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We denote by S l ( N , ψ ) (resp. S k ( N , χ ) ) the space of cusp forms of weight l (resp. k) for Γ 0 ( N ) with nebentypus ψ (resp. χ). We give an arithmetic expression for the critical values of the Rankin–Selberg L-series D ( s , f ⊗ g ) associated with a pair ( f , g ) of modular forms for S L 2 ( Z ) . It is also shown that a cusp form g ∈ S l ( N , ψ ) is uniquely determined by certain critical values of the family of Rankin–Selberg L-series D ( s , f ⊗ g ) , where f runs over a fixed orthogonal basis of S k ( N , χ ) .

Iwaniec, Luo, and Sarnak proposed Hypothesis S and its generalization which predicts non-trivial bounds for a smooth sum of the product of an arithmetic sequence { a n } and a fractional exponential function. When a n is the Fourier coefficient λ f ( n ) of a fixed holomorphic cusp form f, however, a resonance phenomenon prohibits any improvement of the bound beyond a barrier. It is believed that this resonance barrier could be overcome when the weight k of f tends to infinity. The present paper is a first step toward this goal by proving non-trivial bounds for this sum when k and the summation length X both tend to infinity. No such non-trivial bounds are previously known if the form f is allowed to move. Similar bounds are also proved for linear phases and for Maass forms. The main technology is improved large sieve inequalities over a short interval.

We prove that for a Hecke cuspform f ∈ S k ( Γ 0 ( N ) , χ ) and a prime l > max ⁡ { k , 6 } such that l ∤ N , there exists an infinite family { k r } r ≥ 1 ⊆ Z such that for each k r , there is a cusp form f k r ∈ S k r ( Γ 0 ( N ) , χ ) such that the Deligne representation ρ f k r , l is a crystalline and potentially diagonalizable lift of ρ ‾ f , l . When f is l-ordinary, we base our proof on the theory of Hida families, while in the non-ordinary case, we adapt a local-to-global argument due to Khare and Wintenberger in the setting of their proof of Serre's modularity conjecture, together with a result on existence of lifts with prescribed local conditions over CM fields, a flatness result due to Bockle and a local dimension result by Kisin. We discuss the motivation and tentative future applications of our result in ongoing research on the automorphy of GL 2 n -type representations in the higher level case.

Let $$k\ge 2$$ be an even integer, and let M be a positive integer. We prove a hybrid subconvexity bound for $$L(1/2,\text {sym}^2 f\otimes \chi )$$ with the associated parameters k, M explicitly determined, where f is a primitive holomorphic cusp form of weight k and level 1, and $$\chi $$ is a primitive Dirichlet character modulo M satisfying $$1< k\le M^{1/26}$$ .

Suppose that f is a primitive Hecke eigenform or a Mass cusp form for Γ0(N) with normalized eigenvalues λf (n) and let X > 1 be a real number. We consider the sum $${{\cal S}_k}(X): = \sum\limits_{n < X} {\sum\limits_{n = {n_1},{n_2}, \ldots ,{n_k}} {{\lambda _f}({n_1}){\lambda _f}({n_2}) \ldots {\lambda _f}({n_k})}}$$ and show that $${{\cal S}_k}(X){\ll _{f,\varepsilon }}{X^{1 - 3/(2(k + 3)) + \varepsilon}}$$ for every k ⩾ 1 and e > 0. The same problem was considered for the case N = 1, that is for the full modular group in Lu (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for k ⩾ 5. Since the result is valid for arbitrary level, we obtain, as a corollary, estimates on sums of the form $${{\cal S}_k}(X)$$ , where the sum involves restricted coefficients of some suitable half integral weight modular forms.

We examine an arithmetical function defined by recursion relations on the sequence {f(pk)}k∈ℕ and obtain sufficient condition(s) for the sequence to change sign infinitely often. As an application we give criteria for infinitely many sign changes of Chebyshev polynomials and that of sequence formed by the Fourier coefficients of a cusp form.

