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 newforms on $GL_2 (\mathbb A_{\mathbb F_q(t)})$ of squarefree level, leading to bounds on the sup-norm that are stronger than what is known in the analogous problem for newforms on $GL_2(\mathbb A_{\mathbb Q})$ (i.e. classical holomorphic and Maass modular forms.)
We show that if a positive integer $q$ has $s(q)$ odd prime divisors $p$ for which $p^2$ divides $q$, then a positive proportion of the Laplacian eigenvalues of Maass newforms of weight $0$, level $q$, and principal character occur with multiplicity at least $2^{s(q)}$. Consequently, the new part of the cuspidal spectrum of the Laplacian on $Γ_0(q) \backslash \mathbb{H}$ cannot be simple for any odd non-squarefree integer $q$. This generalises work of Stromberg, who proved this for $q = 9$ by different methods.
Let $p\geq 5$ be prime. For elliptic modular forms of weight 2 and level $Γ_0(N)$ where $N>6$ is squarefree, we bound the depth of Eisenstein congruences modulo $p$ (from below) by a generalized Bernoulli number with correction factors and show how this depth detects the local non-principality of the Eisenstein ideal. We then use admissibility results of Ribet and Yoo to give an infinite class of examples where the Eisenstein ideal is not locally principal. Lastly, we illustrate these results with explicit computations and give an interesting commutative algebra application related to Hilbert-…
We show that there are only finitely many nonconstant reflective automorphic forms $Ψ$ on even lattices of squarefree level splitting two hyperbolic planes and give a complete classification in the case where the zeros of $Ψ$ are simple and $Ψ$ has singular weight.
Let f be a newform of weight at least 2 and squarefree level with Fourier coefficients in a number field K. We give explicit bounds, depending on congruences of f with other newforms, on the set of primes lambda of K for which the deformation problem associated to the mod lambda Galois representation of f is obstructed. We include some explicit examples.
We compute the space $S_2(K(N))$ of weight $2$ Siegel paramodular cusp forms of squarefree level $N<300$. In conformance with the paramodular conjecture of A. Brumer and K. Kramer, the space is only the additive (Gritsenko) lift space of the Jacobi cusp form space $J_{2,N}^{\text{cusp}}$ except for $N=249,295$, when it further contains one nonlift newform. For these two values of $N$, the Hasse-Weil $p$-Euler factors of a relevant abelian surface match the spin $p$-Euler factors of the nonlift newform for the first two primes $p\nmid N$.
We relate the study of Landau-Siegel zeros to the ranks of Jacobians $J_0(q)$ of modular curves for large primes $q$. By a conjecture of Brumer-Murty, the rank should be equal to half of the dimension. Equivalently, almost all newforms of weight two and level $q$ have analytic rank $\leq 1$. We show that either Landau-Siegel zeros do not exist, or that almost all such newforms have analytic rank $\leq 2$. In particular, almost all odd newforms have analytic rank equal to one. Additionally, for a sparse set of primes $q$ we show the rank of $J_0(q)$ is asymptotically equal to the rank predicted…
For a newform $f=\sum a_n q^n$ of weight $k \geq 3$ and a prime $λ$ of $\mathbf{Q}(a_n)$, the deformation problem for its associated mod $λ$ Galois representation is unobstructed for all primes outside some finite set. Previous results gave an explicit bound on this finite set for $f$ of squarefree level; we modify this bound and remove the squarefree hypothesis. We also show that if the $λ$-adic deformation problem for $f$ is unobstructed, then $f$ is not congruent mod $λ$ to a newform of lower level.
A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on $Γ_0(N)$ to a simpler function of $k$ and $N$, showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on $Γ_0(N)$. It follows that a hypothetical fast algorithm for computing the number of such automorphic repre…
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 $H^{\pm}_{2k} (N^3)$ denote the set of modular newforms of cubic level $N^3$, weight $2 k$, and root number $\pm 1$. For $N > 1$ squarefree and $k>1$, we use an analytic method to establish neat and explicit formulas for the difference $|H^{+}_{2k} (N^3)| - |H^{-}_{2k} (N^3)|$ as a multiple of the product of $φ(N)$ and the class number of $\mathbb{Q}(\sqrt{- N})$. In particular, the formulas exhibit a strict bias towards the root number $+1$. Our main tool is a root-number weighted simple Petersson formula for such newforms.
