Questions about modular forms and related areas

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Modular group modulo $N$

Let $N\geq2$ be a positive integer. Is the canonical homomorphism $\pi$ from $SL_2(\mathbb{Z})$ to $SL_2(\mathbb{Z}/N\mathbb{Z})$ surjective? What if we ask the same question for $SL_n$?
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Petersson product of newforms to different level

Let $\text{S}_k^{new}(\Gamma_0(N),\chi)$ be the space of newforms. We call $f\in\text{S}_k^{new}(\Gamma_0(N))$ a newform if $f$ is a Hecke eigenform i.e $\text{T}_nf=\lambda_nf$ ($\text{T}_n$ hecke ...
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2answers
153 views

Real character modular forms: Fourier coefficient real?

Let $f$ be a modular form of level $N$ and real character $\chi$ of mod $N$ and weight $k$. Does the Fourier coefficient or hecke-eigenvalue of $f$ have to be real? What I knew is that if $N=1$ and ...
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1answer
127 views

Examples of Maass forms with eigenvalue 1/4

For what I have heard, Maass forms of (Laplacian) eigenvalue $1/4$ on modular surfaces are somewhat special. But I don't know where to look for explicit examples. (In fact, one form came here on MO ...
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1answer
118 views
+50

Equivalence of Ramanujan's complete series with modular forms

In his notebooks Ramanujan mentions something called a "complete series" which is some power series $\sum_{n = 0}^{\infty}a_{n}q^{n}$ in terms of $q = e^{-y}$ with $y = \pi K'/K$ and $z = 2K/\pi$ such ...
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101 views

Ramanujan's series for $(1/\pi)$ and modular equation of degree $29$

In his famous paper "Modular Equations and Approximations to $\pi$", Ramanujan gives the following famous series for $1/\pi$: $$\frac{1}{2\pi\sqrt{2}} = \frac{1103}{99^{2}} + ...
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2answers
193 views

Interpolation of periods for a Hida family of modular forms

Let $\mathbf{f}$ be a Hida family of ordinary $p$-adic modular forms, and let $V(\mathbf{f})$ be the corresponding $\Lambda$-adic Galois representation (a quotient of the inverse limit $$ ...
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139 views

eisenstein part of the theta function

If $Q:\mathbb{Z}^{2k}\to \mathbb{Z}$ is any positive definite integer -valued quadratic form in $2k$ variables, then it is well known, that the $\textbf{theta series}$ ...
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1answer
150 views

What are the modular transformation properties of q-Pochhammer symbols?

Do q-Pochhammer symbols, defined as $(a;q) = \prod_{k=0}^{\infty} (1- a q^k)$ have known modular transformation properties? That is, if we write $q = q[z] = e^{2\pi i z}$, is there any reasonably ...
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1answer
108 views

Properties of representations attached to p-adic modular forms

I found an old MOF post about representations attached to p-adic modular forms: Representations attached to p-adic modular forms and I have some follow up questions on the same topic. If we have a ...
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115 views

Is there a version of Serre's modularity conjecture for projective representations?

Serre's modularity conjecture asserts that a continuous odd irreducible representation $$\overline{\rho} : G_\mathbb{Q} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$$ must be modular, in the sense that ...
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1answer
210 views

A conjecture related to the Cohen-Oesterlé dimension formula of spaces of modular forms for half-integer weights

Denote $\operatorname{dim}(M_k(\Gamma_0(N)))$ by $m(k,N)$ and $\operatorname{dim}(S_k(\Gamma_0(N)))$ by $s(k,N)$. Let $N$ any positive multiple of $4$ and $j \ge 1$. $$ a(N) := \frac1j \left(m ...
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1answer
497 views

Ramanujan's pi formulas with a twist

Given the binomial function $\binom{n}{k}$, define the following sequences, $$\begin{aligned} u_1(k) &= \tbinom{2k}{k}\tbinom{3k}{k}\tbinom{6k}{3k} = 1, 120, 83160, 81681600,\dots \\ u_2(k) ...
6
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2answers
659 views

Ramanujan's tau function

Why was Ramanujan interested in the his tau function before the advent of modular forms? The machinery of modular forms used by Mordel to solve the multiplicative property seems out of context until I ...
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1answer
170 views

Theta series for the Leech lattice

The Leech lattice, found by John Leech in 1965, is a fascinating combinatorial object and gives the best possible lattice sphere packing in $\mathbb{R}^{24}$. This result was proved by Cohn and Kumar ...
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1answer
207 views

What can the theory of automorphic forms for $SL(n,\mathbb{Z})$ say about $SL(n,\mathbb{Z})$?

