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Questions tagged [modular-forms]

Questions about modular forms and related areas

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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_{\...
QU Binggang's user avatar
4 votes
1 answer
159 views

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 ...
yhb's user avatar
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214 views

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 $gz=(az+b)/(cz+d)$...
Emmanuel Royer's user avatar
1 vote
0 answers
43 views

$ \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 ...
user967210's user avatar
1 vote
0 answers
87 views

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 ...
yhb's user avatar
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1 vote
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68 views

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 ...
User0829's user avatar
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4 votes
2 answers
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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\...
Nomas2's user avatar
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2 votes
2 answers
307 views

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 ...
Nomas2's user avatar
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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)...
Wolfgang's user avatar
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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 ...
John Baez's user avatar
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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 ...
Joe's user avatar
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27 votes
2 answers
973 views

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 $...
Terry Tao's user avatar
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8 votes
1 answer
162 views

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 $(r,s)$...
Henri Cohen's user avatar
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2 votes
2 answers
218 views

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^{...
Christopher-Lloyd Simon's user avatar
6 votes
1 answer
440 views

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 ...
user15243's user avatar
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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 $1$ ...
Zakariae.B's user avatar
3 votes
0 answers
111 views

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 (...
Ancient Antagonist's user avatar
4 votes
2 answers
1k views

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 ...
Daigaku no Baku's user avatar
11 votes
2 answers
628 views

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 ...
KStarGamer's user avatar
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 ...
LWW's user avatar
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3 votes
1 answer
120 views

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_{\...
Spoilt Milk's user avatar
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(\...
Henri Cohen's user avatar
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2 votes
1 answer
163 views

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 $p < 11$...
Offlaw's user avatar
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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! ...
ModularForms's user avatar
4 votes
0 answers
95 views

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})}}...
Student's user avatar
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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: ...
user15243's user avatar
  • 474
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?
Cosimo's user avatar
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4 votes
1 answer
202 views

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 $|q|\lt 1$....
Nomas2's user avatar
  • 367
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})...
Nomas2's user avatar
  • 367
2 votes
1 answer
362 views

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 ...
AlphaNotKnows's user avatar
11 votes
1 answer
212 views

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 ...
Henri Cohen's user avatar
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2 votes
1 answer
130 views

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}...
hofnumber's user avatar
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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 ...
Adithya Chakravarthy's user avatar
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 ...
Alexander's user avatar
1 vote
0 answers
81 views

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),\...
Kevin's user avatar
  • 111
4 votes
0 answers
257 views

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 ...
Dendrit's user avatar
  • 41
2 votes
0 answers
129 views

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 $C_i$, ...
ayan's user avatar
  • 21
9 votes
1 answer
632 views

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 ...
GuoJi's user avatar
  • 245
4 votes
0 answers
128 views

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 ...
Rayane B.'s user avatar
17 votes
3 answers
2k views

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 ...
Coherent Sheaf's user avatar
8 votes
2 answers
2k views

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 ...
ShyamalSayak's user avatar
1 vote
0 answers
122 views

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 ...
Jan-Willem van Ittersum's user avatar
1 vote
0 answers
146 views

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)...
hofnumber's user avatar
  • 553
5 votes
0 answers
123 views

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 ...
Seewoo Lee's user avatar
  • 1,911
7 votes
1 answer
383 views

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 ...
ModularForms's user avatar
2 votes
0 answers
281 views

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\{...
T. Amdeberhan's user avatar
3 votes
0 answers
139 views

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 ...
stillconfused's user avatar
4 votes
1 answer
185 views

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 ...
User0829's user avatar
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1 vote
0 answers
61 views

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 $...
yhb's user avatar
  • 390
3 votes
1 answer
167 views

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 ...
swati setia's user avatar

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