Let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing squarefree level q tending to infinity. We prove that the pushforward of the mass of f to the modular curve of level 1 equidistributes with respect to the Poincaré measure. Our result answers affirmatively the squarefree level case of a conjecture spelled out by Kowalski, Michel and Vanderkam (2002) in the spirit of a conjecture of Rudnick and Sarnak (1994). Our proof follows the strategy of Holowinsky and Soundararajan (2008) who show that newforms of level 1 and large weight have equidistributed mass. …
We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$ where $D$ is an indefinite quaternion division algebra over $\mathbb{Q}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times θ_χ)$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $θ_χ$ is an (essentially fixed) automorphic form on $\mathrm{GL}_2$ obtained as the theta lift of a Hecke character $χ$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-na…
Iwaniec and Sarnak showed that at the minimum 25% of L-values associated to holomorphic newforms of fixed even integral weight and level $N \rightarrow \infty$ do not vanish at the critical point when N is square-free and $ϕ(N)\sim N$. In this paper we extend the given result to the case of prime power level $N=p^ν$, $ν\geq 2$.
This paper studies the Fourier expansion of Hecke-Maass eigenforms for $GL(2, \mathbb Q)$ of arbitrary weight, level, and character at various cusps. Translating well known results in the theory of adelic automorphic representations into classical language, a multiplicative expression for the Fourier coefficients at any cusp is derived. In general, this expression involves Fourier coefficients at several different cusps. A sufficient condition for the existence of multiplicative relations among Fourier coefficients at a single cusp is given. It is shown that if the level is 4 times (or in some…
With the method of the relative trace formula and the classification of simple supercuspidal representations, we establish some Fourier trace formulas for automorphic forms on $PGL(2)$ of cubic level. As applications, we obtain a non-vanishing result for central $L$-values of holomorphic newforms and a weighted Weyl's law for Maass newforms.
Let $f$ be an $L^2$-normalized Hecke--Maass cuspidal newform of level $N$, character $χ$ and Laplace eigenvalue $λ$. Let $N_1$ denote the smallest integer such that $N|N_1^2$ and $N_0$ denote the largest integer such that $N_0^2 |N$. Let $M$ denote the conductor of $χ$ and define $M_1= M/\gcd(M,N_1)$. In this paper, we prove the bound $|f|_\infty$ $\ll_ε$ $N_0^{1/6 + ε} N_1^{1/3+ε} M_1^{1/2} λ^{5/24+ε}$, which generalizes and strengthens previously known upper bounds for $|f|_\infty$. This is the first time a hybrid bound (i.e., involving both $N$ and $λ$) has been established for $|f|_\infty$…
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.
Recently, the problem of bounding the sup norms of $L^2$-normalized cuspidal automorphic newforms $ϕ$ on $\text{GL}_2$ in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character $χ$ of $ϕ$ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general $χ$. If the level $N$ is a square, our result reduces to $$\|ϕ\|_\infty \ll N^{\frac14+ε},$$ at least under the Ramanujan Conjecture. In particular, when $χ$ has conductor $N$, this improves upon the previous best known bound…
In this paper we study the Fermat equation $x^n+y^n=z^n$ over quadratic fields $\mathbb{Q}(\sqrt{d})$ for squarefree $d$ with $26 \leq d \leq 97$. By studying quadratic points on the modular curves $X_0(N)$, $d$-regular primes, and working with Hecke operators on spaces of Hilbert newforms, we extend work of Freitas and Siksek to show that for most squarefree $d$ in this range there are no non-trivial solutions to this equation for $n \geq 4$.