Recently Luo and Zhou showed that for a fixed GL(2) Hecke-Maass cusp form, the natural density of primes at which the Satake parameters fail the Ramanujan Conjecture does not exceed 1/35. In this short note, we investigate the GL(3) case and obtain two similar (conditional) results.

We prove that given any $$\epsilon >0$$ , a non-zero adelic Hilbert cusp form $${\mathbf {f}}$$ of weight $$k=(k_1,k_2,\ldots ,k_n)\in ({\mathbb {Z}}_+)^n$$ and square-free level $$\mathfrak {n}$$ with Fourier coefficients $$C_{{\mathbf {f}}}(\mathfrak {m})$$ , there exists a square-free integral ideal $$\mathfrak {m}$$ with $$N(\mathfrak {m})\ll k_0^{3n+\epsilon }N(\mathfrak {n})^{\frac{6n^2+1}{2}+\epsilon }$$ such that $$C_{{\mathbf {f}}}(\mathfrak {m})\ne 0$$ . The implied constant depends on $$\epsilon , F$$ .

We consider the Fourier expansion of a Hecke (resp. Hecke–Maaß) cusp form of general level N at the various cusps of $$\Gamma _{0}(N)\backslash \mathbb {H}$$ Γ 0 ( N ) \ H . We explain how to compute these coefficients via the local theory of p-adic Whittaker functions and establish a classical Voronoï summation formula allowing an arbitrary additive twist. Our discussion has applications to bounding sums of Fourier coefficients and understanding the (generalised) Atkin–Lehner relations.

In this paper we prove that the first Fourier-Jacobi coefficient of a non-zero Siegel cusp form and a Hecke eigenform of degree 2, weight k, for S p 4 ( Z ) is not identically zero.

We obtain certain estimates for averages of shifted convolution sums involving the Fourier coefficients of a normalized Hecke–Maass eigenform and holomorphic cusp form.

We study the argument of L(s, f) associated with holomorphic cusp form in both weight aspect and t-aspect. We prove that for −1 ≤ σ ≤ 2 and t ≥ 4, argL(s, f) ≪ log kt, where s = σ + it. Assuming the generalized Riemann hypothesis (GRH), we have arg L(s, f) ≪ log kt/log log kt for σ ≥ 1/2 and t ≥ 4.

For a positive integer [Formula: see text], we say that [Formula: see text] is a Weierstrass point on the modular curve [Formula: see text] if there is a non-zero cusp form of weight [Formula: see text] on [Formula: see text] which vanishes at [Formula: see text] to order greater than the genus of [Formula: see text]. If [Formula: see text] is a prime with [Formula: see text], Ogg proved that [Formula: see text] is not a Weierstrass point on [Formula: see text] if the genus of [Formula: see text] is [Formula: see text]. We prove a similar result for even weights [Formula: see text]. We also study the space of weight [Formula: see text] cusp forms on [Formula: see text] vanishing to order greater than the dimension.

Let $$\lambda _{f}(n)$$ be the normalized Fourier coefficients of a holomorphic Hecke cusp form of full level. We study a generalized divisor problem with $$\lambda _{f}(n)$$ over some special sequences. More precisely, for any fixed integer $$k\ge 2$$ and $$j\in \{1,2,3,4\},$$ we are interested in the following sums $$\begin{aligned} S_{k}(x,j):=\sum _{n\le x}\lambda _{k,f}(n^{j})=\sum _{n\le x}\sum _{n=n_{1}n_{2}\cdots n_{k}}\lambda _{f}(n_{1}^{j})\lambda _{f}(n_{2}^{j})\cdots \lambda _{f}(n_{k}^{j}). \end{aligned}$$

Let $$S(t,F):=\pi ^{-1}\arg L\big (\frac{1}{2}+it,F\big ),$$ where F is a Hecke–Maass cusp form for $$\mathrm {SL}_3({\mathbb {Z}}).$$ We establish an asymptotic formula for the spectral moments of S(t, F), and obtain several other results on S(t, F).

