Questions about modular forms and related areas

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3
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118 views

Moduli interpretation of Eisenstein series

Let $N \geq 11$ be an integer and consider the basis of Eisenstein series for $M_2(\Gamma_0(N))$ described in Theorem $4.6.2$ of Diamond--Shurman's book. Pick and Eisenstein series $F$ in this basis. ...
6
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0answers
76 views

Weyl law for Maass forms with nontrivial character

The classical Weyl law for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$ counts the number of Maass cusp forms on $\Gamma \backslash \mathbb{H}$ with Laplace eigenvalue less than $T$. This is originally due to ...
3
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0answers
106 views

Fourier expansions of newforms at width-1 cusps

Let $f_E$ be the newform attached to the Elliptic Curve $E$ with cremona label $\textbf{100a1}$ and let $\alpha = \left[\begin{matrix} 1&0 \\ 10&1 \end{matrix}\right] \in SL_2(\mathbb{Z})$. ...
6
votes
1answer
116 views

Cuspidal modular forms - toroidal or minimal compactification?

Let $Y$ be a Siegel variety and let $X$ be a toroidal compactification of $Y$. For any tuple of integers $\underline k$ we have the usual sheaf $\omega^{\underline k}$. The space of modular forms of ...
9
votes
1answer
329 views

Holomorphic cusp forms and cohomology of GL(2,Z)

Let $V_{k}$ denote the complex representation of $\mathrm{GL}(2)$ given by $\mathrm{Sym}^k(V)$, where $V$ is the defining 2-dimensional representation. Assume that $k$ is even. I would like to compute ...
2
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0answers
65 views

Modular forms related to $G(q)$ and $H(q)$

If $G(q),H(q)$ are the functions appearing in Rogers-Ramanujan identities $$G(q)=\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{(q;q)_{n}}=\prod_{n=1}^{\infty}\frac{1}{(1-q^{5n-1})(1-q^{5n-4})}$$ and ...
8
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1answer
285 views

Adelic open image for modular forms?

There's a famous theorem of Serre that if $E$ is a non-CM elliptic curve over $\mathbf{Q}$, and $\rho_{E, \ell} : Gal(\overline{\mathbf{Q}}/{\mathbf{Q}}) \to GL_2(\mathbf{Z}_\ell)$ is its $\ell$-adic ...
4
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2answers
147 views

Images of the fundamental domain of $\text{SL}_2(\mathbb{Z})\backslash \mathbb{H}$ whose Euclidean area is large

Let $S$ be a compact subset of the closure of the upper half plane. (Assume that $S$ is a (Euclidean) rectangular box, if you wish.) Let $D$ be the standard fundamental domain of ...
6
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2answers
221 views

Holomorphic Hoffstein-Lockhart

In the article Hoffstein, Jeffrey; Lockhart, Paul "Coefficients of Maass forms and the Siegel zero." Ann. of Math. (2) 140 (1994), no. 1, 161–181, it is stablished a good bound for the Petersson norm ...
6
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2answers
274 views

Field of definition of Galois representations of weight 1 modular forms

Let $f$ be a weight 1 modular form (let's say cuspidal, new, normalized, and a Hecke eigenform). Then there's an associated Artin representation $\rho_f: \operatorname{Gal}(\overline{\mathbf{Q}} / ...
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0answers
43 views

Algebraic Relations between $G(q)$ and $H(q)$

I had posted this originally on MSE, but got no response at all hence posting the same here. In his paper "Algebraic Relations between Certain Infinite Products" (Proceedings of the London ...
5
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2answers
202 views

Why is the supersingular locus the zero locus of a modular form?

This question is related to my other question here: Examples of subspaces singled out by modular forms. Here I am wondering if there is a philosophical explanation about why the supersingular locus ...
5
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1answer
306 views

Ternary quadratic form theta series as Hecke eigenforms and class number one

At Simple comparison of positive ternary quadratic form representation counts Jeremy answered: "The reason is that the theta series for the sums of three squares form is an eigenfunction for all the ...
3
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1answer
90 views

Simple comparison of positive ternary quadratic form representation counts

Something came up yesterday in a referee request and I was surprised to find that I did not know the facts in full generality. This is about positive quadratic forms in three variables with integer ...
1
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0answers
40 views

Semi-simple controlling operator

I've just come across this paper by Coleman and Edixhoven called "On the semi-simplicity of the $U_p$ operator on modular forms", where (as the title says) they show that the $U_p$ operator is ...
3
votes
1answer
140 views

Existence of real modular function with specific behavior as $q\to 0$

I am looking for a real modular function $F(q,\bar{q})$ such that in the limit of small $q,\bar{q}$ it behaves as: $F(q,\bar{q})=(a_0 + a_1 (q + \bar q)+...)\log q \bar q+ (b_0 + b_1 (q + \bar ...
2
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0answers
79 views

