Generalizing and unifying prior results, we solve the subconvexity problem for the $L$-functions of $\GL_{1}$ and $\GL_{2}$ automorphic representations over a fixed number field, uniformly in all aspects. A novel feature of the present method is the softness of our arguments; this is largely due to a consistent use of canonically normalized period relations, such as those supplied by the work of Waldspurger and Ichino--Ikeda.
We prove the subconvexity conjecture for sup-norms of automorphic forms for congruence subgroups of SL(n, Z) that satisfy the Ramanujan conjecture at infinity.
From a spectral identity we obtain asymptotics with error term for the second integral moments of families of automorphic L-functions for GL(2) over an arbitrary number field according to twists by idele characters with arbitrary ramification at a fixed finite place. The power-saving in the error term breaks convexity at this non-archimedean place.
We prove a new (conditional) result towards the subconvexity problem for certain automorphic $L$-functions for $\mathrm{GL}_2 \times \mathrm{GL}_3$. This follows from the computation of new $\mathrm{SL}_2$-period integrals associated with newforms $f$ and $g$ of even weight and odd squarefree level. The same computations also lead to a central value formula for degree $6$ complex $L$-series of the form $L(f \otimes \mathrm{Ad}(g), s)$, extending previous work in arXiv:1804.02352.
A non-symmetric reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q and primitive central character twisted by the l-th Hecke eigenvalue as a twisted mixed moment of automorphic L-functions of level l and trivial central character. As an application, uniform subconvexity bounds for L-functions in the level and the eigenvalue aspect are derived.
We introduce a ``geometric'' method to bound periods of automorphic forms. The key features of this method are the use of equidistribution results in place of mean value theorems, and the systematic use of mixing and the spectral gap. Applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of CM points; to subconvexity of the triple product period in the level aspect over number fields, which implies subconvexity for certain standard and Rankin-Selberg $L$-functions; and to bounding Fourier coefficients of automorphic forms.
We describe a new method to obtain weak subconvexity bounds for $L$-functions with mild hypotheses on the size of the Dirichlet coefficients. We verify these hypotheses for all automorphic $L$-functions and (with mild restrictions) the Rankin-Selberg $L$-functions attached to two automorphic representations. The proof relies on a new unconditional log-free zero density estimate for Rankin-Selberg $L$-functions.
In this paper we establish a very flexible and explicit Voronoi summation formula. This is then used to prove an almost Weyl strength subconvexity result for automorphic $L$-functions of degree two in the depth aspect. That is, looking at twists by characters of prime power conductor. This is the natural $p$-adic analogue to the well studied $t$-aspect.
We study the subconvexity problem for $GL_{3}(R)$ $L$-functions in the t-aspect using integral representations by combining techniques employed by Michel-Venkatesh in their study of the corresponding problem for $GL_{2}$ with ideas from recent works of Munshi, Holowinsky-Nelson and Lin. Our main objective is to give - from the perspective of integral representations of $L$-functions and automorphic representation theory - a possible explanation of the origin of the "key identity" arising in the latter series of works.
We improve the error term in the asymptotic formula for the twisted fourth moment of automorphic L functions of prime level and weight two proved by Kowalski, Michel and Vanderkam. As a consequence, we obtain a new subconvexity bound in the level aspect and improve the lower bound on proportion of simultaneous non-vanishing.
Let $π$ be a $SL(3,\mathbb Z)$ automorphic form. Let $χ=χ_1χ_2$ be a Dirichlet character with $χ_i$ primitive modulo $M_i$. Suppose $M_1$, $M_2$ are primes such that $\sqrt{M_2}M^{4δ}<M_1<M_2M^{-3δ}$, where $M=M_1M_2$ and $0<δ<1/28$. In this paper we will prove the following subconvex bound $$ L(\t1/2,π\otimesχ)\ll_{π,\varepsilon} M^{3/4}-δ+\varepsilon}. $$
Let $q \in \mathbb{Z} [i]$ be prime and $χ$ be the primitive quadratic Hecke character modulo $q$. Let $π$ be a self-dual Hecke automorphic cusp form for $\mathrm{SL}_3 (\mathbb{Z} [i] )$ and $f$ be a Hecke cusp form for $Γ_0 (q) \subset \mathrm{SL}_2 (\mathbb{Z} [i])$. Consider the twisted $L$-functions $ L (s, π\otimes f \otimes χ) $ and $L (s, π\otimes χ)$ on $\mathrm{GL}_3 \times \mathrm{GL}_2$ and $\mathrm{GL}_3$. We prove the subconvexity bounds \begin{equation*} L \big(\tfrac 1 2, π\otimes f \otimes χ\big) \ll_{\, \varepsilon, π, f } \mathrm{N} (q)^{5/4 + \varepsilon}, L \big(\tfrac 1 2…
We describe a new method to estimate the trilinear period on automorphic representations of PGL(2,R). Such a period gives rise to a special value of the triple L-function. We prove a bound for the triple period which amounts to a subconvexity bound for the corresponding special value of the triple L-function. Our method is based on the study of the analytic structure of the corresponding unique trilinear functional on unitary representations of PGL(2,R).