While reading "Automorphic Forms and L-functions for the Group $GL(n,R)$" by D. Goldfeld, I've got a feeling that linear groups over $\mathbb{R}$ and $\mathbb{Z}$ are considered only as technical ...
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0answers
263 views

More on Moonshine for the Thompson group and weakly holomorphic weight one half modular forms

This question is a follow-up to Monstrous Moonshine for Thompson group $Th$? and is based on various comments to that question, in particular S. Carnahan's mention of the connection to known ...
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1answer
776 views

Is the Modularity Theorem (currently) effective?

The Modularity Theorem says every elliptic curve over $\mathbb{Q}$ can be gotten from the classic modular curve $X_0(N)$ by a rational map. Here $N$ is the conductor, easily calculable from a ...
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3answers
385 views

Modular form on $\Gamma_0(N)$

I recently asked this question on Math.StackExchange with no answer so far. So I thought maybe I can find an answer here. Let $M(k,\Gamma_0(N))$ be a space of modular forms of weight $k$ on ...
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Modular form on $\Gamma_0(N)$ [duplicate]

Let $M(k,\Gamma_0(N))$ be a space of modular forms of weight $k$ on $\Gamma_0(N)$. Each $f \in M(k,\Gamma_0(N))$ has a Fourier expansion of the form $$f(\tau)=\sum_{n\in ...
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649 views

Monstrous Moonshine for Thompson group $Th$?

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2, $$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, ...
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1answer
117 views

On Universal Abelian surfaces over a Shimura curve.

Let ${\cal O}, {\cal O}'$ be two order in ${\mathrm M}_2({\Bbb R})$ that are sets of all $2 \times 2$ matrices over real number ${\Bbb R}$. Assume that we have the relation ${\cal O}' = a{\cal ...
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1answer
183 views

Under which constraints are there only finite numbers of irreducible eta product identities?

For the Dedekind eta function, defined as usual by $\eta(q) = q^{\frac1{24}} \prod\limits_{n=1}^{\infty} (1-q^{n})$, let for brevity $e_k:=\eta(q^k)$. An eta product identity (or eta identity for ...
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4answers
260 views

Extending a function from $\mathbb{Q}$ to the upper half plane $\mathbb{H}\cup\mathbb{Q}\cup\{i\infty\}$

Define the extended upper half plane $$\overline{\mathbb{H}}:=\{z\in\mathbb{C}: \mathrm{Im}(z)>0\} \cup \mathbb{Q} \cup \{i\infty\}.$$ To what extent can an arbitrary function on the rationals ...
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1answer
139 views

How to obtain the Period matrix from the Igusa Invariants of a genus two curve?

I am looking for an algorithm to obtain the period matrix tau= (tau1 & tau12\ tau12 & tau2) from the Igusa invariants of a genus two curve. More precisely: I consider a family of genus two ...
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about lemma 5.9 of Mazur's famous Eisenstein ideal paper

In Lemma 5.9 of Chapter II of his famous Eisenstein ideal paper, Mazur proved that when $1/N$ is invertible in the ring $R$, if $\phi$ is a holomorphic modular form in $\omega^k$ over $\Gamma_0(N)$ ...
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another question on the Manin-Drinfeld theorem

A few days ago I asked a question about possible higher dimensional generalizations of the Manin-Drinfeld theorem. Let me come back to the classical statement. It says that a divisor on the modular ...
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Solvable parametric $7$th and $13$th degree equations using $\eta(q)/\eta(q^p)$?

Q: Why is that some polynomial relations between eta quotients have a solvable Galois group, even if the deg is $n>4$? For example, we have the well-known modular equation, $$u^6 - v^6 + ...
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Liftability of mod p Hilbert modular forms of parellel weight

If $f$ is a mod $p$ Katz modular form of weight $k$ for $k \geq 2$, there is a classical modular form $g$ such that reduction of $g$ is $f$. Is there a similar property holds for Hilbert modular ...
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1answer
219 views

computing spaces of $p$-adic modular forms

Let $p$ be a prime, and $\alpha$ a positive integer. How do you compute the space of $p$-ordinary $p$-adic modular forms (in the sense of Serre) of weight 2 on $\Gamma_0(p^\alpha)$? I'm really only ...
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1answer
1k views

There's something strange about $\sqrt{d\big(j(\tau)-1728\big)}$

Given the j-function $j(\tau)$, I was looking at, $$F(\tau) = \sqrt{d\big(j(\tau)-1728\big)}$$ which appears in Ramanujan-type pi formulas. Let $C_d$ be the prime factors of the constant term of the ...
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2answers
432 views

Etymology of cuspidal representations

In the literature on representation theory of $GL_2(\Bbb F_p)$ and $GL_2(\Bbb Q_p)$, the irreducible representations with trivial Jacquet module are often called "cuspidal" or "supercuspidal". Why are ...
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1answer
112 views

