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An automorphic form is a well-behaved function from a topological group $G$ to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup $\Gamma \subset G$ of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.

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Given a holomorphic cusp form $f$ with weight $k \geq 2$, level $N$ and trivial nebentypus. I am wondering if $f$ can be a CM (dihedral) cusp form, i.e. $f$ is isomorphic to its quadratic twist. ...
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I have a question on the large sieve inequality involving $GL(2)$ harmonics. Recall that one has the analog for $GL(1)$ harmonics that, for any complex numbers $\alpha_m,\beta_n$, one has $$\sum_{q\le ...
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Let $q\in \mathbb{N}$. Denote by ${B}(q,\chi)$ an orthogonal basis of $GL(2)$-Maass cusp forms of level $q$ with nebentypus $\chi\bmod q$. Does there exist a corresponding version for the following ...
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Let $L(s,f)$ and $L(s,g)$ be two $L$-functions of degrees $d_{f}$ and $d_{g}$ with local roots $\alpha_{j}(p)$ and $\beta_{\ell}(p)$ at $p$ and $\kappa_{j}$ and local roots $\nu_{\ell}$ at $\infty$. ...
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In Stevens's paper "Poincaré Series on $\operatorname{GL}(r)$ and Kloostermann Sums", he gave a nontrivial estimate for the $\operatorname{GL}(3)$ local Kloosterman sum corresponding to the ...
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Let $F$ be a fixed number field, and $S$ be a non-empty fixed finite set of places of $F$. We further assume that $S$ contains all the infinity places and at least one finite place. Suppose that for ...
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This is a slightly philosophical question. I was wondering if there is some structure that explains why Hecke eigenvalues and Whittaker coefficients are, essentially, the same for classical ...
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find $\rm{dim} (\Theta_k)$ : $\mathbb H=\{x+iy|y>0;x,y\in \mathbb R\}$is the upper half plane. $z=x+iy$ and $k$ can be any positive real number If $f:\mathbb H\to\mathbb C$ ,which satisfies: 1.$...
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$\DeclareMathOperator\Mp{Mp}$Let $F$ be a p-adic local field and $W$ be a 2-dimensional symplectic space and $V$ is a 1-dimensional orthogonal space over $F$, respectively. Let $e$ be an element with $...
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I have some queries regarding the proof of the following lemma. In the paper 'Euler subgroups' by I.I. Piatetski-Shapiro, the author states the following lemma: Lemma 1: For any $\varphi(p) \in S_n$, ...
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Let $f(z)= \sum_{n \geq 1} \lambda_f(n)n^{(k-1)/2}e^{2\pi i nz}$ be a holomorphic cusp newform of level one, weight $k$ and trivial nebentypus. Let $p$ be an odd prime and $\chi$ be the principal ...
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I am now reading Langlands' Beyond Endoscopy. In Section 2.3, Langlands claimed that in Arthur's paper: On the Fourier transforms of weighted orbital integrals, one has the following estimate \begin{...
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I read some paper and get that Kuznetsov formula can transform $\sum_c \frac{S(m, n ; c)}{c} g\left(\frac{4 \pi \sqrt{m n}}{c}\right)$ into some information from automorphic form. I wonder if we do ...
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I have a puzzle which perhaps looks naive for many experts here. Let $k \ge 2$ be an even integer and $N > 0$ be an integer. Let $\chi$ be a primitive character to modulus $q$ such that $N|q$, ...
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In M. Ram Murty's paper "Oscillations of Fourier coefficients of modular forms", Math. Ann. 262, 431-446 (1983), MR696516, Zbl 0489.10020 (an offprint can be found here), I see a conjecture (...
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Let K be an imaginary quadratic field $Q(\sqrt{-3})$. Let $p$ be a prime which split in $K$ and Let $\varpi$ be one of the factors of $p$ in $K$. Let $E$ be an elliptic curve $y^2=x^3+\varpi^2/4$ with ...