We describe a new method to estimate the trilinear period on automorphic representations of PGL(2,R). Such a period gives rise to a special value of the triple L-function. We prove a bound for the triple period which amounts to a subconvexity bound for the corresponding special value. Our method is based on the study of the analytic structure of the corresponding unique trilinear functional on unitary representations of PGL(2,R).
A reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q twisted by the ell-th Hecke eigenvalue as the fourth moment of automorphic L-functions of level ell twisted by the q-th Hecke eigenvalue. Direct corollaries include subconvexity bounds for L-functions in the level aspect and a short proof of an upper bound for the fifth moment of automorphic L-functions.
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 …
Let f be a cusp form for the group SL(3, Z) with Langlands parameter mu and associated L-function L(s, f). If mu is in generic position, i.e. away from the Weyl chamber walls and away from the self-dual forms, we prove the subconvexity bound L(1/2, f) << || mu || ^{3/4 - 1/120000}.
Fix $g$ a self-dual Hecke-Maass form for $SL_3(\mathbb{Z})$. Let $f$ be a holomorphic newform of prime level $q$ and fixed weight. Conditional on a lower bound for a short sum of squares of Fourier coefficients of $f$, we prove a subconvexity bound in the $q$ aspect for $L(s, g\times f)$ at the central point.
A simple proof of the classical subconvexity bound $ζ(1/2+it) \ll_εt^{1/6+ε}$ for the Riemann zeta-function is given, and estimation by more refined techniques is discussed. The connections between the Dirichlet divisor problem and the mean square of $|ζ(1/2+it)|$ are analysed.
We extend the results of Watson, which link quantum unique ergodicity on arithmetic hyperbolic surfaces with subconvexity for the triple product L function, to the case of arithmetic hyperbolic three manifolds. We work with the full unitary dual of SL(2,C), and consider QUE for automorphic forms of arbitrary fixed weight and growing spectral parameter. We obtain our results by constructing microlocal lifts of nonspherical automorphic forms using representation theory, and quantifying the generalised triple product formula of Ichino in the case of complex places.
Let $π$ be a cuspidal automorphic representation of $PGL_2(\mathbb{A}_\mathbb{Q})$ of arithmetic conductor $C$ and archimedean parameter $T$, and let $ϕ$ be an $L^2$-normalized automorphic form in the space of $π$. The sup-norm problem asks for bounds on $\| ϕ\|_\infty$ in terms of $C$ and $T$. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the $L^2$-mass $|ϕ|^2 (g) \, d g$ of $ϕ$. All previous work on these problems in the conductor-aspect has focused on the case that $ϕ$ is a newform. In this work, we study these problems for a class of automorphic forms that a…
Let $π$ be a cuspidal automorphic representation of $\operatorname{GL}_2(\mathbb{A}_{\mathbb{Q}})$ and $d$ be a fundamental discriminant. Hoffstein and Kontorovich ask for a bound on the least $|d|$ (if it exists) such that the central value $L(1/2, π\otimes χ_d) \neq 0$. The bound should be given in terms of the weight, Laplace eigenvalue and/or level of $π$. Let $f$ be a holomorphic twist-minimal newform of even weight $\ell$, odd cubefree level $N$, and trivial nebentypus. When $π\cong π_f$ and the squarefree part of $N$ is of appropriate size, we conditionally improve upon level aspect res…
When $s\ge k\ge 3$ and $n_1,\ldots ,n_k$ are large natural numbers, denote by $A_{s,k}(\mathbf n)$ the number of solutions in non-negative integers $\mathbf x$ to the system \[ x_1^j+\ldots +x_s^j=n_j\quad (1\le j\le k). \] Under appropriate local solubility conditions on $\mathbf n$, we obtain an asymptotic formula for $A_{s,k}(\mathbf n)$ when $s\ge k(k+1)$. This establishes a local-global principle in the Hilbert-Kamke problem at the convexity barrier. Our arguments involve minor arc estimates going beyond square-root cancellation.