Relation between Hecke Operator and Trace map

Let $f\in S_k(\Gamma_0(N))$, $(p,N)=1$ and $T_r:S_k(\Gamma_0(Np))\longrightarrow S_k(\Gamma_0(N))$ the trace map. Does then $T_r(f\mid V_p) = f\mid T_p$ hold?
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118 views

Are the Fourier coefficients of a new form real

Let $f\in S^\text{new}_k(\Gamma_0(N))$ be a newform . Are all its Fourier coefficients real? Of course the Hecke operators $T_n$ are selfadjoint for $(n,N)=1$, but is it also true for all $n$?
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A problem involving the converse of Ramanujan's theta operation

Let $A := \mathbb{C}[E_2;E_4,E_6]$ be the algebra of (almost) modular forms for $\mathrm{SL}_2(\mathbb{Z})$. (The weight $2$ Eisenstein series $E_2 := 1 - 24\sum_{n \geq 1} \sigma_1(n)q^n$ satisfies ...
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2answers
387 views

Using Eichler-Selberg trace formula to compute dimension of modular forms?

Is it possible to use Eichler-Selberg trace formula to compute the dimension of modular forms of weight $k$ for $SL(2,\mathbb Z)$? This was computed by classical methods such as Riemann-Roch.
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1answer
360 views

The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13

When $N > 1$, the modular forms $\Delta(z)$ and $\Delta(Nz)$ are algebraically independent over the complexes, and the same then is true of their expansions at infinity. But using the fact that the ...
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2answers
1k views

Why should I care about topological modular forms?

There seems to be a lot of recent activity concerning topological modular forms (TMF), which I gather is an extraordinary cohomology theory constructed from the classical theory of modular forms on ...
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0answers
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Explicit generators of maximal ideals in completed Hecke algebras

A particular question: Let M be the subset of Z/3[[x]] consisting of those power series that are reductions of elements of Z[[x]] that arise as expansions of modular forms for Gamma_0 (2). ...
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Cube-root of j-invariant [closed]

Cube-root of j-invariant is a modular function of level 3, does it have similar property as j-invariant? How about the its minimal polynomial? In particular, are it's coefficients much smaller like ...
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1answer
137 views

Real cusp forms

Most literature on modular functions (invariant or covariant with weight k under the full modular PSL_2(Z) group) treats holomorphic functions and introduce the notion of cusp forms (modular functions ...
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309 views

Representation-theoretic operations on modular forms

Let $A$ and $B$ be Hecke eigenforms of some weight $k$ and level $N$. We know that there are irreducible representations $\rho_a$, $\rho_b$ of the absolute Galois group of $\mathbb{Q}$ whose trace of ...
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547 views

Eichler-Shimura over Totally Real Fields

By Eichler-Shimura over totally real fields I mean the following conjecture. Conjecture. Let $K$ be a totally real field. Let $f$ be a Hilbert eigenform with rational eigenvalues, of parallel weight ...
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2answers
285 views

Adelic methods for classical modular forms

Many conjectures about properties of automorphic forms on $\mathrm{GL}(2)$ can be formulated in the basic language of classical modular forms (i.e. Hecke forms that are holomorphic on $\mathbb{H}$ or ...
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4D polytope analogues of the icosahedron/Rogers-Ramanujan continued fraction relationship?

The formula for the j-function which employs polynomial invariants of the icosahedron, $$j(\tau)=-\frac{(r^{20} - 228r^{15} + 494r^{10} + 228r^5 + 1)^3}{r^5(r^{10} + 11r^5 - 1)^5}$$ where, ...
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1answer
265 views

What does the Jacquet-Langlands correspondence say about quaternion algebras of class number one?

If F is a totally real number field of degree n, and A is a definite quaternion algebra over F, I understand (not really) the Jacquet Langlands correspondence to construct a modular form in n ...
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401 views

Gross's paper on Heegner points

I try to read Gross's paper on Heegner points and it seems ambiguous for me on some points: Gross (page 87) said that $Y=Y_{0}(N)$ is the open modular curve over $\mathbb{Q}$ which classifies ordered ...
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319 views

Benedict Gross's paper on companion forms

In the page 458 of his paper(A tameness criterion for Galois representations associated to modular forms), Gross wrote the following "A detailed analysis of $U_p(Af)+V_p(<p>f)$ shows that it ...
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1answer
148 views

Real modular form, inverse transform

The real Eisenstein series $G_s^* = \frac{\Gamma(s)}{\pi^s} \sum'_{m,n}\frac{Im(\tau)}{|m+n \tau|^{2s}}$ admits the following integral representation (their Mellin transform): $G_s^* = ...
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1answer
227 views

Does the following operation on modular forms yield something modular?

Let $f(z) = \sum_{n=0}^\infty a_nq^n$ be the fourier expansion of a (quasi-)modular form (with $q = e^{2\pi i z}$). Consider the following related functions: $$f_{m,k}(z) = \sum_{n=0}^\infty a_{mn + ...