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Questions about modular forms and related areas

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I am investigating a system of congruences related to the elliptic curve $y^2 = x^3 - n^2x $ (congruent number curve) with the following conditions: System: $$k^2 ≡ 64(x^3 - n^2x) \pmod{p}$$ where 'p' ...
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In their paper p-adic families of Siegel modular cuspforms, Andreatta, Iovita and Pilloni defined operators $U_{p,i},i=1,2,...g$ on the spaces of overconvergent modular forms of arbitary $p-$adic ...
Yuheng 's user avatar
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Given a lattice $\Lambda=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$ in $\mathbb{C}$. It's well known that given $n_i\in\mathbb{Z}, z_i\in \mathbb{C}$ satisfying $\sum n_i z\in\Lambda$ and $\sum n_i=0$, ...
Frederick's user avatar
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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. ...
JACK's user avatar
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Recently, I was interested in the large sieve inequalities. A few days ago, I came up with a question on the large sieve inequality involving 𝐺𝐿(2); see On the large sieve inequality involving $GL(2)...
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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 $$\sum_{q\le ...
hofnumber's user avatar
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In Ogg, Andrew P., Hyperelliptic modular curves, Bull. Soc. Math. Fr. 102, 449-462 (1974). ZBL0314.10018. p. 450, there is the sentence: “As LEHNER and NEWMAN noted in a page of corrections attached ...
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Let $q\in \mathbb{N}$. Denote by ${B}(q,\chi)$ an orthogonal basis of $GL(2)$-Maass cusp forms of level $q$ with nebentypus $\chi\bmod q$. Does there exist a corresponding version for the following ...
hofnumber's user avatar
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Let $\mathfrak{P}$ be a place of $\bar{\mathbb{Q}}$ above a prime number $p$, $k \geq 1$, $N \geq 1$ prime to $p$ and $\varepsilon$ a Dirichlet character mod $N$. It is well known that a modular form ...
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As far as I know it seems that quantum modular forms lack a clean operator-theoretic foundation parallel to that of classical and Maass forms. The spectral theory of automorphic Laplacians and the ...
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What's the definition for a Siegel-Eisenstein series with 2 characters? I know in the $\mathrm{SL}_2$ case it's something like this: let $\psi, \chi$ be two characters with modulus $u,v$ respectively, ...
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Let $k\geq 3$ be an integer, fix a level $N$, consider the critical $L$-values of all cusp forms in this level and weight with algebraic Fourier coefficients: $$\mathbb{L}_{N,k}:= \{(2\pi)^{k-1-s} L(f,...
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Consider Siegel upper half space, consisting of symmetric matrices $X+iY$ such that $Y$ is positive definite. This has an action of $\operatorname{Sp}_{2n}(\mathbb{Z})$ on it by generalized Möbius ...
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I. Quintic The general quintic was reduced to the Bring form $x^5+ax+b=0$ in the 1790s, while the Baby Monster was found in the 1970s. We combine the two together using the McKay-Thompson series (...
Tito Piezas III's user avatar
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Consider the modular forms $$\theta(q):=\sum_{n\in\mathbb Z}q^{n^2},\qquad E_4(q):=1+240\sum_{n\ge1}\sigma_3(n)q^n,$$ $$\sum_{n\ge1}\alpha(n)q^n:=\frac1{16}(\theta^{10}(q)-\theta^2(q)E_4(q^2)).\tag{$\...
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My first goal is trying to understand the congruence between Fourier coefficients of $\Delta(z)$ and weight 1 modular form $\eta(z) \eta(23z) \pmod{23}$ given by the following: $a(p) \equiv$ \begin{...
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I. Definitions. Given the nome $q = e^{\pi i\tau}\,$ and $\tau=\sqrt{-n}\,$ for positive integer $n$, then the Ramanujan G and g functions are, $$\begin{align}2^{1/4}G_n &= q^{-\frac{1}{24}}\prod_{...
Tito Piezas III's user avatar
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I. Let $q = e^{2\pi i\tau}$ and $r=R(q)$ be the Rogers-Ramanujan continued fraction. Then the j-function $j(\tau)$ has the formula using polynomial invariants of the icosahedron, $$j(\tau) = -\frac{(r^...
Tito Piezas III's user avatar
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There is a powerful theorem in modular form, which says Valence theorem Let $f \neq 0$ be a modular form of weight $k \geqslant 0$. We have $$ m_f(\infty) + \frac{1}{2}m_f(i) + \frac{1}{3}m_f(\rho) + \...
Aolin HAN's user avatar
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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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I've been trying to understand these 1959 results of Newman giving necessary and sufficient conditions for an eta quotient $f(z) = \prod_{0 < d \mid N} \eta(dz)^{r_d} $ of integer weight $k = \...
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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 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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I am reading Swinnerton Dyer's paper on "On $\ell$-adic representations and congruences for coefficients of modular forms". It defines a prime $\ell$ to be exceptional for an eigenform $f \...
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In 1914 Ramanujan [Quart. J. Math. (Oxford) 45 (1914)] discovered the following irrational series for $1/\pi$: $$\sum_{k=0}^\infty\left(k+\frac{31}{270+48\sqrt5}\right)\binom{2k}k^3\left(\frac{(\sqrt5-...
