# Questions tagged [modular-forms]

Questions about modular forms and related areas

1,309
questions

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### The product of Petersson norm of Hecke eigenforms

Let $f_1,\dots,f_k$ be the normalized Hecke eigenforms in $S_{12k}(\operatorname{SL}_2(\mathbb{Z}))$. Do we have asymptotic formula for the quantity $\prod_{i=1}^k \langle f_i,f_i \rangle_{\...

4
votes

1
answer

159
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### Atkin-Lehner involution on the modular abelian varieties

Let $f$ be a CM modular forms with coefficients in the imaginary qudaratic field $K=Q(\sqrt{-3}))$ which correspond to an elliptic curve $E$ defined over $K$. Then Shimura constructed an abelian ...

4
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0
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214
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### Holomorphic non vanishing modular form

Let $\mathcal{O}(\mathcal{H})^\times$ be the multiplicativee group of holomorphic functions on the Poincaré half-plane $\mathcal{H}$ that do not vanish there.
Let $j(g,z)=(cz+d)$ and $gz=(az+b)/(cz+d)$...

1
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0
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### $ \lambda^{*}(n) $ minimal polynomials

I already asked a closely related question on MO, but didn't received any answer.
Considering the modular lambda function, the values of $ \lambda^{*}(n) $ for some integers are given on here.
Is ...

1
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0
answers

87
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### Period lattice of cubic twists of CM modular forms

Let $K=\mathbb{Q}(\sqrt{-3})$ be a CM field. Let $E_1:y^2=x^3+1/4$ and $E_p:y^2=x^3+p^2/4$ where $p\equiv 1\mod 3$ is a prime. Let $f_1$ and $f_p$ be the modular forms of $E_1$ and $E_p$. They are ...

1
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0
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68
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### Global minimal discriminants of elliptic curves and Galois representations

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $\Delta$ be the global minimal discriminant. By A. Wiles et al. $E$ is modular so that there is a corresponding modular form $f$. I am wondering if ...

4
votes

2
answers

160
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### Eisenstein $E_2$ at imaginary quadratic arguments

In the paper On Epstein's zeta-function, Chowla and Selberg give a formula for evaluating the Dedekind eta function
$$\eta (\tau)=e^{\pi i\tau/12}\prod_{n=1}^\infty (1-e^{2\pi i n\tau}),\quad \Im\tau\...

2
votes

2
answers

307
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### How did Ramanujan find $\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}?$

The formula
$$\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}$$
(in older notation) appears as eq. 38 in Ramanujan's paper Modular equations ...

1
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0
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94
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### How to prove this peculiar relationship between minimal polynomials of Ramanujan class invariants?

The Ramanujan class invariants (a.k.a. "Ramanujan-Weber class invariants")
are defined for $n>0$ by
$$G_n=2^{-1/4}e^{\pi\sqrt{n}/24}\prod_{k=0}^\infty \left(1+e^{-(2k+1)\pi\sqrt{n}}\right)...

6
votes

1
answer

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### What is this huge generalization of the Modularity Theorem?

A friend of mine wrote:
The point is of course that the Modularity Theorem (as I stated it) is/should be really just a special case of some much bigger theorem which sets up a bijection between ...

3
votes

0
answers

90
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### How to get $U(N)_k$ Kac-Moody modules and characters from $N \cdot k$ Dirac Fermions using $U(N \cdot k)_1 / SU(k)_N$?

It is known that the $U(N)_k$ Kac-Moody algebra can be written as the coset $U(N)_k = U(N \cdot k)_1 / SU(k)_N$. (This fact is related to the level-rank duality of $U(N)_k \leftrightarrow U(k)_N$.) A ...

27
votes

2
answers

973
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### Proofs of the valence formula that avoid tricky contours?

$\DeclareMathOperator\ord{ord}\DeclareMathOperator\Im{Im}$The valence formula for a modular form asserts that if $f: \mathbf{H} \to \mathbf{C}$ is a modular form of weight $k$ on the upper half-plane $...

