Questions about modular forms and related areas

**3**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**2**answers

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**

votes

**2**answers

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}} / ...

**1**

vote

**0**answers

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

vote

**0**answers

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

**1**answer

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**

votes

**0**answers

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

**2**answers

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

**1**answer

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

**3**answers

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**

votes

**1**answer

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 ...

**1**

vote

**0**answers

97 views

### 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**

votes

**0**answers

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 ...

**5**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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**

votes

**1**answer

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

**1**answer

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

**1**answer

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**

votes

**1**answer

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

**1**answer

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**

votes

**2**answers

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**

votes

**5**answers

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**

votes

**1**answer

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

**1**answer

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}, ...

**4**

votes

**0**answers

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

**1**answer

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**

votes

**1**answer

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**

votes

**2**answers

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

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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

**1**answer

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 / ...

**5**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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

**1**answer

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**

votes

**1**answer

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 $S_2(\Gamma_0(N)) = {0}$?