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


                                                                              Add remark

                                                                              Problem 1.38.

                                                                              [org.aimpl.user:keshav.aggarwal@gmail.com] 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.