# Questions tagged [modular-forms]

Questions about modular forms and related areas

1,039
questions

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### Necessary and sufficient condition for a prime to be represented by an arbitrary positive definite binary quadratic form $ax^2+bxy+cy^2$

Given an arbitrary (but fixed) positive definite primitive integral binary quadratic form $g(x, y)=ax^2+bxy+cy^2$, and let $m$ be an arbitrary integer. We will denote the discriminant of $g$ by $D=D_g=...

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226 views

### Surjectivity in Deligne-Serre

Let $f$ be a newform of weight $k$ and level $N$ with integer coefficients. Deligne-Serre theorem theorem says there exist a nice associated representation $\rho_{f}^{(\ell)}:\text{Gal}(\overline{\...

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116 views

### Level vs. conductor of a supercuspidal representation

What is the relation between level and conductor of a supercuspidal representation of $\operatorname{GL}_2(\mathbb{Q}_p)$ for some prime $p$?
Proposition 3.4 in Loeffler and Weinstein - On the ...

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136 views

### Half integral weight modular forms that reduce to a nonzero constant modulo a given prime

Let $B_k = \frac{N_k}{D_k}$ be the reduced numerator and denominator of the $k$-th Bernoulli number. For a given prime $p>2$, the (unconventionally normalized) Eisenstein series $E_{p-1}(z) = N_{p-...

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248 views

### A general question on Sato-Tate

What is the most general version of Sato-Tate ? Like, I know when $f$ is an eigenform (lying in the space of new-forms), without CM, of weight $k$ and level $N,$ then $\frac{a(p)}{p^{(k-1)/2}}'s$ are ...

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69 views

### Slopes of classical newforms

Let $f$ be a newform in $S_k(\Gamma_1(Np^r))$ with $r\geq 1$, and let $a_p$ be the $U_p$ eigenvalue of $f$.
If $f$ is in $S_k(\Gamma_0(Np^r))$, it seems to be a well known consequence of Atkin-...

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115 views

### A complex analytic version of the eigencurve

I am very much a beginner to the theory of eigencurves so there might be many mistakes in what follows, especially since it is all very speculative.
My understanding of the eigencurve $\mathcal C_{N,...

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52 views

### Recurrence relation between $a(p^{2n})$'s

Let $f$ be a Hecke eigenform of weight $k,$ and $a(p^n)$ be the $p^{n}$th Fourier co-efficient of $f$. I need to a determine a recurrence relation (Hecke type) between $s(n)=a(p^{2n})'s$. We can write ...

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145 views

### Hecke correspondence and the trace map of differential forms

Let $k$ be a field, $X$, $Y$, $Z$ smooth geometrically connected curves, and $f: Z \to X$, $g : Z \to Y$ finite morphisms.
Suppose that $f$ is separable.
Then we have $f_* \circ g^* : \Gamma(Y, \...

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239 views

### How to prove some identities about infinite product?

Recently, I read one paper titled Modular equations and approximations to π by Ramanujan, in which there are some formulas for $q=\pi i \tau$( where $\tau=x+yi, y>0$, hence $|q|<1)$ :
$$\prod_{...

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69 views

### Sato-Tate for modular forms

Let $f$ be non-CM holomorphic modular forms of weight $k\geq 2$, and $a(p)$ be its $p^{th}$ Fourier-coefficient. the Sato–Tate conjecture is known to be true in this case, I want to know whether such ...

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253 views

### Explanation for Deligne-Rapoport

Now I'm reading chapter 2 of Deligne-Rapoport's Les schemas de modules de courbes elliptiques.
But its proof is too brief for me.
Are there some good references to understand this paper (or theory ...

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91 views

### (Explicit) Basis for Kohnen's plus-space of modular forms of half integral weight

Sorry if this is trivial, but I could not find any reference.
Let $k,a,b$ be integers. The space of modular forms of integer weight $M_k(\text{SL}_2(\mathbb{Z}))$ admits a basis of the form $\{ E_4^...

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86 views

### How canonical are integral lifts of Hasse invariants and other mod-$p$ modular forms?

Let $p$ be an odd prime. Recall that the mod $p$ Hasse invariant $A$ of an elliptic curve is an $\mathrm{SL}(2,\mathbb Z)$-modular form of weight $p-1$ defined over $\mathbb{F}_p$. Writing $\overline{\...

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61 views

### A weakly holomorphic modular form is a harmonic maass form

It is known that for $\Gamma_0(N)$, a weakly holomorhpic modular form is a harmonic maass form. Here is the definitions.
A weakly modular form $f$ for $\Gamma_0(N)$ is a meromorphic function on the ...

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279 views

### An explicit description of $X(3)$ and its universal generalized elliptic curve

I'm struggling with the proof of 2.21 of Saito's "Fermat's Last Theorem".
Let $\omega$ be a primitive 3rd root of unity, $X(3) = \mathbb{P}^1_{\mathbb{Q}(\omega)}$, and $E = \{ X^3 + Y^3 + Z^3 - 3 \...