8
votes

1
answer

162
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### Finding a rational point of large height on an elliptic curve knowing a real approximation

Let $y^2=x(x^2+n)$ be an elliptic curve with $n\in\Bbb Z$ (the same question can of course
be asked for a general e.c). I know (e.g. it has rank 1) that there exists a nontrivial
rational point $(r,s)$...

2
votes

2
answers

218
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### Conditional convergence of exponential sums related to a Hecke modular form

Definition
Consider the Fourier coefficients $\psi(n)$ of the modular form $\eta^4(6\tau)$,
which are defined in terms of $q=\exp(i2\pi\tau)$ by the identity:
$$\eta^4(6\tau) = q \prod_1^\infty (1-q^{...

6
votes

1
answer

440
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### Symmetric power lift of modular forms

Let $f_1$ and $f_2$ be two cuspforms of weights $k_1$ and $k_2$ and nebentypus $\epsilon_1$ and $\epsilon_2$ respectively such that $f_1 \neq f_2 \otimes \chi$ for some Dirichlet character $\chi$ of ...

3
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0
answers

155
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### Congruences between Eisenstein series and cusp forms

Let $k\geq 4$ be an even integer. Let $p>k$ be a prime
such that $p\mid B_k$, the $k$th Bernoulli number. Then there is a primitive cusp form $f=\sum_{n\geq1}c(n, f)q^n$
of weight $k$ and level $1$ ...

3
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0
answers

111
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### Is there any notion of Poincaré series for Hermitian modular forms?

I have been studying modular forms and their generalisations for a year or so. It is a very interesting fact that the space of cusp forms $S_k$ is generated by the Poincaré series of exponential type (...

4
votes

2
answers

1k
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### Are umbral moonshine and umbral calculus connected?

In a 2013 article, Cheng, Duncan and Harvey introduce the concept of umbral moonshine as a generalization of monstrous moonshine. The terminology they use, starting with the title, is common in umbral ...

11
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2
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628
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### Does the number of roots of the modular form associated to an elliptic curve, on the positive imaginary axis, equal the analytic rank?

Recently I've been playing around with elliptic curves and have seemingly come up with a conjecture that I could not find elsewhere:
Let $E$ be an elliptic curve, and $f(q)$ its associated modular ...

3
votes

1
answer

195
views

### $p$th Fourier coefficients of newforms for ramified primes $p$

This question is about some basic(classical) results on Atkin-Lehner-Li theory of newforms. Let $f$ be a (normalized) newform of level $N$ and character $\epsilon$. Denote the $n$th Fourier ...

3
votes

1
answer

120
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### Fourier expansion of half-integral weight Eisenstein series associated with Kohnen's plus space

The Eisenstein series associated with Kohnen's plus space in $\Gamma_{0}(4)$ is expressed as follows,
\begin{align}
\begin{split}
E_{k + \tfrac{1}{2}}^{\infty}(\tau) =& \sum\limits_{\...

2
votes

0
answers

63
views

### Simultaneous computation of the three Weber modular functions

Recall that the three classical Weber modular functions are defined by
$f(\tau)=e^{-\pi i/24}\eta((\tau+1)/2)/\eta(\tau)$,
$f_1(\tau)=\eta(\tau/2)/\eta(\tau)$, and
$f_2(\tau)=\sqrt{2}\eta(2\tau)/\eta(\...

2
votes

1
answer

163
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### Cusp forms of weight 2 and level $\Gamma_0(p)$ where $p < 11$

Using Hida theory, we can prove that there is a cusp form of weight 2 and level $\Gamma_0(11)$. Are there ways to prove that there is no cusp forms of weight 2 and level $\Gamma_0(p)$ where $p < 11$...