Let $$\lambda _f (n)$$ denote the normalized n-th Fourier coefficient of a holomorphic Hecke eigencuspform or a Hecke–Maass cusp form for the full modular group. In this paper we shall exhibit cancellations in the following sum: $$\begin{aligned} \sum _{N

Let $g$ be a fixed Hecke cusp form for $\mathrm{SL}(2,\mathbb{Z})$ and $\chi$ be a primitive Dirichlet character of conductor $M$. The best known subconvex bound for $L(1/2,g\otimes \chi)$ is of Burgess strength. The bound was proved by a couple of methods: shifted convolution sums and the Petersson/Kuznetsov formula analysis. It is natural to ask what inputs are really needed to prove a Burgess-type bound on $\rm GL(2)$. In this paper, we give a new proof of the Burgess-type bounds ${L(1/2,g\otimes \chi)\ll_{g,\varepsilon} M^{1/2-1/8+\varepsilon}}$ and $L(1/2,\chi)\ll_{\varepsilon} M^{1/4-1/16+\varepsilon}$ that does not require the basic tools of the previous proofs and instead uses a trivial delta method., Comment: 17 pages; referee comments incorporated; to appear in the Ramanujan Journal

We prove an explicit integral representation—involving the pullback of a suitable Siegel Eisenstein series—for the twisted standard L-function associated to a holomorphic vector-valued Siegel cusp form of degree n and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to L-functions of vector-valued Siegel cusp forms. The key new ingredient in our method is a novel choice of local vectors at the archimedean place which allows us to exactly compute the archimedean local integral. By specializing our integral representation to the case $$n=2$$ n = 2 we are able to prove a reciprocity law—predicted by Deligne’s conjecture—for the critical values of the twisted standard L-function for vector-valued Siegel cusp forms of degree 2 and arbitrary level. This arithmetic application generalizes previously proved critical-value results for the full level case. By specializing further to the case of Siegel cusp forms obtained via the Ramakrishnan–Shahidi lift, we obtain a reciprocity law for the critical values of the symmetric fourth L-function of a classical newform.

We present a set of diagonal matrices which index enough Fourier coefficients for a complete characterization of all Siegel cusp forms of degree 2, weight k, level N and character χ, where k is an even integer ≥4, N is an odd, square-free positive integer, and χ has conductor equal to N. As an application, we show that the Koecher-Maass series of any F ∈ S k 2 twisted by the set of Maass waveforms whose eigenvalues are in the continuum spectrum of the hyperbolic Laplacian determines F. We also generalize a result due to Skogman about the non-vanishing of all theta components of a Jacobi cusp form of even weight and prime index, which may have some independent interest.

The zeta-function ζ(s, F), s = σ + it of a cusp form F of weight κ in the half-plane σ > (κ + 1)/2 is defined by the Dirichlet series whose coefficients are the coefficients of the Fourier series of the form F. The compositions V(ζ(s,F)) with an operator V on the space of analytic functions are considered, and the functional independence of these compositions for certain classes of operators V is proved.

We use a simple delta method to prove the Weyl bound in t-aspect for the L-function of a holomorphic or a Hecke-Maass cusp form of arbitrary level and nebentypus. In particular, this extends the results of Meurman and Jutila for the t-aspect Weyl bound, and the recent result of Booker, Milinovich and Ng to Hecke cusp forms of arbitrary level and nebentypus.