We introduce a new trace formula of Kuznetsov type involving the central standard L-values and the Whittaker periods of cuspidal automorphic representations of PGL_n(Q) which are spherical at the archimedean place. As an application, we show a simultaneous non-vanishing of standard L-values at n-1 points on the critical strip for infinitely many Hecke-Maass cuspidal newforms.
Let f be a classical holomorphic newform of level q and even weight k. We show that the pushforward to the full level modular curve of the mass of f equidistributes as qk -> infinity. This generalizes known results in the case that q is squarefree. We obtain a power savings in the rate of equidistribution as q becomes sufficiently "powerful" (far away from being squarefree), and in particular in the "depth aspect" as q traverses the powers of a fixed prime. We compare the difficulty of such equidistribution problems to that of corresponding subconvexity problems by deriving explicit extensions…
Let f be an L^2-normalized Hecke--Maass cuspidal newform of level N and Laplace eigenvalue λ. It is shown that |f|_\infty <<_{λ, ε} N^{-1/12 + ε} for any ε>0. The exponent is further improved in the case when N is not divisible by "small squares". Our work extends and generalizes previously known results in the special case of N squarefree.
We refine known dimension formulas for spaces of cusp forms of squarefree level, determining the dimension of subspaces generated by newforms both with prescribed global root numbers and with prescribed local signs of Atkin-Lehner operators. This yields precise results on the distribution of signs of global functional equations and sign patterns of Atkin-Lehner eigenvalues, refining and generalizing earlier results of Iwaniec, Luo and Sarnak. In particular, we exhibit a strict bias towards root number +1 and a phenomenon that sign patterns are biased in the weight but perfectly equidistributed…
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…
Let $F$ be a totally real field, and $\mathbb{A}_F$ be the adele ring of $F$. Let us fix $N$ to be a positive integer. Let $π_1=\otimesπ_{1,v}$ and $π_2=\otimesπ_{2,v}$ be distinct cohomological cuspidal automorphic representations of $\mathrm{GL}_n(\mathbb{A}_{F})$ with levels less than or equal to $N$. Let $\mathcal{N}(π_1,π_2)$ be the minimum of the absolute norm of $v \nmid \infty$ such that $π_{1,v} \not \simeq π_{2,v}$ and that $π_{1,v}$ and $π_{2,v}$ are unramified. We prove that there exists a constant $C_N$ such that for every pair $π_1$ and $π_2$, $$\mathcal{N}(π_1,π_2) \leq C_N.$$ T…
The goal of this paper is to explain certain experimentally observed properties of the (cuspidal) spectrum and its associated automorphic forms (Maass waveforms) on the congruence subgroup $Γ_{0}(9)$. The first property is that the spectrum possesses multiplicities in the so-called new part, where it was previously believed to be simple. The second property is that the spectrum does not contain any "genuinely new" eigenvalues, in the sense that all eigenvalues of $Γ_{0}(9)$ appear in the spectrum of some congruence subgroup of lower level. The main theorem in this paper gives a precise decompo…
Let $\mathitΠ$ be a cohomological irreducible cuspidal automorphic representation of ${\rm GL}_2(\mathbb{A}_{\mathbb F})$ with central character $ω_{\mathitΠ}$ over a totally real number field ${\mathbb F}$. In this paper, we prove the algebraicity of the near central non-critical value of the symmetric fourth $L$-function of $\mathitΠ$ twisted by $ω_{\mathitΠ}^{-2}$. The algebraicity is expressed in terms of the Petersson norm of the normalized newform of $\mathitΠ$ and the top degree Whittaker period of the Gelbart-Jacquet lift ${\rm Sym}^2\mathitΠ$ of $\mathitΠ$.