2
votes

1
answer

209
views

### Generating function over primes in an arithmetic progression

Given a newform $\sum_{n=1}^{\infty}a(n)q^n$. Is the generating function
$$
\sum_{p\equiv a\pmod{m}}a(p)q^p
$$
over the primes $p\equiv a\pmod{m}$ still a modular form? Any help is highly appreciated! ...

4
votes

0
answers

95
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### Elliptic integral as quantity associated with Riemann surface?

There are many elliptic integrals, so to show my point let me
just pick one of them (complete elliptic integral of the first
kind [1]):
$$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})}}...

4
votes

1
answer

429
views

### Automorphic representation of GL(1)

These might be very silly questions, but somehow I am not able to understand it or I might have misunderstood something.
I am reading automorphic forms from this book.
What I have understood till now:
...

0
votes

0
answers

221
views

### Reference book on the relation between modular forms and elliptic curves

What is a modern reference book to understand the relation between modular forms and elliptic curves after the proof of the Taniyama–Shimura theorem?

4
votes

1
answer

202
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### Abscissa of convergence of the $\tau$ Dirichlet series

Define the $\tau$ Dirichlet series $L$ by
$$L(s)=\sum_{n=1}^\infty \frac{\tau (n)}{n^s}$$
where $\tau$ is defined by
$$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$
where $|q|\lt 1$....

0
votes

1
answer

288
views

### Uniqueness of the $J$ invariant

It seems that
The $J$ invariant is the unique modular function of weight zero for $\operatorname{SL}(2,\mathbb{Z})$ which is holomorphic away from a simple pole at the cusp such that
$$J(e^{2\pi i/3})...

2
votes

1
answer

362
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### How to prove Siegel upper half plane is a hermitian symmetric space

There is a statement that is Siegel upper half plane of genus g, $\mathbb{H}_g:=\left\{Z=X+i Y \in M_n(\mathbb{C}) \mid X, Y \text { real }, Z=Z^{T}, Y=\operatorname{Im} Z>0\right)$ is isomorphic ...

11
votes

1
answer

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### Properties of the ring of all holomorphic modular forms

Let $R$ be the ring of modular forms on congruence subgroups, say
of integral or half integral weight. In other words
$$R=\bigcup_{N\ge1}\bigoplus_{k\in(1/2)\Bbb Z}M_k(\Gamma(N))\;.$$
The important ...

2
votes

1
answer

130
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### On the square mean of Fourier coefficients of cusp forms

I have a question which may look naive for many experts here:
For any primitive holomorphic form $f$ of level $M$ ($M\in \mathbb{N}$), whether or not one has the lower bound that:
$$\sum_{X<n\le 2X}...

2
votes

0
answers

142
views

### When do Fourier coefficients vanish in Hida families?

Suppose you have a Hida family with $q$-expansion $F = \sum_{n=1}^{\infty} a_n(T) q^n$, where the coefficients $a_n(T)$ are power series in $\mathbb{Z}_p [[T]]$. Assume that $F$ is a cuspidal ...

4
votes

1
answer

190
views

### Identity related to Ramanujan's congruences

A very simple question: how do you prove the following identity:
$$\sum_{k=0}^\infty p_{5k+4}x^k=5\frac{\phi(x^5)^5}{\phi(x)^6},$$
where
$$\phi(x)=\prod_{n=1}^\infty 1-x^n,$$
and $p_n$ is the ...

1
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0
answers

81
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### Effective bound of Fourier coefficients of weakly modular forms

Assuming $$f=\sum_{n=n_0}^\infty c_f(n/m)e^{{2\pi inz}/{m}},\quad (n_0\in\mathbb Z, m\in\mathbb Z_{\geq1})$$ is a weakly modular form with weight $k$ and congruence subgroup $\Gamma=\Gamma_0(N),\...

4
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0
answers

257
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### Special case of Eichler–Shimura

I'm reading ‘Rational Points on Elliptic Curves’ by Silverman and Tate, and the exercise 4.6 is about the following special case of the Eichler–Shimura theorem. Let $C$ be the elliptic curve given by ...