Let $f$ and $g$ be holomorphic cusp forms of weights $k_1$and $k_2$ for the congruence subgroups$\Gamma_0(~\mathcal{N}_1~)$ and $\Gamma_0(~\mathcal{N}_2~)$,respectively. In this paper the square moment of theRankin-Selberg $L$-function for $f$ and $g$ in the aspect of both weights in short intervals is bounded, when$k_1^\varepsilon\ll~k_2\ll~k_1^{1-\varepsilon}$.These bounds are the mean Lindelof hypothesis in one caseand subconvexity bounds on average in other cases. Thesesquare moment estimates also imply subconvexity bounds forindividual $L(\frac12+{\rm~i}t,f\times~g)$ for all $g$ when $f$is chosen outside a small exceptional set. In the bestcase scenario the subconvexity bound obtained reaches theWeyl-type bound proved by Lau et al. (2006) inboth the $k_1$ and $k_2$ aspects.

Let f be a holomorphic Hecke cusp form with even integral weight k ⩾ 2 for the full modular group, and let χ be a primitive Dirichlet character modulo q. Let Lf (s, χ) be the automorphic L-function attached to f and χ. We study the mean-square estimate of Lf (s, χ) and establish an asymptotic formula.

Blomer and Maga [2] recently proved that, if F is an L 2 -normalized Hecke-Maass cusp form for SL n ( Z ) , and Ω is a compact subset of PGL n ( R ) / PO n ( R ) , then we have ‖ F | Ω ‖ ∞ ≪ Ω λ F n ( n − 1 ) / 8 − δ n for some δ n > 0 , where λ F is the Laplacian eigenvalue of F. In the present paper, we prove an explicit version of their result.

We prove the elliptic transformation law of Jacobi forms for the generating series of Pandharipande--Thomas invariants of an elliptic Calabi--Yau 3-fold over a reduced class in the base. This proves part of a conjecture by Huang, Katz, and Klemm. For the proof we construct an involution of the derived category and use wall-crossing methods. We express the generating series of PT invariants in terms of low genus Gromov--Witten invariants and universal Jacobi forms. As applications we prove new formulas and recover several known formulas for the PT invariants of $\mathrm{K3} \times E$, abelian 3-folds, and the STU-model. We prove that the generating series of curve counting invariants for $\mathrm{K3} \times E$ with respect to a primitive class on the $\mathrm{K3}$ is a quasi-Jacobi form of weight -10. This provides strong evidence for the Igusa cusp form conjecture.

This work contains a proof of a non-trivial explicit quantitative bound in the eigenvalue aspect for the sup-norm of a SL(3,Z) Hecke-Maass cusp form restricted to a compact set.

Let [Formula: see text] be a normalized holomorphic cusp form for [Formula: see text] of weight [Formula: see text] with [Formula: see text]. By the Kuznetsov trace formula for [Formula: see text], we obtain the twisted first moment of the central values of [Formula: see text], where [Formula: see text] varies over Hecke–Maass cusp forms for [Formula: see text]. As an application, we show that such [Formula: see text] is determined by [Formula: see text] as [Formula: see text] varies.

We introduce a new type of summation formula for central values of GL ( 4 ) × GL ( 2 ) L-functions, when varied over Maas forms. By rewriting such a sum in terms of GL ( 4 ) × GL ( 1 ) L-functions and applying a new “balanced” Voronoi formula, the sum can be shown to be equal to a differently-weighted average of the same quantities. By controlling the support of the spectral weighting functions on both sides, this reciprocity formula gives estimates on spectral sums that were previously obtainable only for lower rank groups. The “balanced” Voronoi formula has Kloosterman sums on both sides, and can be thought of as the functional equation of a certain double Dirichlet series involving Kloosterman sums and GL ( 4 ) Hecke eigenvalues. As an application we show that for any self-dual cusp form Π for SL ( 4 , Z ) , there exist infinitely many Maas forms π for SL ( 2 , Z ) such that L ( 1 / 2 , Π × π ) ≠ 0 .

Let f(z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth normalized Fourier coefficient (Hecke eigenvalue) by λf (n). Let a(n) be the function with squarefull kernel. In this paper, we establish that $$ {\sum}_{n\leqslant x}a(n){\lambda}_f^2\left(n+1\right)= Cx+O\left({x}^{13/14+\upepsilon}\right) $$, where C is a constant that can be explicitly evaluated.