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41 views

### Meaning of extended principal part of weakly holomorhpic modular forms

In p.312 of 'Rhoades, Robert C., Linear relations among Poincaré series via harmonic weak Maass forms. Ramanujan J. 29 (2012), no. 1-3, 311–320', the author defines the extended principal part at ...

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274 views

### Siegel's bad character

Let $K$ be an imaginary quadratic field with discriminant $d_K$. Suppose that $d_K=gt$, where either $g,t$ are discriminants or have the value $g=1,t=d$. Let $f$ be an additional discriminant of a ...

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71 views

### Computing all eta quotients of given weight and level

I have written a rather naive program for finding all holomorphic eta quotients of
given weight and level (and varying character). When the level has few divisors it is
very fast, but incredibly slow ...

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190 views

### Weight 3 modular form associated to singular abelian surfaces?

Given an extremal K3 surface $S$ over $\mathbb{Q}$ (i.e. a K3 surface with maximal Picard rank) there is a 2-dimensional Galois representation on the transcendental lattice $T(S)$, and an associated ...

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139 views

### Modularity of elliptic curves with only minimal lifting

I have been trying to understand a bit of the basics of deformations of Galois representations. One point which leaves me curious now is that proving modularity lifting with arbitrary ramification on ...

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104 views

### Atkin-Lehner operator on supercuspidals

Suppose $f$ is a normalized cuspidal eigenform of level $p^2N$ ($p\nmid N$) and trivial character, such that the corresponding representation at $p$ is supercuspidal. Now suppose $\chi$ is primitive ...

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48 views

### Paramodular forms with level and Iwahori subgroups?

Given an integer $N>0$, not necessarily prime, we have the paramodular group $K(N) \subset \text{Sp}_{4}(\mathbb{Q})$, which consists of matrices of the form
$$\begin{bmatrix} * & *N & * &...

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276 views

### Integrality of Atkin-Lehner operator for $\Gamma_1(N)$

A result due to B. Conrad (http://math.stanford.edu/~conrad/papers/prasanna-inv.pdf, Theorem A.1) states that the Atkin-Lehner operator $w_{Q,k}$ is $\mathbb{Z}[1/Q]$-integral on $M_k(\Gamma_0(N))$. ...

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383 views

### Why is modular forms applicable to packing density bounds from linear programming at $n\in\{8,24\}$?

Sphere packing problem in $\mathbb R^n$ asks for the densest arrangement of non-overlapping spheres within $\mathbb R^n$. It is now know that the problem is solved at $n=8$ and $n=24$ using modular ...

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64 views

### Nature of Fourier coefficient of a modular form after applying a certain map (trace operator)

Asking this here because of no response at (MathStackExchange).
Let $N|M$, and consider the trace operator $Tr^M_N$ defined on $M_k(\Gamma_0(M))$ - vector space of modular forms of weight $k$ for ...

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75 views

### “ Laurent expansion” of quasi-periodic complex complex function

Suppose a complex function $f(z)$ depends only on $z$, and satisfies the quasi-periodicity in both directions:
$$f(z+ a_x)= e^{i \theta_{a_x}} f(z)$$
$$f(z+ i a_y)= e^{i \theta_{a_y}} f(z)$$
where $\...

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158 views

### Analogous theorem for Hilbert modular forms

I have studied modular forms and saw a correspondence like a newform correspond to a automorphic representation of $\mathrm{GL}_n(\mathbb{A_Q})$. Does any similar result holds for Hilbert modular ...

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74 views

### Conceptual meaning of a non-linear relation connecting $6$ Mordell integrals?

Define Mordell integral by
$$
\phi_\alpha(\theta)=\int\limits_0^\infty\frac{\cos\pi \theta x}{\cosh \pi x}\,e^{-\pi \alpha x^2}dx.\tag{1}
$$
There are a lot of linear relations connecting integrals ...

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110 views

### Properties of Mod $\ell^m$ Galois representation associated to modular form

(Sorry for my poor english..)
Let $F(z)\in S_{2k}(SL_2(\mathbb{Z})$) be a newform and $\ell$ be a prime larger than $3$. Let $K$ be a some number field and $v$ be a prime of $K$ over $\ell$. Let $K_v$...

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146 views

### Corollaries of the halo conjecture that do not involve the eigencurve

In the theory of p-adic modular forms there is a certain construction called the Coleman-Mazur eigencurve. The spectral halo conjecture roughly states that if you remove a closed subdisc of the weight ...

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159 views

### Algebra of Hecke operators on $M_k(\mathrm{SL}_2\mathbb{Z})$ is an integral domain?

Let $M_k(\mathrm{SL}_2\mathbb{Z})$ be the space of modular forms of (integer) weight for the full modular group. Let $\mathbf{H}$ denote the Algebra generated by the Hecke operators $T_n$. Is $\mathbf{...

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308 views

### Are the L-functions of a normalized newform and the corresponding cuspidal representation equal?

Let $f \in S_k(\Gamma_0(N))$ be a normalized newform with Fourier expansion
$$f(z) = \sum\limits_{n=1}^{\infty} a_n e^{2\pi i z n}$$
and $a_1 = 1$. Then $f$ is an eigenfunction of all Hecke ...