2
votes

0
answers

129
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### Isom-functor for generalized elliptic curves is representable

I am studying Deligne-Rapoport's 'Les Schémas de Modules de Courbes Elliptiques'. The following excerpt is from the proof of Theorem 2.5, Chapter III, page DeRa-61,
(page DeRa-61) (*) For $C_i$, ...

9
votes

1
answer

632
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### What is the value of $j(2\sqrt{-163})$?

My question is how to calculate the value of $j(2\sqrt{-163})$ and its minimal polynomial, where the $j$ is elliptic modular function (see https://mathworld.wolfram.com/j-Function.html). The class ...

4
votes

0
answers

128
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### Road map for learning about the computational/general theory of modular curves/isogenies of abelian varieties for cryptography

I am a graduate math/crypto student. So I've had some free time last year and I heard about elliptic curves in cryptography and how a resilient cryptosystem got demolished by a spectacular attack ...

17
votes

3
answers

2k
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### Are some congruence subgroups better than others?

When I first started studying modular forms, I was told that we can consider any congruence subgroup $\Gamma\subset\operatorname{SL}_2(\mathbb{Z})$ as a level, but very soon the book/lecturer begins ...

8
votes

2
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### Trivial homomorphism from a non-abelian group to an abelian group

I am stuck on this problem and cannot seem to find a good reasoning for drawing the required conclusion. The problem is as follows:
Let $m\in \mathbb{N}$ and $n>3$. I want to show that there can be ...

1
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0
answers

122
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### Invariant polynomials under a non-standard group action

There is a whole theory of finding the invariant polynomials for matrix groups $\Gamma$ acting on the polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$. I would be interested in finding invariant ...

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0
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146
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### On two formulas involving the $k$-fold divisor function $d_k$ and the function $r_k$

I have a puzzle which needs some help form the experts here. Let $d(n)$ be the divisor function, and $d_k(n)$ the $k$-fold divisor function.
I) It is known that, for any positive integer $h$,
$$d(n+h)...

5
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0
answers

123
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### Sphere packing and modular forms in known dimensions (maybe 2)

Viazovska constructed magic functions via integral transforms of (quasi-)modular forms that gives a tight bound for linear programming bounds in 8 and 24 dimensions (with other mathematicians after ...

7
votes

1
answer

383
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### Lacunary weight one modular forms

By a result of Serre, it’s known that a cusp form of weight $k\geq2$ and level $\Gamma_0(N)$ with some $\chi$ is lacunary if and only if it is in the space of CM newforms. Is there a similar result ...

2
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281
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### Is Sturm's theorem able to do these?

$\newcommand{\Ord}{\operatorname{Ord}}$Let $p$ be a positive integer and $F(q)=\sum A(m)q^m$ be a formal power series with integer coefficients. Then $\Ord_p(F(q))$ is defined by
$$\Ord_p(F(q)):=\min\{...

3
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0
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139
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### Explicit relationship between Gross--Zagier's On Singular Moduli, and Heegner Points and Derivatives of L-series

In various places in the literature surrounding the Gross--Zagier formula, the results in Heegner points and the derivatives of $L$-series (hereafter, Heegner points) are referred to as a ...

4
votes

1
answer

185
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### Modular interpretation of the stalks of modular curves

One may see the modular interpretation of (points of) modular curves in the very first course on modular forms and modular curves. I am wondering if it is well-known that modular interpretation of the ...

1
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0
answers

61
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### The modular forms of cubic twist of elliptic curves [duplicate]

I want to ask the same question with Does the modular form associated to cubic twist of a elliptic curve $E$ corresponds to some twist of $f_E$?
Let $E$ be an elliptic curve defined over $\Bbb Q$ and $...

3
votes

1
answer

167
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### Computations of half-integer forms in SAGE/Magma

I am currently going through Shimura's paper on half-integer weight modular forms. I would like to understand given a 𝑞-expansion of half-integral weight modular forms of arbitrary level and ...