We complement and offer a new perspective of the proof of a Motohashi-type formula relating the fourth moment of $L$-functions for $\mathrm{GL}_1$ with the third moment of $L$-functions for $\mathrm{GL}_2$ over number fields, studied earlier by Michel-Venkatesh and Nelson. Our main tool is a new type of pre-trace formula with test functions on $\mathrm{M}_2(\mathbb{A})$ instead of $\mathrm{GL}_2(\mathbb{A})$, on whose spectral side the matrix coefficients are replaced by the standard Godement-Jacquet zeta integrals. This is also a generalization of Bruggeman-Motohashi's other proof of Motohashi's formula. We give a variation of our method in the case of division quaternion algebras instead of $\mathrm{M}_2$, yielding a new spectral reciprocity, for which we are not sure if it is within the period formalism given by Michel-Venkatesh. We also indicate a further possible generalization, which seems to be beyond what the period method can offer.
We derive a Motohashi-type formula for the cubic moment of central values of $L$-functions of level $q$ cusp forms twisted by quadratic characters of conductor $q$, previously studied by Conrey and Iwaniec and Young. Corollaries of this formula include Weyl-subconvex bounds for $L$-functions of weight two cusp forms twisted by quadratic characters, and estimates towards the Ramanujan-Petersson conjecture for Fourier coefficients of weight 3/2 cusp forms.
We prove an analogue of Selberg's explicit formula for Motohashi's product (see arXiv:1104.1358v3 [math.NT]). We also provide a zero-density theorem for the product, which follows from Soundararajan's theorem for moments of the Riemann zeta-function on the critical line.
Motohashi established an explicit identity between the fourth moment of the Riemann zeta function weighted by some test function and a spectral cubic moment of automorphic L-functions. By an entirely different method, we prove a generalization of this formula to a fourth moment of Dirichlet L-functions modulo q weighted by a non-archimedean test function. This establishes a new reciprocity formula. As an application, we obtain sharp upper bounds for the fourth moment twisted by the square of a Dirichlet polynomial of length q^{1/4}. An auxiliary result of independent interest is a sharp upper …
We establish a Weyl-type subconvexity of $L(\tfrac{1}{2},f)$ for spherical Hilbert newforms $f$ with level ideal $\mathfrak{N}^2$, in which $\mathfrak{N}$ is required to be cube-free, and at any prime ideal $\mathfrak{p}$ with $\mathfrak{p}^2 \mid \mathfrak{N}$ the local representation generated by $f$ is not supercuspidal. The proof exploits a distributional version of Motohashi's formula over number fields developed by the first author, as well as Katz's work on hypergeometric sums over finite fields in the language of $\ell$-adic cohomology.
We establish a relative trace formula on $\mathrm{GL}(n+1)$ weighted by cusp forms on $\mathrm{GL}(n)$ over number fields. The spectral side is a weighted average of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over the \textit{full spectrum}, and the geometric side consists of Rankin-Selberg $L$-functions for $\mathrm{GL}(n)\times\mathrm{GL}(n),$ and certain explicit holomorphic functions. The formula yields new results towards central $L$-values for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ (over number fields): the second moment evaluation, and simultaneous nonvnaish…
Let $E/\mathbb{Q}$ be a semistable elliptic curve such that $\mathrm{ord}_{s=1}L(E,s) = 1$. We prove the $p$-part of the Birch and Swinnerton-Dyer formula for $E/\mathbb{Q}$ for each prime $p \geq 5$ of good reduction such that $E[p]$ is irreducible: $$ \mathrm{ord}_p \left (\frac{L'(E,1)}{Ω_E\cdot\mathrm{Reg}(E/\mathbb{Q})} \right ) = \mathrm{ord}_p \left (\#\mathrm{Sha}(E/\mathbb{Q})\prod_{\ell\leq \infty} c_\ell(E/\mathbb{Q}) \right ). $$ This formula also holds for $p=3$ provided $a_p(E)=0$ if $E$ has supersingular reduction at $p$.
Let $E/F$ be an extension of number fields with $\mathrm{Gal}(E/F)$ simple and nonabelian. In [G] the first named author suggested an approach to nonsolvable base change and descent of automorphic representations of $\mathrm{GL}_2$ along such an extension. Motivated by this we prove a trace formula whose spectral side is a weighted sum over cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A}_E)$ that are isomorphic to their $\mathrm{Gal}(E/F)$-conjugates.