Let f(z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth Hecke eigenvalue or normalized Fourier coefficient by \(\lambda_{f}(n)\). Let g(z) be another distinct primitive holomorphic cusp form of even integral weight \(\ell\) for the full modular group. In this paper, we establish that the set $$\{p| \lambda_f(p^j)

Let F be a self-dual Hecke-Maass cusp form for S L ( 3 , Z ) and let a F ( 1 , n ) denote the n-th coefficient of the Godement-Jacquet L-function L ( s , F ) . Then we show that there exists an absolute constant c 0 > 0 such that ∑ n ⩽ X a F ( 1 , n ) μ ( n ) e ( n α ) ≪ X exp ⁡ ( − c 0 log ⁡ X ) . Here the implied constant depends only on the form F and the bound is uniform in α ∈ R . Moreover, we notice that the aforementioned result generalises to self-dual automorphic cuspidal representations of G L 3 ( A Q ) , with unitary central character.

Let $\mathfrak{q}>2$ be a prime number, $\chi$ a primitive Dirichlet character modulo $\mathfrak{q}$ and $f$ a primitive holomorphic cusp form or a Hecke-Maass cusp form of level $\mathfrak{q}$ and trivial nebentypus. We prove the subconvex bound $$ L(1/2,f\otimes \chi)\ll \mathfrak{q}^{1/2-1/12+\varepsilon}, $$ where the implicit constant depends only on the archimedean parameter of $f$ and $\varepsilon$. The main input is a modifying trivial delta method developed in [1]., Comment: 12 pages

In this article, we have considered the problem of first sign change of Fourier coefficients of a primitive cusp form at a sparse sequence, where elements of the sparse sequence can be written as a sum of two squares. Following the methods of Kohnen–Sengupta, we give an effective upper bound on elements of such a sparse sequence to have first sign change.

Given a fixed Hermitian Jacobi cusp form, we define a family of linear operators between spaces of Hermitian Jacobi cusp forms using Rankin--Cohen brackets. We compute the adjoint maps of such a family with respect to the Petersson scalar product. The Fourier coefficients of the Hermitian Jacobi cusp forms constructed using this method involve special values of certain Dirichlet series associated to Hermitian Jacobi cusp forms.

For a semisimple Lie group G satisfying the equal rank condition, the most basic family of unitary irreducible representations is the discrete series found by Harish-Chandra. In this paper, we study some of the branching laws for these when restricted to a subgroup H of the same type by combining the classical results with the recent work of T. Kobayashi. We analyze aspects of having differential operators being symmetry-breaking operators; in particular, we prove in the so-called admissible case that every symmetry breaking (H-map) operator is a differential operator. We prove discrete decomposability under Harish-Chandra's condition of cusp form on the reproducing kernels. Our techniques are based on realizing discrete series representations as kernels of elliptic invariant differential operators.

We prove the meromorphic continuation and the functional equation of a twisted real-analytic Hermitain Eisenstein series of Klingen type, and as a consequence, deduce similar properties for the twisted Dirichlet series associated to a pair of Hermitian modular forms involving their Fourier–Jacobi coefficients. As an application of our result, we prove that infinitely many of the Fourier–Jacobi coefficients of a non-zero Hermitian cusp form do not vanish in any non-trivial arithmetic progression.

Let f ( z ) be a holomorphic normalized Hecke-eigen cusp form of weight κ for S L 2 ( Z ) with Fourier coefficients λ f ( n ) . In this paper, we estimate the sum of λ f ( n ) at primes twisted with exponential functions e ( α p θ ) for 0 θ 1 . New upper bounds are proved for fixed α. Similar results are also given for the analogous situation of non-holomorphic forms.