Examples of subspaces singled out by modular forms

I am wondering what subspaces of modular varieties defined as the zero locus of modular forms have been studied in the literature. To be more clear let me explain the example I have in mind. Let ...
7
votes
2answers
390 views

modular forms, invertible sheaves, and quotients

I'm very confused about some contradicatory statements, and I hope someone can help me clarify this. Let $\Gamma$ be a congruence subgroup. It is well known that modular forms of weight $k$ for ...
7
votes
1answer
188 views

Atkin--Lehner operators in Hida theory

Let $p$ be a prime, and $F$ a $p$-adic Hida family of ordinary modular forms (of some tame level $N \ge 1$). I'd like to know whether, for $q$ a prime factor of $N$, the actions of the Atkin--Lehner ...
4
votes
3answers
179 views

Expression for the derivative of Eisenstein series $G_2$

I am new to number theory, so I am guessing this is a standard formula. I would be grateful for a reference: We know that the Eisenstein series $G_2$ is quasimodular of level $SL_2(\mathbb Z)$, so ...
4
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1answer
147 views

Reference or proof for the fact that $J(X_0(N))$ splits into abelian varieties with real multiplication

It´s known that $J_0(N) = J(X_0(N))= \bigoplus_f E(f)$ splits as a sum of abelian varieties parametrized by the Hecke eingenfunctions and that it´s an elliptic curve iff the Hecke eingenvalue is an ...
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0answers
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Modular form, number of divisors [duplicate]

The Fourier expansion of Eisenstein series $E_k$ $(k \ge 4)$, which are modular forms, as well as the quasimodular $E_2$, involves powers-of-divisors $\sigma_{k-1}(n) = \sum_{d|n} d^{k-1}$. Is there ...
6
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93 views

A partition congruence modulo 13

In the paper "Note on certain modular relations considered by Messrs Ramanujan, Darling and Rogers" (Proceedings of London Mathematical Society (1922) s2-20 (1): 408-416) Mordell gives proofs of the ...
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96 views

Density of p-ordinary modular forms

Fix an odd prime $p$. For concreteness, let $N$ be coprime to $p$, and let $2 \leq k \leq p$. Let $S^+(N,k)$ be the newforms in $S_k(\Gamma_1(N))$. Let $f = \sum a_n q^n \in S^+(N,k)$. We say that ...
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0answers
73 views

Are they mock theta functions or mock modular forms?

would anyone help me to show if the following $q-$series have modular property? Thank you very much. \begin{eqnarray} B(\phi^{*};q)=\sum_{n\in \mathbb{Z}}{\frac{q^{n}}{(-q^2;q^2)_n}} \end{eqnarray} ...
2
votes
1answer
127 views

About the restriction of a modular representation to a decomposition subgroup II

This question is a variant of this one. Let $f$ be as in the other question, but suppose that we look at the $\ell$-adic representiation attached to $f$: $$ \rho_f : G_{\mathbb Q} \to ...
2
votes
1answer
168 views

About the restriction of a modular representation to a decomposition subgroup

Let $f$ be an eigenform of level $\Gamma_1(N)$ and let $p$ be a prime that does not divide $N$. It is well know that there is a $2$-dimensional representation $$ \rho_f \colon G_{\mathbb Q} \to ...
14
votes
1answer
392 views

A sum by Ramanujan for $\coth^{2}(5\pi)$

Ramanujan mentions in one of his letters to Hardy that $$\frac{1^{5}}{e^{2\pi} - 1}\cdot\frac{1}{2500 + 1^{4}} + \frac{2^{5}}{e^{4\pi} - 1}\cdot\frac{1}{2500 + 2^{4}} + \cdots = ...
8
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1answer
232 views

Overconvergent cohomology and overconvergent modular forms

I've been reading a preprint by David Hansen (with appendix by James Newton) called Universal eigenvarieties, trianguline Galois representations and p-adic Langlands functoriality. In it he talks ...
4
votes
1answer
186 views

looking for reference on dihedral, tetrahedral, or octahedral forms

I am looking for a reference on dihedral, tetrahedral, or octahedral forms. As far as I read, they are some cuspidal automorphic forms on $GL(2)$ induced from $GL(1)$. Dihedral is from $GL(1)/K$ to ...
3
votes
1answer
245 views

Is there a nice way to write the generating function obtained by taking the quadratic coefficients of another one?

Suppose that you have a generating function $$ f(q) = \sum_{k=0}^\infty a_k q^k $$ It's not too hard to obtain the generating function $$ f_{n,m}(q) = \sum_{k=0}^\infty a_{nk + m}q^k $$ by taking a ...
12
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1answer
505 views

Curves on K3 and modular forms

The paper of Bryan and Leung "The enumerative geometry of $K3$ surfaces and modular forms" provides the following formula. Let $S$ be a $K3$ surface and $C$ be a holomorphic curve in $S$ representing ...
10
votes
1answer
248 views

Example of a non-smooth irreducible component of the generic fibre of a Hida family?

