Questions tagged [modular-forms]
Questions about modular forms and related areas
1,298
questions
25
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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
0
answers
121
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 $(r,s)$...
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^{...
6
votes
1
answer
408
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 ...
3
votes
0
answers
140
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$ ...
3
votes
0
answers
104
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 (...
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
votes
2
answers
612
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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
186
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
114
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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
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(\...
2
votes
1
answer
144
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$...
2
votes
1
answer
199
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
$$
over the primes $p\equiv a\pmod{m}$ still a modular form? Any help is highly appreciated! ...
4
votes
0
answers
86
views
Elliptic integral as quantity associated with Riemann surface?
There are many elliptic integrals, so to show my point let me
just pick one of them (complete elliptic integral of the first
kind [1]):
$$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})}}...
4
votes
1
answer
388
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
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?
4
votes
1
answer
197
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 $|q|\lt 1$....
0
votes
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
$$J(e^{2\pi i/3})...
2
votes
1
answer
312
views
How to prove Siegel upper half plane is a hermitian symmetric space
There is a statement that is Siegel upper half plane of genus g, $\mathbb{H}_g:=\left\{Z=X+i Y \in M_n(\mathbb{C}) \mid X, Y \text { real }, Z=Z^{T}, Y=\operatorname{Im} Z>0\right)$ is isomorphic ...
11
votes
1
answer
208
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Properties of the ring of all holomorphic modular forms
Let $R$ be the ring of modular forms on congruence subgroups, say
of integral or half integral weight. In other words
$$R=\bigcup_{N\ge1}\bigoplus_{k\in(1/2)\Bbb Z}M_k(\Gamma(N))\;.$$
The important ...
2
votes
1
answer
127
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On the square mean of Fourier coefficients of cusp forms
I have a question which may look naive for many experts here:
For any primitive holomorphic form $f$ of level $M$ ($M\in \mathbb{N}$), whether or not one has the lower bound that:
$$\sum_{X<n\le 2X}...
2
votes
0
answers
139
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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
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 ...
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),\...
4
votes
0
answers
251
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 ...
2
votes
0
answers
127
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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
621
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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
121
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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
answers
2k
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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
vote
0
answers
115
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 ...
1
vote
0
answers
140
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 $h$,
$$d(n+h)...
5
votes
0
answers
116
views
Sphere packing and modular forms in known dimensions (maybe 2)
Viazovska constructed magic functions via integral transforms of (quasi-)modular forms that gives a tight bound for linear programming bounds in 8 and 24 dimensions (with other mathematicians after ...
7
votes
1
answer
381
views
Lacunary weight one modular forms
By a result of Serre, it’s known that a cusp form of weight $k\geq2$ and level $\Gamma_0(N)$ with some $\chi$ is lacunary if and only if it is in the space of CM newforms. Is there a similar result ...
2
votes
0
answers
281
views
Is Sturm's theorem able to do these?
$\newcommand{\Ord}{\operatorname{Ord}}$Let $p$ be a positive integer and $F(q)=\sum A(m)q^m$ be a formal power series with integer coefficients. Then $\Ord_p(F(q))$ is defined by
$$\Ord_p(F(q)):=\min\{...
3
votes
0
answers
131
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 ...
4
votes
1
answer
182
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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
58
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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
159
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 ...
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 ...
1
vote
0
answers
88
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:
$$
\sum_{m=-\infty}^{\infty} \ln \left|1\pm e^{-2\pi \tau_1 \sqrt{m^2+x^2/(4\pi^2\tau_1)}-2 \pi i \tau_0 ...
6
votes
2
answers
1k
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Fourier coefficients of modular forms
Given any nonzero modular form $f$ (of any weight, any level, any character),
consider its $q$-expansion $f(z) = \sum_n a(n) q^n$, where $q=\exp(2\pi iz)$.
Proposition: infinitely many of the ...
4
votes
1
answer
240
views
The Wilton-type bounds involving half-integral weight cusp forms
There is a basic question which puzzles me for a while, and maybe look naive for some experts here. The question is the following:
Let $f(z)=\sum_{n\ge 1} a_f(n) n^{k/2-1/4}e(nz)\in S_{k+1/2}(4N)$ be ...
1
vote
0
answers
86
views
On level-$12$ of the McKay-Thompson series of the Monster and the Domb numbers
(This continues from level 10.) Given some moonshine functions $j_{N}$. There are nice descending and consistent relations for levels $6m$ with $m= 2,3,5,$
$$j_{12A} = \left(\sqrt{j_{12H}} + \frac{\...
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 ...
2
votes
1
answer
222
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 ...
2
votes
1
answer
237
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 ...
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
$$ \...
3
votes
0
answers
80
views
Shimura lift is isomorphic iff twisted Hecke $L$ function does not vanish at central point
Let $S_{2k}(1)$ and $S_{k+1/2}(4)$ denote the set of modular forms of weight $2k$ for $SL_2(\mathbb{Z})$ and weight $k+1/2$ for the congruence subgroup $\Gamma_0(4)$, respectively. Consider the Kohnen ...
3
votes
0
answers
204
views
Using Ramanujan-type "Legendrian" sequences to find new formulas for $\frac1{\pi}$?
I. Recurrences
(Continued from this post.) In Cooper's 2012 paper, "Sporadic sequences, modular forms and new series for 1/π", he did a computer search for the recurrence relation,
$$(n+1)^3 ...