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Forgive my ignorance, I am new to the subject. I have a need for determining the existence of certain lifts of elliptic modular forms to Hilbert modular forms (over a real quadratic field). The ...
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Recently, I am reading the paper *"Twisted moments of L-functions and spectral reciprocity" by Blomer, Valentin and Khan, Rizwanur in 2019 (arXiv, DOI). I wonder what is the Ramanujan-...
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Has anyone devloped the spectral theory of $L^{2}$-integrable $1/2$-integral weight automorphic forms? I know that "The subconvexity problem for Artin L–functions" discusses the spectral ...
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Let $(G,X)$ be a Shimura datum and $K \subset G(\mathbb{A}_f)$ an open compact subgroup. For simplicity, assume that the largest anisotropic subtorus of $Z(G)$ remains anisotropic over $\mathbb{R}$. ...
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Recently, I am very interested in Voronoi summation for high ranks, which a very important tool in the theory of automorphic forms and is frequently applied in many analyses. Note that the Voronoi ...
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$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SU{SU}$Let $F$ be a characteristic zero $p$-adic field, $\varpi$ a uniformizer of $\mathfrak{p}_{F}$, the maximal ideal of $\mathfrak{o}_{F}$, let $G=\...
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Let $G$ be a quasisplit unitary group in three variables, in the paper https://link.springer.com/article/10.1007/BF02784154, page 234 theorem 4, it is proven that a everywhere generic cuspidal ...
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I am interested in modular forms on classical groups e.g. Hilbert, Siegel and Hermitian modular forms. The general question is the following: Let $G$ be a reductive group over $\mathbb Q$, and $g_\...
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This may be reminiscent to this question from some years ago, that received no answer. When one develops a Kuznetsov-type trace formula for $\operatorname{GSp}(4)$, it is natural to select the ...
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Let $G$ be a reductive group over $\mathbb Q$ such that $G(\mathbb R)$ admits a holomorphic discrete series $\pi_\infty$. Maybe well-known to experts doing analytic number theory (or vertical Sato–...
Zhiyu's user avatar
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every expert on MO. Really sorry to disturb. Recently, I was reading the paper of Corbett--Voronoi summation for $\rm{GL}_n$: collusion between level and modulus ( https://arxiv.org/pdf/1807.00716). I ...
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I'm interested in the best-known bounds on coefficients of a weight $k$ holomorphic cusp form on a noncongruence subgroup. Write $$ f(z) = \sum_{n \geq 1} a(n) \exp((n - \alpha)z/q).$$ The best result ...
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$\DeclareMathOperator\GL{GL}$Let $F$ be a local field and $\GL_n$ and $\GL_m$ defined over $F$. Let $\pi$ and $\sigma$ be an irreducible smooth representation of $\GL_n$ and $\GL_m$, respectively. Let ...
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Let $G=\mathrm{GL}_n$ ($n \geq 2$) over $\mathbb Q$, with $\mathfrak g_\mathbb C=\mathfrak{gl}_n(\mathbb C)$ and $K_\infty=O(n)$. Let $\pi_\infty$ be a cohomological irreducible unitary $(\mathfrak g_\...
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Let $K$ be a number field and let $\pi=\otimes'\pi_v$ be a regular cuspidal algebraic automorphic representation of $\mathrm{GL}_n(\mathbb{A}_K)$. If it makes any difference, I'm also happy to assume ...
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Let $e(z)=e^{2\pi i z} $ ; $ \chi_q$ is a primitive Dirichlet character mod q , $\nu=\frac{1-\chi(-1)}{2}$ and $ \theta(z,\chi_q)$ is the Dirichlet theta function : $$\theta(z,\chi_q)=\frac12 \...
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Let $F$ be a p-adic field. Set $X=C_c^\infty(M_n(F))$ to be the space of locally constant compactly supported functions on $M_n(F)$, where $M_n(F)$ denotes the space of $n\times n$ matrices over $F$. ...