In the 80s, Zagier and Jacquet-Zagier tried to derive the Selberg trace formula by applying the Rankin-Selberg method to the automorphic kernel function. Their derivation was incomplete due to a puzzle of the computation of a residue. We solve this puzzle and complete the derivation. The main input is an extension of the theory of regularized integrals invented by Zagier, which is of independent interest.
We stratify the $\mathrm{SL}_3$ big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. Thus the $\mathrm{SL}_3$ long word Kloosterman sum is decomposed into finer parts, and we write it as a finite sum of a product of two classical Kloosterman sums. The fine Kloosterman sums end up being the correct pieces to consider in the Bruggeman-Kuznetsov trace formula on the congruence subgroup $Γ_0(N)\subseteq \mathrm{SL}_3(\mathbb{Z})$. Another application is a new explicit formula, expressing the triple divisor sum functi…
Let $F$ be a number field and let $i\geq 2$ be an integer. In this paper, we study the positive étale wild kernel $\mathrm{WK}^{\mbox{ét},+}_{2i-2}F$, which is the twisted analogue of the $2$-primary part of the narrow class group. If $E/F$ is a Galois extension of number fields with Galois group $G$, we prove a genus formula relating the order of the groups $ (\mathrm{WK}^{\mbox{ét},+}_{2i-2}E)_{G}$ and $\mathrm{WK}^{\mbox{ét},+}_{2i-2}F$.
We prove a cohomological formula for non-critical residues of degree eight automorphic $L$-functions of $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ in the spirit of Beilinson conjecture. We rely on the cohomological interpretation of an automorphic period integral and on the study of Novodvorsky's integral representation of the $L$-functions.
We give a new formula for the Chebotarev densities of Frobenius elements in Galois groups. This formula is given in terms of smallest prime factors $p_{\mathrm{min}}(n)$ of integers $n\geq2$. More precisely, let $C$ be a conjugacy class of the Galois group of some finite Galois extension $K$ of $\mathbb{Q}$. Then we prove that $$-\lim_{X\rightarrow\infty}\sum_{\substack{2\leq n\leq X\\[1pt]\left[\frac{K/\mathbb{Q}}{p_{\mathrm{min}}(n)}\right]=C}}\frac{μ(n)}{n}=\frac{\#C}{\#G}.$$ This theorem is a generalization of a result of Alladi from 1977 that asserts that largest prime divisors $p_{\mathr…
We study a special case of Kashio's $p$-adic $\logΓ$-function, that we call ${\mathrm{Log}}Γ_{\mathit{p}}$, which combines these of Morita and Diamond. It agrees with each of these on large parts of its domain and has the advantage of being a locally analytic function. We prove a distribution formula for ${\mathrm{Log}}Γ_{\mathit{p}}$ which generalizes and links the known distribution formulas for Diamond's and Morita's functions.
We prove an explicit central value formula for a family of complex $L$-series of degree $6$ for $\mathrm{GL}_2 \times \mathrm{GL}_3$ which arise as factors of certain Garret--Rankin triple product $L$-series associated with modular forms. Our result generalizes a previous formula of Ichino involving Saito--Kurokawa lifts, and as an application, we prove Deligne's conjecture stating the algebraicity of the central values of the considered $L$-series up to the relevant periods.
In this paper, we establish an asymptotic formula with an effective bound on the error term for the Andrews smallest parts function $\mathrm{spt}(n)$. We use this formula to prove recent conjectures of Chen concerning inequalities which involve the partition function $p(n)$ and $\mathrm{spt}(n)$. Further, we strengthen one of the conjectures, and prove that for every $ε>0$ there is an effectively computable constant $N(ε) > 0$ such that for all $n\geq N(ε)$, we have \begin{equation*} \frac{\sqrt{6}}π\sqrt{n}\,p(n)<\mathrm{spt}(n)<\left(\frac{\sqrt{6}}π+ε\right) \sqrt{n}\,p(n). \end{equation*} …
We prove an asymptotic formula for the number of $n$-dimensional representations of the group $\mathrm{SU}(3)$. Main tools for the proof are Wright's Circle Method and the Saddle Point Method.