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87 views

### Does the following have an expression in terms of modular form?

Let f(q) be the following power series:
$$f(q)=\sum_{k\geq0}q^{k+\frac{1}{2}}\frac{(4k+1)!}{(2k+1)!(k!)^2}$$
How to relate it to quasi modular form ?

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402 views

### conductor formula

Let $\pi_p$ be an irreducible representation of $GL_2(\mathbb{Q}_p)$. Assume $\pi_p$ is ramified,hence it will have a positive conductor. Consider $sym^3(\pi_p)$ which is a representation of $GL_4(\...

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100 views

### Eichler Shimura in higher genera

Let $\mathcal{M}$ denote the moduli stack of smooth elliptic curves over $\mathbb{C}$. There is a local system, $\mathcal{H}$, on $\mathcal{M}$ with fibre $H^{1}(E,\mathbb{C})$ at the $\mathbb{C}$-...

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194 views

### Generalisation of modular forms

I am looking for a generalisation of a modular form that transforms as something like:
$f(\frac{a \tau+b}{c \tau+d}) = (c \tau+d)^k c^k f(\tau)$
I understand this cannot be literally true, as the ...

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98 views

### Conductor formula for symmetric square transfer

Let $\pi$ be an automorphic cuspidal representation of $\operatorname{GL}_2(\mathbb A_{\mathbb Q})$, and let $\Pi = \operatorname{sym}^2(\mathbb\pi)$, which is a representation of $\operatorname{GL}_3(...

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138 views

### Is weakly holomorphic modular form finitely generated as module of modular function?

Let's use $M_k^{!}(\Gamma_{0}(N))$ to denote weakly holomorphic modular form with weight k, level $\Gamma_0(N)$(in particular, k might be negative). Then obviously $M_0^{!}(\Gamma_{0}(N))$ acts on it ...

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### Adjoint Selmer groups and Deformation rings

Let $\rho:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}_2(\bar{\mathbb{Z}}_p)$ be a $p$-adic Galois representation associated to a $p$-ordinary Hecke eigencuspform, let $...

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72 views

### Hodge-Tate map in families

I've been working with the Hodge-Tate map in the context of modular forms, but I don't understand really what it is. For an elliptic curve $E$ over $K$, a number field, the Hodge-Tate decomposition is ...

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167 views

### Modular form attached to a hecke character of cubic/quartic extension of rational numbers

We know that for each Hecke character of a quadratic extension of $\mathbb{Q}$, we can define a modular form (in fact a cusp form). We can find this construction in the book Topics in Classical ...

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164 views

### Shortest possible reasonably self-contained formulation of the modularity theorem

This is question in mathematical exposition, not research, I hope this is ok.
I am writing a book about great theorems. My question is: what is the shortest formulation of the modularity theorem, ...

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143 views

### Can the Petersson inner product $\langle f(z), f(2z) \rangle$ be zero?

Suppose $f$ is a weight $k$ cuspidal Hecke eigenform on $\Gamma_0(N)$. Then $f(2z)$ is a weight $k$ cuspform on $\Gamma_0(2N)$.
Is it possible that $f(z)$ and $f(2z)$ can be orthogonal (regarded as ...

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129 views

### Congruences of modular forms modulo other modular forms

Congruences between modular forms are certainly a big topic in number theory, maybe with
$$E_{p-1}\equiv 1 \mod p \qquad \text{for a prime }p\geq 5$$
as the easiest example. Sometimes, $p$ might be ...

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86 views

### Classical Hecke operators and Hecke algebra of type $A_1$

What's the relation between the classical Hecke operators (as defined in J. P. Serre's A course in arithmetic chapter 6) and the Hecke algebra of type $A_1$, i.e. the algebra generated by the vertices ...

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226 views

### Inside the construction of the Frey curve

Consider the frey curve $E\mathrel: y^2=x(x-a^{p})(x+b^{p})$ with conductor $N =2\prod_{p|(abc)^{2p}}p $. Frey assume that $p$ does not divide $(abc)^{2p} $ so the level of the cusp form predict by ...

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149 views

### Hecke operators - why is it well-defined?

I have a basic question concerning Hecke Operators in spaces of Modular Forms. I am followinf these notes: http://www.few.vu.nl/~sdn249/modularforms16/Notes.pdf. In page 49, the author writes
...

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108 views

### Decomposition of a sum of holomorphic squares into modular forms

Say there is a (non-holomorphic) real function $Z(\tau,\bar{\tau})$ which obeys the (non-holomorphic) modularity conditions of some weight $k$
$$Z(\tau+1,\bar{\tau}+1)=Z(\tau,\bar{\tau}),\qquad Z(-1/\...

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220 views

### Upper bound of summation $\sum_{m < \frac{1}{2}X} \frac{|a(m_1m_2^2)|}{m_1m_2^2} \log\frac{X}{m}$

I am studying the paper M. Ram Murty, V. Kumar Murty: Mean values of derivatives of modular $L$-series, Ann. of Math. (2) 133 (1991), no. 3, 447-475.
Let $L(s)=\sum_{n=1}^{\infty} \frac{a(m)}{m^s}$ ...