arXiv subjects
Publications and source records attributed to Wing Hong Leung.
We give a new proof of the converse theorem for Maass forms on ${\rm GL}(3)$ using a technique that is inspired by Langlands' philosophy of "beyond endoscopy", thereby implementing these ideas for the first time in a higher rank setting.
We establish power saving asymptotics for the sum of the divisor function along a binary quartic form, improving on work of Daniel. The proof involves an application of a recent two dimensional delta method due to Li, Rydin-Myerson, and Vishe and an exploitation of $\mathrm{GL}_2$ automorphic forms arising from the factorization of varying cubic Dedekind zeta functions.
We provide a power-saving bound for certain smoothed shifted convolution sums for Fourier coefficients of Siegel cusp forms. This result is the first nontrivial estimate for a shifted convolution sum with two cusp forms on a group of higher rank than $\GL_2$. Our approach is based on a novel automorphic reinterpretation of the delta method of Duke, Friedlander, and Iwaniec. The method reduces the problem to the estimation of Fourier coefficients of Siegel Poincare series, which is ultimately based on the Weil bound.
Let $A(1,m)$ be the Fourier coefficients of a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form $π_1$ and $λ(m)$ be those of a $SL(2,\mathbb{Z})$ Hecke holomorphic or Hecke-Mass cusp form $π_2$. Let $H\subset[\![ -X^{1-\varepsilon},X^{1+\varepsilon}]\!]$ and $\{a(h)\}_{h\in H}\subset\mathbb{C}$ be a sequence. We show that if $H\subset \ell+[\![ 0,X^{1/2+\varepsilon}]\!] $ for some $\ell\geq0$, \begin{align*} D_{a,H}(X):=\frac{1}{|H|}\sum_{h\in H}a(h)\sum_{m=1}^\infty A(1,m)λ(rm+h)V\left(\frac{m}{X}\right)\ll_{π_1,π_2,\varepsilon} \frac{X^{1+\varepsilon}}{|H|}\|a\|_2 \end{align*} for any $\varepsilon>0$, and a similar bound when $|H|$ is big. This improves Sun's bound and generalizes it to an average with arbitrary weights. Moreover, we demonstrate how one can recover the factorizable moduli structure given by the Jutila's circle method via studying a shifted sum with weighted average. This allows us to recover Munshi's bound on the shifted sum with a fixed shift without using the Jutila's circle method.
We use a trivial delta method with multiplicative characters for congruence detection to prove the Weyl bound for GL(2) in $t$-aspect for a holomorphic or Hecke-Maass cusp form of arbitrary level and nebentypus. This parallels the work of Aggarwal in 2018, with the difference being multiplicative character has a more natural connection to the twisted $L$-function. This provides another view point to understand and explore the trivial and other delta methods.
Let $π$ be a Hecke cusp form for $\mathrm{SL}_3(\mathbb{Z})$. We bound the second moment average of $L(s,π)$ over a short interval to obtain the subconvexity estimate $$ L(1/2+it, π) \ll_{π, \varepsilon} (1+|t|)^{3/4-1/8+\varepsilon}. $$
In 1992, Erd$ő$s and Hegyv$á$ri showed that for any prime p, there exist infinitely many length 3 weakly prime-additive numbers divisible by p. In 2018, Fang and Chen showed that for any positive integer m, there exists infinitely many length 3 weakly prime-additive numbers divisible by m if and only if 8 does not divide m. Under the assumption (*) of existence of a prime in certain arithmetic progression with prescribed primitive root, which is true under the Generalized Riemann Hypothesis (GRH), we show for any positive integer m, there exists infinitely many length 4 weakly prime-additive numbers divisible by m. We also present another related result analogous to the length 3 case shown by Fang and Chen.
We prove a novel four-variable character sum identity which serves as a twisted, non-archimedean counterpart to Weber's integrals for Bessel functions. Using this identity and ideas from Venkatesh's thesis, we present a new, spectral proof of the Voronoi formula for classical modular forms.
We present a "beyond-endoscopic" treatment of the functional equation for the standard $L$-function of a holomorphic cusp form with level and nebentypus. We use Petersson's formula and methods from Venkatesh's thesis and "spectral reciprocity".
Let $f$ be a newform of prime level $p$ with any central character $χ\, (\bmod\, p)$, and let $g$ be a fixed cusp form or Eisenstein series for $\hbox{SL}_{2}(\mathbb{Z})$. We prove the subconvexity bound: for any $\varepsilon>0$, \begin{align*} L(1/2, \, f \otimes g) \ll p^{1/2-1/524+\varepsilon}, \end{align*} where the implied constant depends on $g$, $\varepsilon$, and the archimedean parameter of $f$. This improves upon the previously best-known result by Harcos and Michel. Our method ultimately relies on non-trivial bounds for bilinear forms in Kloosterman fractions pioneered by Duke, Friedlander, and Iwaniec, with later innovations by Bettin and Chandee.
We obtain a strong bound on the second moment of the $GL_3$ standard $L$-function on the critical line. The method builds on the recent work of Aggarwal, Leung, and Munshi which treated shorter intervals. We deduce some corollaries including an improvement on the error term in the Rankin-Selberg problem, and on certain subconvexity bounds for $GL_3 \times GL_2$ and $GL_3$ $L$-functions. As a byproduct of the method of proof, we also obtain an estimate for an average of shifted convolution sums of $GL_3$ Fourier coefficients.
We bound non-linear additive twists of $\mathrm{GL}_{3}$ Hecke eigenvalues, improving upon the work of Kumar-Mallesham-Singh (2022). The proof employs the DFI circle method with standard manipulations (Voronoi, Cauchy-Schwarz, lengthening, and additive reciprocity). The main novelty includes the conductor lowering mechanism, albeit sacrificing some savings to remove an analytic oscillation, followed by the iteration ad infinitum of Cauchy-Schwarz and Poisson. The resulting character sums are estimated via the work of Adolphson-Sperber (1993). As an application, we prove nontrivial bounds for the first moment of $\mathrm{GL}_{3}$ Hardy's function, which corresponds to the cubic moment of Hardy's function studied by Ivić (2012).
Let $P$ be a prime and $k$ be an even integer. Let $f$ be a full level holomorphic cusp form of weight $k$ and $ρ$ be a primitive level $P$ holomorphic cusp form with arbitrary nebentypus and fixed weight $κ$. We prove a hybrid subconvexity bound for $L\left(\frac{1}{2},\mathrm{Sym}^2 f\otimes ρ\right)$ when $P^{\frac{1}{4}+η}<k<P^{\frac{21}{17}-η}$ for any $0<η<\frac{67}{136}$. This extends the range of $P$ and $k$ achieved by Holowinsky, Munshi and Qi. The result is established using a new variant of the delta method.
Let $k$ be an algebraically closed field of characteristic zero and let $H$ be a noetherian cocommutative Hopf algebra over $k$. We show that if $H$ has polynomially bounded growth then $H$ satisfies the Dixmier-Moeglin equivalence. That is, for every prime ideal $P$ in ${\rm Spec}(H)$ we have the equivalences $$P~{\rm primitive}\iff P~{\rm rational}\iff P ~{\rm locally~closed~in}~{\rm Spec}(H).$$ We observe that examples due to Lorenz show that this does not hold without the hypothesis that $H$ have polynomially bounded growth. We conjecture, more generally, that the Dixmier-Moeglin equivalence holds for all finitely generated complex noetherian Hopf algebras of polynomially bounded growth.