Zhi-Wei Sun's user avatar
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I was reading a paper on Bessel functions appearing in number theory including modular forms, and I found an identity reminiscent of Guinand's formula: $$ \frac{1}{s}+4\sum_{n=1}^\infty \frac{n}{\...
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Does $\Phi(s) := 4 \sum_{t=1}^\infty \frac{t}{\sqrt{s}} K_1(2t\sqrt{s})$ satisfy $\Phi(1/s)=s^4\Phi(s)?$ Here $K_1$ is the modified Bessel function. I'm interested in this because I want to further ...
John McManus's user avatar
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Let $F$ be a totally real field of even degree and let $f$ be a Hilbert cusp form of level $\mathfrak{n}$ that is an eigenform for Hecke operators. Let $p$ be an odd prime number and $\mathfrak{p}$ a ...
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I am curious about a certain sign distribution question regarding the Fourier coefficients of modular forms. I will describe a particular concrete case of my curiosity, from where variants and ...
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An integral point $P$ of a Grassmannian $Gr(k,n)$ is a $k$-dimensional subspace such that $P \cap \mathbb{Z}^n$ is a rank $k$ sublattice of $\mathbb{Z}^n$. Its height $H$ is given by the determinant ...
Peter Liu's user avatar
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As part of my investigations into modular forms, I want to prove that the commutator subgroup $\mathrm{SL}_2(\mathbb Z)'$ is a congruence subgroup of level $12$. That is: $\Gamma(12)\subseteq\mathrm{...
Zongshu Wu's user avatar
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What is the étale-$\pi_1$ of the modular curve $Y$ with level structure $\Gamma \subset \operatorname{PSL}(2,\mathbb{Z})$? How is that related to $\Gamma$? I have found some related discussions about ...
Michael Cheng's user avatar
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Let $\bar{\rho}:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}_2(\bar{\mathbb{F}}_p)$ be an irreducible modular Galois representation and consider the set $S(\bar{\rho})$ ...
Anwesh Ray's user avatar
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I'm trying to understand the theorem predicting the existence of a normalized newform $f \in S_2(\Gamma_0(N))$ associated to an elliptic curve $E/\mathbb{Q}$, such that there is a modular ...
DEBAJYOTI DE's user avatar
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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-...
Aolin HAN's user avatar
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Consider the $\mathbb{Z}$-lattice consisting of modular forms $f=\sum_{n\geq0}a_n q^n$ of weight $24$ and level $1$ such that $a_n$ is an integer for $n\geq1$. This is a full rank lattice of the space ...
Zakariae.B's user avatar
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Update: Two additional unusual continued fraction behaviors have been observed for expressions involving zeta(3) and log(pi) and are documented in the second script below. I've recently encountered ...
brr's user avatar
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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 ...
Laan Morse's user avatar
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Assume $p\neq 3$ is a prime. Let $f$ be a modular function on $X_0(3p)$ whose divisor is support on the upper half plane (i.e. $f$ has no zeros and poles at the cusps). If the divisor $D(f)=3D$ where $...
yhb's user avatar
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The Dedekind $\eta$ function has the transformation law $$\eta(-1/\tau) = \sqrt{-i\tau}\,\eta(\tau)\ .\tag{1}$$ The Jacobi $\vartheta$ functions obey very similar laws, e.g. $$\vartheta_{01}(z; -1/\...
Manuel Eberl's user avatar
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If $f \in M_k(N, \chi)$ is a normalized integer weight Hecke eigenform for all $T_p$, is there a way to obtain $g \in M_k \left(4N, \chi \left( \frac{-4}{\cdot} \right) \right)$ via the use of ...
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Let $r_Q(n)=\#\{x\in \mathbb{Z}^m|Q(x)=n\}$ be the number of representations of an integer $n$ by a definite quadratic form Q, and define the L-function $$\zeta_Q(s)=\sum_{n=1}^\infty r_Q(n)n^{-s}$$ ...
Alexander's user avatar
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Let $f(X) = a X^2 + b X + c \in \mathbb{Z}[X]$ be an irreducible quadratic polynomial with integer coefficients. For a large prime $p$, the polynomial $f$ has either two or zero roots in $\mathbb{F}_p$...
Jakub Konieczny's user avatar
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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 ...
yhb's user avatar
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12 votes
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I am trying to understand the famous paper by the Chudnovsky brothers, "Approximations and complex multiplication according to Ramanujan" (reprinted in Pi: A Source Book), which (among other ...
Timothy Chow's user avatar
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Let $f = \displaystyle{\sum_{n}} a(n) q^{n} \in M_{k}(N, \chi)$ be a normalized Hecke eigenform. We know that the coefficients a(n) follow a multiplicative relation, $a(n) a(m) = \displaystyle{\sum_{d ...
user554145's user avatar
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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_\...
Zhiyu's user avatar
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How does U-operator act on Rankin-Cohen brackets of two modular forms $[f_{1},f_{2}]$ where $f_{1}, f_{2} \in M_{k_{i}}(N, \chi)$ for $i \in \{1,2\}$? Here, $U_{m}(f) = \frac{1}{m} \displaystyle{\...
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