Questions tagged [modular-forms]

Questions about modular forms and related areas

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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 $...
Terry Tao's user avatar
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7 votes
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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 $(r,s)$...
Henri Cohen's user avatar
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2 votes
2 answers
208 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
372 views
+50

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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3 votes
0 answers
135 views

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
103 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
606 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
185 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
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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_{\...
Spoilt Milk's user avatar
2 votes
0 answers
60 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
140 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
198 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
83 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
379 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
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0 answers
218 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
196 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
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1 answer
285 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
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2 votes
1 answer
274 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
207 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
127 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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0 answers
139 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
186 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
76 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
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250 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
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2 votes
0 answers
127 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
620 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
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4 votes
0 answers
120 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
112 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
138 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
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5 votes
0 answers
114 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
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7 votes
1 answer
379 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
130 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
181 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
58 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
  • 338
3 votes
1 answer
158 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
1 vote
0 answers
96 views

Reference for modularity of the Andrews–Gordon–Rogers–Ramanujan identities?

The right-hand side of the identity https://mathworld.wolfram.com/Andrews-GordonIdentity.html is a $q$-series $\frac{(q^i,q^{2k+1-i},q^{2k+1};q^k)_\infty}{(q;q)_\infty}$; is there a reference of its ...
Yifeng Huang's user avatar
1 vote
0 answers
87 views

Numerical strategies for evaluating a modular invariant infinite sum

I'm working on a problem that involves the numerical evaluation of the following infinite sum: $$ \sum_{m=-\infty}^{\infty} \ln \left|1\pm e^{-2\pi \tau_1 \sqrt{m^2+x^2/(4\pi^2\tau_1)}-2 \pi i \tau_0 ...
Ganymed_'s user avatar
6 votes
2 answers
1k views

Fourier coefficients of modular forms

Given any nonzero modular form $f$ (of any weight, any level, any character), consider its $q$-expansion $f(z) = \sum_n a(n) q^n$, where $q=\exp(2\pi iz)$. Proposition: infinitely many of the ...
Erich Selder's user avatar
4 votes
1 answer
238 views

The Wilton-type bounds involving half-integral weight cusp forms

There is a basic question which puzzles me for a while, and maybe look naive for some experts here. The question is the following: Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be ...
hofnumber's user avatar
  • 553
1 vote
0 answers
86 views

On level-$12$ of the McKay-Thompson series of the Monster and the Domb numbers

(This continues from level 10.) Given some moonshine functions $j_{N}$. There are nice descending and consistent relations for levels $6m$ with $m= 2,3,5,$ $$j_{12A} = \left(\sqrt{j_{12H}} + \frac{\...
Tito Piezas III's user avatar
1 vote
0 answers
96 views

On level $6$ of the McKay–Thompson series of the Monster and Apéry numbers, et al

After the McKay–Thompson series of levels $1$, $2$, $3$, $4$ of the Monster were mentioned in this MO post, level $6$ has very interesting relations as well. (Level 10 is in this post.) I. Level-6 ...
Tito Piezas III's user avatar
2 votes
1 answer
218 views

Multiplicity one for newforms modulo $p$

The strong multiplicity one theorem for newforms says the following. Suppose that $f_1 \in S_k(\Gamma_0(N_1))$ and $f_2 \in S_k(\Gamma_0(N_2))$ are newforms, where $N_1, N_2 \geq 1$ are arbitrary ...
Adithya Chakravarthy's user avatar
2 votes
1 answer
232 views

$\pi$-adic Galois representations of attached to newforms at $p \nmid N$ are crystalline

Is [Scholl, Motives for modular forms, Theorem 1.2.4 (ii)] proven for any $p$ independent of the weight? Concretely, let $f$ be a normalized eigenform of weight $w$. Let $p$ be a prime not dividing ...
user471019's user avatar
3 votes
0 answers
77 views

Additional symmetries in Theta-like function

cross-posted from https://math.stackexchange.com/questions/4708694/curious-symmetry-in-a-theta-like-function Let $\Theta : \mathfrak{h}\times \mathfrak{h} \to \mathbb{R}$ be defined as follows $$ \...
Testcase's user avatar
  • 541
3 votes
0 answers
80 views

Shimura lift is isomorphic iff twisted Hecke $L$ function does not vanish at central point

Let $S_{2k}(1)$ and $S_{k+1/2}(4)$ denote the set of modular forms of weight $2k$ for $SL_2(\mathbb{Z})$ and weight $k+1/2$ for the congruence subgroup $\Gamma_0(4)$, respectively. Consider the Kohnen ...
1.414212's user avatar
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3 votes
0 answers
204 views

Using Ramanujan-type "Legendrian" sequences to find new formulas for $\frac1{\pi}$?

I. Recurrences (Continued from this post.) In Cooper's 2012 paper, "Sporadic sequences, modular forms and new series for 1/π", he did a computer search for the recurrence relation, $$(n+1)^3 ...
Tito Piezas III's user avatar

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