Let $q$ be a large prime, and $\chi$ the quadratic character modulo $q$. Let $\phi$ be a self-dual Hecke--Maass cusp form for $SL(3,\mathbb{Z})$, and $u_j$ a Hecke--Maass cusp form for $\Gamma_0(q)\subseteq SL(2,\mathbb{Z})$ with spectral parameter $t_j$. We prove the hybrid subconvexity bounds for the twisted $L$-functions \[ L(1/2,\phi\times u_j\times\chi)\ll_{\phi,\varepsilon} (qt_j)^{3/2-\theta+\varepsilon},\quad L(1/2+it,\phi\times\chi)\ll_{\phi,\varepsilon} (qt)^{3/4-\theta/2+\varepsilon}, \] for any $\varepsilon>0$, where $\theta=1/23$ is admissible., Comment: 35 pages. Comments are welcome! Accepted for publication in Sci China Math

Let f be a full-level cusp form for GLm(ℤ) with Fourier coefficients Af(cm-2,..., c1, n): Let λ(n) be either the von Mangoldt function Λ(n) or the k-th divisor function τk(n): We consider averages of shifted convolution sums of the type Σ|h|⩽H | ΣX

Let $\phi$ denote a primitive Hecke-Maass cusp form for $\Gamma_o(N)$ with the Laplacian eigenvalue $\lambda_\phi=1/4+t_{\phi}^2$. In this work we show that there exists a prime $p$ such that $p\nmid N$, $|\alpha_{p}|=|\beta_{p}| = 1$, and $p\ll(N(1+|t_{\phi}|))^c$, where $\alpha _{p},\;\beta _{p}$ are the Satake parameters of $\phi$ at $p$, and $c$ is an absolute constant with $0, Comment: Version 2: typos corrected and a new section on natural density added

Let φ be a spherical Hecke–Maas cusp form on the non-compact space PGL3(ℤ)PGL3(ℝ). We establish various pointwise upper bounds for φ in terms of its Laplace eigenvalue λφ. These imply, for φ arithmetically normalized and tempered at the archimedean place, the bound $$||\phi |{|_\infty } \ll_\varepsilon \lambda _\phi ^{39/40 + \varepsilon }$$ for the global sup-norm (without restriction to a compact subset). On the way, we derive a new uniform upper bound for the GL3 Jacquet–Whittaker function.

We provide applications to studying the behavior of Selmer groups under specialization. We consider Selmer groups associated to four dimensional Galois representations coming from (i) the tensor product of two cuspidal Hida families $F$ and $G$, (ii) its cyclotomic deformation, (iii) the tensor product of a cusp form $f$ and the Hida family $G$, where $f$ is a classical specialization of $F$ with weight $k \geq 2$. We prove control theorems to relate (a) the Selmer group associated to the tensor product of Hida families $F$ and $G$ to the Selmer group associated to its cyclotomic deformation and (b) the Selmer group associated to the tensor product of $f$ and $G$ to the Selmer group associated to the tensor product of $F$ and $G$. On the analytic side of the main conjectures, Hida has constructed one variable, two variable and three variable Rankin-Selberg $p$-adic $L$-functions. Our specialization results enable us to verify that Hida's results relating (a) the two variable $p$-adic $L$-function to the three variable $p$-adic $L$-function and (b) the one variable $p$-adic $L$-function to the two variable $p$-adic $L$-function and our control theorems for Selmer groups are completely consistent with the main conjectures., Incorporated changes suggested by the referee. Accepted for publication in Annales de l'Institut Fourier

Let Δk be the unique normalized cusp form of level one and weight k with k = 16 , 18 , 20 , 22 , 26 . In this paper, we describe a method to achieve the explicit bounds Bk of n such that the Fourier coefficients an(Δk) ≠ 0 for all n

Let f be a Hecke-Maass cusp form for SL(3,ℤ) with Fourier coeffcients Af (m, n), and let ϕ(x) be a C∞-function supported on [1, 2] with derivatives bounded by ϕ(j)(x)≪j 1: We prove an asymptotic formula for the nonlinear $$\Sigma_{n\equiv l \rm{mod} \it{q}}$$ Af (m, n)ϕ(n/X)e(3(kn)1/3/q), where e(z) = e2πiz and k ∈ℤ+.