We give a two variable Rankin-Selberg integral inspired by consideration of Garrett's pullback formula. For a globally generic cusp form on $\mathrm{GL}_2\times \mathrm{GSp}_4$, the integral represents the product of the $\mathrm{Std}\times \mathrm{Spin}$ and $\mathbf{1} \times \mathrm{Std}$ $L$-functions. We prove a result concerning an Archimedean principal series representation in order to verify a case of Jiang's first-term identity relating certain non-Siegel Eisenstein series on symplectic groups. Using it, we obtain a new proof of a known result concerning possible poles of these $L$-fu…
We establish an explicit Plancherel decomposition for $\mathrm{GL}_n(F)\backslash \mathrm{GL}_n(E)$ where $E/F$ is a quadratic extension of local fields of characteristic zero by making use of a local functional equation for Asai $γ$-factors. We also give two applications of this Plancherel formula: first to the global Ichino-Ikeda conjecture for unitary groups by completing a comparison between local relative characters that was left open by W. Zhang and secondly to the Hiraga-Ichino-Ikeda conjecture on formal degrees in the case of unitary groups.
On the Gross-Prasad conjecture with its refinement for $\left(\mathrm{SO}\left(5\right),\mathrm{SO}\left(2\right)\right)$ and the generalized Böcherer conjecture
Topologically twisted $\mathcal{N} = 4$ super Yang-Mills theory has a partition function that counts Euler numbers of instanton moduli spaces. On the manifold $\mathbb{P}^2$ and with gauge group $\mathrm{U}(3)$ this partition function has a holomorphic anomaly which makes it a mock modular form of depth two. We employ the Circle Method to find a Rademacher expansion for the Fourier coefficients of this partition function. This is the first example of the use of Circle Method for a mock modular form of a higher depth.
Let F be a number field and $\widetilde{\mathrm{Sp}}(2n)$ be the metaplectic covering of Weil of $\mathrm{Sp}(2n, \mathbb{A}_F)$. We stabilize the elliptic semi-simple terms of the genuine part of trace formula for $\widetilde{\mathrm{Sp}}(2n)$, by using the transfer of test functions and the fundamental lemma established in a previous article. We obtain therefore an expression in terms of stable orbital integrals along certain $F$-elliptic classes in $\mathrm{SO}(2n'+1) \times \mathrm{SO}(2n''+1)$, which we call equisingular, for pairs $(n', n'')$ satisfying $n'+n''=n$. A simple formula for t…
We give new explicit formulas for Grassmannian and Aomoto polylogarithms in terms of iterated integrals, for arbitrary weight. We also explicitly reduce the Grassmannian polylogarithm in weight 4 and in weight 5 each to depth 2. Furthermore, using this reduction in weight 4 we obtain an explicit, albeit complicated, form of the so-called 4-ratio, which gives an expression for the Borel class in continuous cohomology of $\mathrm{GL}_4$ in terms of $\mathrm{Li}_4$.
We show that Fourier coefficients of automorphic forms attached to minimal or next-to-minimal automorphic representations of ${\mathrm{SL}}_n(\mathbb{A})$ are completely determined by certain highly degenerate Whittaker coefficients. We give an explicit formula for the Fourier expansion, analogously to the Piatetski-Shapiro-Shalika formula. In addition, we derive expressions for Fourier coefficients associated to all maximal parabolic subgroups. These results have potential applications for scattering amplitudes in string theory.
For a split reductive group $G$ over a number field $k$, let $ρ$ be an $n$-dimensional complex representation of its complex dual group $G^\vee(\mathbb{C})$. For any irreducible cuspidal automorphic representation $σ$ of $G(\mathbb{A})$, where $\mathbb{A}$ is the ring of adeles of $k$, in \cite{JL21}, the authors introduce the $(σ,ρ)$-Schwartz space $\mathcal{S}_{σ,ρ}(\mathbb{A}^\times)$ and $(σ,ρ)$-Fourier operator $\mathcal{F}_{σ,ρ}$, and study the $(σ,ρ,ψ)$-Poisson summation formula on $\mathrm{GL}_1$, under the assumption that the local Langlands functoriality holds for the pair $(G,ρ)$ at…