We provide an explicit integral representation for L-functions of pairs (F,g) where F is a holomorphic genus 2 Siegel newform and g a holomorphic elliptic newform, both of squarefree levels and of equal weights. When F,g have level one, this was earlier known by the work of Furusawa. The extension is not straightforward. Our methods involve precise double-coset and volume computations as well as an explicit formula for the Bessel model for GSp(4) in the Steinberg case; the latter is possibly of independent interest. We apply our integral representation to prove an algebraicity result for a cri…
The question about modular forms have recently received a lot of attention; concerning the non-vanishing of automorphic L-functions Michel, Kowalski and Vanderkam proved (among others results) that there's positive proportion of non-vanishing of primitives forms at the critical point. This result was proved by these authors in the prime level case; on the othe hand, Iwaniec, Luo and Sarnak showed the same result for the squarefree level case. In order to understand the influence of the arithmetic shape of level forms in their vanishing, this paper studies a generalisation to the primitives for…
Let E/Q be a real quadratic field and pi_0 a cuspidal, irreducible, automorphic representation of GL(2,A_E) with trivial central character and infinity type (2,2n+2) for some non-negative integer n. We show that there exists a non-zero Siegel paramodular newform F with weight, level, Hecke eigenvalues, epsilon factor and L-function determined explicitly by pi_0. We tabulate these invariants in terms of those of pi_0 for every rational prime p.
The Katz-Sarnak density conjecture states that the scaling limits of the distributions of zeros of families of automorphic $L$-functions agree with the scaling limits of eigenvalue distributions of classical subgroups of the unitary groups $U(N)$. This conjecture is often tested by way of computing particular statistics, such as the one-level density, which evaluates a test function with compactly supported Fourier transform at normalized zeros near the central point. Iwaniec, Luo, and Sarnak studied the one-level densities of cuspidal newforms of weight $k$ and level $N$. They showed in the l…
The Manin constant $c$ of an elliptic curve $E$ over $\mathbb{Q}$ is the nonzero integer that scales the differential $ω_f$ determined by the normalized newform $f$ associated to $E$ into the pullback of a Néron differential under a minimal parametrization $ϕ\colon X_0(N)_{\mathbb{Q}} \twoheadrightarrow E$. Manin conjectured that $c = \pm 1$ for optimal parametrizations, and we prove that in general $c \mid \mathrm{deg}(ϕ)$ under a minor assumption at $2$ and $3$ that is not needed for cube-free $N$ or for parametrizations by $X_1(N)_{\mathbb{Q}}$. Since $c$ is supported at the additive reduct…
Let $D$ be an indefinite quaternion division algebra over $\mathbb{Q}$. We approach the problem of bounding the sup-norms of automorphic forms $ϕ$ on $D^\times(\mathbb{A})$ that belong to irreducible automorphic representations and transform via characters of unit groups of orders of $D$. We obtain a non-trivial upper bound for $\|ϕ\|_\infty$ in the level aspect that is valid for arbitrary orders. This generalizes and strengthens previously known upper bounds for $\|ϕ\|_\infty$ in the setting of newforms for Eichler orders. In the special case when the index of the order in a maximal order is …
Let $f$ be a newform of even weight $2κ$ for $D^\times$, where $D$ is a possibly split indefinite quaternion algebra over $\mathbb{Q}$. Let $K$ be a quadratic imaginary field splitting $D$ and $p$ an odd prime split in $K$. We extend our theory of $p$-adic measures attached to the power series expansions of $f$ around the Galois orbit of the CM point corresponding to an embedding $K\hookrightarrow D$ to forms with any nebentypus and to $p$ dividing the level of $f$. For the latter we restrict our considerations to CM points corresponding to test objects endowed with an arithmetic $p$-level str…
Let $q$ be a prime, $P \geq 1$ and let $N_q(P)$ denote the number of rational primes $p \leq P$ that split in the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$. The first part of this paper establishes various unconditional and conditional (under existence of a Siegel zero) lower bounds for $N_q(P)$ in the range $q^{1/4+\varepsilon} \leq P \leq q$, for any fixed $\varepsilon>0$. This improves upon what is implied by work of Pollack and Benli-Pollack. The second part of this paper is dedicated to proving an estimate for a bilinear form involving Weyl sums for modular square roots (equivalen…