The thesis gave a fine study on the distribution of the coefficients of automorphic L-functions for GL(m) with m>1. In particular we have treated two types of problems: change of signs of these coefficients (when they are real) and their decompensation on the prime numbers. Chapter 2 is devoted to the results known concerning the first problem in the case of classical modular forms and to explain our motivation of research. In Chapter 3, we treated the case of primitive Maass forms and obtained an estimate of the subconvexity-type for the least integer n such that the n-th coefficient is negat…
Modifying a method of Jutila, we prove a t aspect subconvexity estimate for L-functions associated to primitive holomorphic cusp forms of arbitrary level that is of comparable strength to Good's bound for the full modular group, thus resolving a problem that has been open for 35 years. A key innovation in our proof is a general form of Voronoi summation that applies to all fractions, even when the level is not squarefree.
We study a double Dirichlet series of the form $\sum_{d}L(s,χ_{d}χ)χ'(d)d^{-w}$, where $χ$ and $χ'$ are quadratic Dirichlet characters with prime conductors $N$ and $M$ respectively. A functional equation group isomorphic to the dihedral group of order 6 continues the function meromorphically to $\mathbb{C}^{2}$. A convexity bound at the central point is established to be $(MN)^{3/8+\varepsilon}$ and a subconvexity bound of $(MN(M+N))^{1/6+\varepsilon}$ is proven. The developed theory is used to prove an upper bound for the smallest positive integer $d$ such that $L(1/2,χ_{dN})$ does not vanis…
We use the relative trace formula to obtain exact formulas for central values of certain twisted quadratic base change L-functions averaged over Hilbert modular forms of a fixed weight and level. We apply these formulas to the subconvexity problem for these L-functions. We also establish an equidistribution result for the Hecke eigenvalues weighted by these L-values.
A strategy to address the inverse Galois problem over Q consists of exploiting the knowledge of Galois representations attached to certain automorphic forms. More precisely, if such forms are carefully chosen, they provide compatible systems of Galois representations satisfying some desired properties, e.g. properties that reflect on the image of the members of the system. In this article we survey some results obtained using this strategy.
We apply some ideas of Bombieri and Garrett to construct natural self-adjoint operators on spaces of automorphic forms whose only possible discrete spectrum is $λ_{s}$ for $s$ in a subset of on-line zeros of an $L$-function, appearing as a compact period of cuspidal-data Eisenstein series on $GL_{4}$. These ideas have their origins in results of Hejhal and Colin de Verdiére. In parallel with the $GL(2)$ case, the corresponding pair-correlation and triple-correlation results limit the fraction of on-the-line zeros that can appear in this fashion.
Let $ϕ$ be an $L^2$-normalized Hecke--Maaß cusp form for $\mathrm{PGL}_n(\mathbb{Z}[i])$ on the locally symmetric space $X:=\mathrm{PGL}_n(\mathbb{Z}[i])\backslash \mathrm{PGL}_n(\mathbb{C}) / \mathrm{PU}_n$. If $Ω$ is a compact subset of $X$, then we prove the bound $\|ϕ|_Ω\|_{\infty}\ll_Ω λ_ϕ^{n(n-1)/4-δ}$ for some $δ>0$ depending only on $n$, where $λ_ϕ$ is the Laplace eigenvalue of $ϕ$.