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Let $(G, X)$ be a Shimura datum with associated Shimura varieties $M_K=Sh_K(G,X)$ over $\mathbb C$ (for compact open subgroups $K \leq G(\mathbb A_f)$). Let $\pi$ be a discrete automorphic ...
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Let $G$ be a quasi-split semi-simple group over $\mathbb Q$. Let $P=MU$ be a parabolic of $G$ over $\mathbb Q$ with unipotent radical $U$ and Levi $M$. Let $\chi: M \to \mathbb G_m$ be a character of $...
Zhiyu's user avatar
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In P.Cartier's article "Representations of $\mathfrak p$-adic groups: a survey" something is written at the very beginning of IV that I don't quite understand. Let $G$ be a $p$-adic ...
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Let $G$ be a connected reductive group over $\mathbb R$. Let $G'$ over $\mathbb R$ be an inner form of $G$ with ${}^LG={}^LG'$. By local Langlands correspondence over $\mathbb R$, if a $L$-packet of $...
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Let $F$ be a totally real field. Let $G$ be a (split) connected reductive group over $F$. Let $\pi$ be an irreducible automorphic representation of $G$. Recall in the work of Buzzard and Gee "The ...
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$\DeclareMathOperator\GL{GL}$Let $f$ be an automorphic form on $\GL_3$ for $\Gamma_0(p)$ with $p$ being a prime (see Bump or Goldfeld's books for definitions). Recall that, in this paper-"The ...
hofnumber's user avatar
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Let $\pi$ be an irreducible non-cuspidal automorphic representation of a classical group $G$ defined over a number field $F$. Let $P$ be a maximal parabolic subgroup of $G$. Let $\pi'$ be the ...
Andrew's user avatar
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Let $\pi$ be a regular, cuspidal, algebraic automorphic representation of $GL_n(\mathbb{A}_K)$ for a totally real field $K$. Then for every embedding $\lambda$ of $E=\mathbb{Q}(\pi)$ in $\overline{\...
Alireza Shavali's user avatar
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Assume a holomorphic function from a product of two upper half planes $Z: \mathbb{H}_+\times \mathbb{H}_+\rightarrow \mathbb{C}$ with an expansion of the form $$ Z(\tau,\tau') = \sum_{(h,h')\in S} a_{...
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Studying Langlands's classification of irreducible admissible representations, I have been rather stunned by the following: Theorem Up to infinitesimal equivalence, all irreducible admissible ...
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Let $D_k$ denote a discrete series representation of $\text{GL}_2(\mathbb{R})$ of weight $k\geq 2$. Consider the parabolically induced representation $D_k \times D_k$, which is a representation of $\...
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I often notice papers and results that assume that the level is squarefree in the setting of modular forms, but have a hard time figuring how where this impacts or simplifies the argument. Is there in ...
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$\DeclareMathOperator\GL{GL}$There is a lemma in the proofs of local converse theorem stated as Suppose $F$ is a non-archimedean local field, $\psi$ is a non-trivial addtive character on $F$. $N_n$ ...
Zhefeng Shen's user avatar
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Let $f$ be a cuspidal Hecke newform of weight $k$ and level $N$, and denote by $a_f(n)$ its $n$-th Fourier coefficient. The newform $f$ is normalized so that $a_f(1) = 1$. As a consequence of Rankin-...
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I have a puzzle on the local factors of Rankin-Selberg $L$-functions. Consider two newforms on $\text{GL}_2$. Let $f$ be a newform of square-free level $N$, and $g$ a newform of trivial level. As ...
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Let $\pi$ be the irreducible cuspidal automorphic representation of $\mathrm{GL}_2$. Let $E/F$ be a quadratic extension with given embedding $E^{\times} \to \mathrm{GL}_2(F)$. For $f_1 \in \pi$, $f_2 \...
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I asked exact same question on MSE but haven't got answer yet, so asking here, too. I may erase the original one once I got an answer here. -- I'm confusing about automorphic representations of $\...
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