We introduce and analyse a general class of not necessarily bounded multiplicative functions, examples of which include the function $n \mapsto \delta^{\omega (n)}$, where $\delta \neq 0$ and where $\omega$ counts the number of distinct prime factors of $n$, as well as the function $n \mapsto |\lambda_f(n)|$, where $\lambda_f(n)$ denotes the Fourier coefficients of a primitive holomorphic cusp form. For this class of functions we show that after applying a `$W$-trick' their elements become orthogonal to polynomial nilsequences. The resulting functions therefore have small uniformity norms of all orders by the Green--Tao--Ziegler inverse theorem, a consequence that will be used in a separate paper in order to asymptotically evaluate linear correlations of multiplicative functions from our class. Our result generalises work of Green and Tao on the M\"obius function., Comment: 95 pages; final version

In the 1960s Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct $$\chi _{5}$$ , the cusp form of lowest weight for the group $${\text {Sp}}(2,\mathbb {Z})$$ . In 2010 Gritsenko found three towers of orthogonal type modular forms which are connected with certain series of root lattices. In this setting Siegel modular forms can be identified with the orthogonal group of signature (2, 3) for the lattice $$A_{1}$$ and Igusa’s form $$\chi _{5}$$ appears as the roof of this tower. We use this interpretation to construct a framework for this tower which uses three different types of constructions for modular forms. It turns out that our method produces simple coordinates.

In this paper, we investigate sign changes of Fourier coefficients of half-integral weight cusp forms. In a fixed square class $$t\mathbb {Z}^2$$ , we investigate the sign changes in the $$tp^2$$ -th coefficient as p runs through the split or inert primes over the ring of integers in a quadratic extension of the rationals. We show that infinitely many sign changes occur in both sets of primes when there exists a prime dividing the discriminant of the field which does not divide the level of the cusp form and find an explicit condition that determines whether sign changes occur when every prime dividing the discriminant also divides the level.

Let π be a \(SL(3,\mathbb Z)\) Hecke-Maass cusp form, and let χ be a primitive Dirichlet character modulo M, which we assume to be prime. We establish the following unconditional subconvex bound for the twisted L-function: $$\displaystyle \begin{aligned} L\left (\tfrac {1}{2},\pi \otimes \chi \right )\ll _{\pi ,\varepsilon } M^{3/4-1/308+\varepsilon }. \end{aligned} $$

Ahlgren and Samart relate three cusp forms with complex multiplication to certain weakly holomorphic modular forms using $p$-adic bounds related to their Fourier coefficients. In these three examples, their result strengthens a theorem of Guerzhoy, Kent, and Ono which pairs certain CM forms with weakly holomorphic modular forms via $p$-adic limits. Ahlgren and Samart use only the theory of modular forms and Hecke operators, whereas Guerzhoy, Kent, and Ono use the theory of harmonic Maass forms. Here we extend Ahlgren and Samart's work to all cases where the cusp form space is one-dimensional and has trivial Nebentypus. Along the way, we obtain a duality result relating two families of modular forms that arise naturally in each case., Comment: Submitted 24 February, 2021. 12 pages

The purpose of this partly expository paper is to give an introduction to modular forms on G2. We do this by focusing on two aspects of G2 modular forms. First, we discuss the Fourier expansion of modular forms, following work of Gan-Gross-Savin and the author. Then, following Gurevich-Segal and Segal, we discuss a Rankin-Selberg integral yielding the standard L-function of modular forms on G2. As a corollary of the analysis of this Rankin-Selberg integral, one obtains a Dirichlet series for the standard L-function of G2 modular forms; this involves the arithmetic invariant theory of cubic rings. We end by analyzing the archimedean zeta integral that arises from the Rankin-Selberg integral when the cusp form is an even weight modular form.