The sup-norm problem in analytic number theory asks for the largest value taken by a given automorphic form. We observe that the function-field version of this problem can be reduced to the geometric problem of finding the largest dimension of the $i$th stalk cohomology group of a given Hecke eigensheaf at any point. This problem, in turn, can be reduced to the intersection-theoretic problem of bounding the "polar multiplicities" of the characteristic cycle of the Hecke eigensheaf, which in known cases is the nilpotent cone of the moduli space of Higgs bundles. We solve this problem for newfor…
This article is the third and last part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part proves the following new result for the inverse Galois problem for symplectic groups. For any even positive integer n and any positive integer d, PSp_n(F_{l^d}) or PGSp_n(F_{l^d}) occurs as a Galois group over the rational numbers for a positive density set of primes l. The result is obtained by showing the existence of a regular, algebraic, self-dual, cuspidal automorphic representation of GL_n(A_Q) with l…
Let $λ_ϕ(n)$ be the Fourier coefficients of a Hecke holomorphic or Hecke--Maass cusp form on ${\rm SL}_2(\mathbb Z)$, and $f$ be any multiplicative function that satisfies two mild hypotheses. We establish a non-trivial upper bound for the correlation $\sum_{n \leq X}f(n)λ_ϕ(n+h)$ uniformly in $0<|h|\ll X$. As applications, we consider some special cases, including $λ_π(n), \,μ(n)λ_π(n)$ and any divisor-bounded multiplicative function. Here $λ_π(n)$ denotes the $n$-th Dirichlet coefficient of $\text{GL}_m$ automorphic $L$-function $L(s,π)$ for an automorphic irreducible cuspidal representation…
We derive a Voronoi-type series approximation for the local weighted mean of an arithmetical function that is associated to Dirichlet series satisfying a functional equation with gamma factors. The series is exploited to study the oscillation frequency with a method of Heath-Brown and Tsang [7]. A by-product is another proof for the well-known result of no element in the Selberg class of degree 0 \textless{} d \textless{} 1. Our major applications include the sign-change problem of the coefficients of automorphic L-functions for GL m , which improves significantly some results of Liu and Wu [1…
We study two closely related problems stemming from the random wave conjecture for Maass forms. The first problem is bounding the $L^4$-norm of a Maass form in the large eigenvalue limit; we complete the work of Spinu to show that the $L^4$-norm of an Eisenstein series $E(z,1/2+it_g)$ restricted to compact sets is bounded by $\sqrt{\log t_g}$. The second problem is quantum unique ergodicity in shrinking sets; we show that by averaging over the centre of hyperbolic balls in $Γ\backslash \mathbb{H}$, quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the …
We obtain uniform lower bounds, true for all automorphic L-functions L(s) associated to cuspidal representations of GL(m,A) where A denotes the adeles of the rationals Q, of the integral on the vertical line (Re(s)=1/2) of the absolute value squared of L(s)/s; and also of L(s)/(s-s_0) when s_0 is a zero of the L-function on the critical line. Several variants are also obtained in small degrees m, for the vertical integrals at different abscissas in the critical strip. For the estimates required to prove convergence, we are led to generalise a result of Friedlander-Iwaniec (Can. J. Math. 57,200…
Let $G$ be a group and $H$ be a subgroup of $G$. The classical branching rule (or symmetry breaking) asks: For an irreducible representation $π$ of $G$, determine the occurrence of an irreducible representation $σ$ of $H$ in the restriction of $π$ to $H$. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation $σ$ of $H$, find an irreducible representation $π$ of $G$ such that $σ$ occurs in the restriction of $π$ to $H$. For automorphic representations of classical groups, the branching problem has been addressed by the well-known globa…
Langlands' beyond endoscopy proposal for establishing functoriality motivates interesting and concrete problems in the representation theory of algebraic groups. We study these problems in a setting related to the Langlands $L$-functions $L(s,π,\,\otimes^3),$ where $π$ is a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_F)$ where $F$ is a global field.
Let $τ$ denote the divisor function, and $f$ be any multiplicative function that satisfies some mild hypotheses. We establish the asymptotic formula or non-trivial upper bound for the shifted convolution sum $\sum_{n \leq X}f(n)τ(n-1)$. We also derive several applications to multiplicative functions in the automorphic context, including the functions $λ_π(n), \,μ(n)λ_π(n)$ and $λ_ϕ(n)^l$. Here $λ_π(n)$ denotes the $n$-th Dirichlet coefficient of $\text{GL}_m$ automorphic $L$-function $L(s,π)$ for an automorphic irreducible cuspidal representation $π$, $λ_ϕ(n)$ denotes the $n$-th Fourier coeffi…
We prove, for many cuspidal automorphic representations for GSp(4), that the local obstructions to the deformation theory of the associated residual Galois representations generically vanish.
We discuss the theory of automorphic descents of Bessel type and its relation to automorphic version of branching problem and its relevant reciprocal branching problem.
Let $m$ be an integer greater than three and $\ell$ be an odd prime. In this paper, we prove that at least one of the following groups: $\mbox{P}Ω^\pm_{2m}(\mathbb{F}_{\ell^s})$, $\mbox{PSO}^\pm_{2m}(\mathbb{F}_{\ell^s})$, $\mbox{PO}_{2m}^\pm(\mathbb{F}_{\ell^s})$ or $\mbox{PGO}^\pm_{2m}(\mathbb{F}_{\ell^s})$ is a Galois group of $\mathbb{Q}$ for infinitely many integers $s > 0$. This is achieved by making use of a slight modification of a group theory result of Khare, Larsen and Savin, and previous results of the author on the images of the Galois representations attached to cuspidal automorp…