Is there a known example of a non-smooth irreducible component of the rigid generic fibre of a Hida family? Let me explain some of the context around this question (but I'm not going to explain Hida ...
10
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2answers
361 views

Langlands' original observation about Ramanujan conjecture

Obviously functoriality of arbitrary high symmetric power lifts of automorphic forms on GL(2) will lead to the Ramanujan conjecture. But I guess that is too strong for Ramanujan. I came across some ...
9
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5answers
1k views

Brief Introduction to Modular Forms

What are the best introductory texts on modular forms that are suited for a brief six week course intended for advanced undergraduates? The students will be quite sharp and as far as prerequisites go, ...
7
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1answer
180 views

standard zero free region of automorphic L-function on GL(N)

Let $L(s,\pi)$ be the standard(Godement-Jacquet) $L$-function of $\pi$, where $\pi$ is a cuspidal automorphic represetation of $GL(m,A_Q)$. What's the standard zero-free region for $L(s,\pi)$? any ...
4
votes
1answer
81 views

Overconvergent Modular Symbols Example

I'm trying to redo some of the computations being done in Section 6 of http://math.bu.edu/people/rpollack/Papers/Overconvergent_modular_symbols.pdf May I ask how is $\varphi_f(D_1)=\frac{1}{5}, ...
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0answers
74 views

Generators of the symplectic subgroup $\Gamma^g(1,2)$

Let $\mathbb{A}^{m\times n}$ denote the set of all $m \times n$ matrices with entries in the set $\mathbb{A}$. For a matrix $M$ we let ${^tM}$ denote its transpose, and $M^{-1}$ its inverse, if it is ...
0
votes
1answer
160 views

Euler product of Asai L-function?

Let $\pi$ be an automorphic form of GL(n)/$\mathbb{Q}$ with standard $L$-function $$L(s,\pi)=\prod_p \prod_{i=1}^n(1-\frac{\alpha_{p,i}}{p^s})^{-1},$$ where $\{\alpha_{p,i}:i=1,\dots,n\}$ are the ...
6
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1answer
329 views

Serre's 1987 letter to Tate about mod p modular forms

In what follows, we have a level $N \geq 3$, and the modular curve $X(N)$, and the invertible sheaf $\omega$ on $X(N)$ such that the global sections of $\omega^{\otimes k}$ correspond to modular forms ...
5
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2answers
352 views

Modularity theorem for abelian varieties

There's alredy two posts on MO about the extension of modularity to elliptic curves over fields other than $\mathbb{Q}$ ([1], [2]), and another one about general algebraic varieties [3]. What is ...
5
votes
1answer
126 views

Functional equation and conductor for a Rankin-Selberg convolution

Let $f$ be a Modular form/Maass form on $GL(2)$ with level $N$ and character $\eta$ and Fourier coefficients $a(n)$. The Rankin-Selberg convolution $$L(s,f\times\bar f)=\sum ...
6
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130 views

Is there an integral pairing between quaternionic Hecke algebras and cusp forms?

Let $F$ be a totally real number field with integers $\mathcal{O}_F$ and $B$ a quaternion algebra over $F$ split at exactly one infinity place.Fix $n\geq 1$ and like in the special case $F=\mathbb{Q}, ...
10
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1answer
223 views

What is our current knowledge on the structure of J_0(N)(Q) and J_1(N)(Q)

The question in the title naturally breaks up in two parts, namely the torsion part and the rank part. I already read about some results on both the torsion and the rank part. And I want to know ...
8
votes
1answer
241 views

Universal deformations of modular Galois representations

Let $\bar\rho$ be an odd, absolutely irreducible, 2-dimensional mod $p$ representation of $\operatorname{Gal}(\overline{\mathbf{Q}} / \mathbf{Q})$ (with coefficients in some finite extension $k / ...
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0answers
73 views

Mock modular forms of higher rank?

It appears that mock modular forms are currently defined as the ``holomorhic part" of a harmonic weak Mass form on the upper half plane (with regard to certain modular subgroup). Is there any ...
3
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0answers
80 views

Functoriality for triple product GL(2) x GL(2) x GL(2)

Let $f$, $g$ and $h$ be three general automorphic forms on GL(2). Do we know that $L(s, f\times g\times h)$ comes from an automorphic form on GL(8)?
5
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1answer
236 views

p-adic L-functions of modular forms: why the condition $v_p(\alpha)<k-1$?

Let $f$ be a modular form (cuspidal, new, eigenform) of weight $k$ and level $N$. Let $p$ be a prime number not dividing $N$. In order to construct a $p$-adic $L$-function $L_p(f, s)$ interpolating ...
2
votes
1answer
94 views

Real weight modular forms as sections of a line bundle

Background: I've been trying to read Baily, Walter L., Jr. The decomposition theorem for V-manifolds. Amer. J. Math. 78 1956 and I have problems with the language used in the paper. Firstly I am ...
3
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1answer
90 views

List of $N$ such that there is no non-zero weight 2 cusp form of level $N$

Is there a list of the currently known $N$ such that $S_2(\Gamma_0(N)) = {0}$?