Questions tagged [modular-forms]
Questions about modular forms and related areas
1,302
questions
6
votes
1
answer
291
views
An explicit equation of the canonical morphism $X_1(N) \to X_0(N)$
I know there are some research about explicit equations for affine models in $\mathbb{A}^2$ of many modular curves over $\mathbb{Q}$, for example of $X_i(N), X(N)$ (where $i = 0, 1, 2$) for small $N$.
...
- 1,352
10
votes
2
answers
939
views
Modular forms with finitely many or very few non-zero Fourier coefficients
I have an elementary question on modular forms, but which I don't know how to solve.
a) Is there a congruence subgroup $\Gamma \leq \mathrm{SL}_2(\Bbb Z)$, an integer $k \in \Bbb Z$ and a non-...
- 255
2
votes
0
answers
219
views
Reference Request - New proof of Ribet's level lowering by Khare and Wintenberger
I'm currently following the note of Sug Woo Shin's course at Berkeley with notes taken by Rong Zhou. In Section 24.3 (Page 86), Ribet's level lowering theorem is stated:
[Theorem 24.7] $E = E_{a^{\...
- 579
4
votes
0
answers
225
views
Diophantine consequences of the Buzzard–Diamond–Jarvis conjecture
Serre's modularity conjecture famously implies Fermat Last Theorem. More generally, Serre's conjecture implies that certain generalized Fermat equations have no non-trivial solutions (see Section 4.1 ...
- 595
2
votes
0
answers
84
views
Ordinary primes for a weak form corresponding to a CM newform
Setup: Let $f$ be a harmonic Maass form of weight $2-k$ ($k \in \mathbb{N}$), level $N$, and character $\chi$. Letting $q := e^{2\pi i z}$ and considering the Fourier expansion of any harmonic Maass ...
- 26
2
votes
0
answers
214
views
"Moonshine" basics?
Having browsed recently a bit about "moonshine", which looks to me like some weird surrealist landscape, I wonder if: 1. sporadic groups could be seen as seemingly isolated special points of ...
- 10.7k
4
votes
1
answer
538
views
Why is Dedekind sum?
The Dedekind function is defined as follows
$$\eta(\tau)=q^{1/24}\prod_{n=1}^\infty(1-q^n),\qquad q=e^{2\pi i\tau}.$$
We have
$$\eta(\tau+1)=\zeta_{24}\eta(\tau),\qquad \eta\left(-\frac{1}{\tau}\right)...
- 2,335
2
votes
0
answers
315
views
A computation of the rank of the Jacobian of a hyperelliptic curve over a number field using MAGMA
In this paper,
the authors says that, in order to show the rank of a Jacobian over $\mathbb{Q}$ is 0, they use the L function.
In the section 3.3, the authors compute the rank of the Jacobian of $X_1(...
- 1,352
2
votes
0
answers
169
views
On Fourier coefficients of Bianchi modular forms, l-ordinary
Let $f\in S_2(\Gamma_1(N))$ be a Hecke eigenform and $\ell$ a prime number does not divide $N$. Let $a_f(\ell)$ be the $\ell$-th Fourier coefficient of $f$. Then $a_f(\ell)$ is is called $\ell$-...
- 21
3
votes
0
answers
186
views
Maximum value of newform from Galois representation
One can attach $\ell$-adic Galois representations to holomorphic cuspidal newforms of weight $2$ on the upper half-plane.
If a newform is $L^2$-normalized, can one extract its maximum value from the ...
- 39
4
votes
1
answer
652
views
Meaning of Atkin-Lehner eigenvalues
Suppose I have $f\in S_2(\Gamma_0(N))$ a classical modular newform of level $N$. I want to understand what information (if any) is carried by its Atkin-Lehner eigenvalues for primes $p\mid N$, as ...
- 51
4
votes
0
answers
251
views
Modular forms on $\Gamma(N)$
I'm wondering where I can find a good reference about what is known about modular forms (especially cuspidal eigenforms) of full principal level $\Gamma(N)$, in terms of their Hecke theory, old/...
- 1,964
7
votes
1
answer
570
views
Hurwitz numbers and $t$-cores
For integers $k \geq 0$ and $d \geq 1$ let $H(k,d)$
be the Hurwitz number which, for the purposes
of this posting, will be defined by:
\begin{equation}
H(k,d)
\, := \ d! \, \sum_{\lambda \, \vdash d}...
- 1,847
3
votes
1
answer
265
views
Modular form not meromorphic at $\infty$
Is there a function $f$ with the following properties
$f$ meromorphic at the upper half plane $\mathfrak h$,
$f$ is of weight $k$ under a congruence subgroup of $\operatorname{SL}_2(\mathbb Z)$,
$f$ ...
- 2,335
10
votes
0
answers
145
views
Is there some precise way in which modular cocycle $c/(c\tau +d)$ "is" the generator of $H^1(\mathcal{M}_{ell}, \omega^2) \cong \mathbb{Z}/12$?
It is semiclassical that the extension class $\text{Ext}^1(\omega^{-1},\omega)\cong H^1(\mathcal{M}_{ell},\omega^2)$ over the modular stack $\mathcal{M}_{ell}/\mathbb{Z}$ is nonvanishing, being cyclic ...
- 1,964
3
votes
1
answer
195
views
Index and weight of Weierstrass $\sigma$-function as a Jacobi form, versus a statement in a note by Zagier
Let
$$\sigma_L(w; \tau):=\frac{w}{\exp\left(\sum_{k\ge 2} 2G_k(q)\frac{w^k}{k!}\right)}$$
be the version of the Weierstrass $\sigma$-function which is used to orient $\text{tmf}$; here $w=2\pi i z$, $...
- 1,964
8
votes
1
answer
746
views
Modular forms over $\mathbb{Z}$ vs modular forms with integral Fourier coefficients
It is well known that the ring of modular forms over $\mathbb{C}$ is $$
\mathbb{C}[c_4,c_6]
$$ where $$
c_4 = 1+240 q + \cdots,\qquad
c_6 = 1-504 q - \cdots
$$ are the standard Eisenstein series, and ...
- 6,063
13
votes
0
answers
312
views
Work of Atkin on the 26th power of eta
The 26th power of the Dedekind $\eta$ function has been mentioned several times here on MO:
A 14th and 26th-power Dedekind eta function identity?
What's the status of the following relationship ...
- 17.5k
0
votes
0
answers
153
views
Holomorphic automorphic/cusp forms on real Lie groups
An automorphic form on a real Lie group $G$ for a discrete subgroup $\Gamma$ is a function $f:G\to\mathbb{C}$ with some properties (see Borel’s definition in Proceedings of Symposia in PURE ...
- 381
4
votes
1
answer
290
views
An explicit equation for $X_1(13)$ and a computation using MAGMA
By a general theory $X_1(13)$ is smooth over $\mathbb{Z}[1/13]$, and so is its Jacobian $J$.
And the hyperelliptic curve given by an affine model $y^2 = x^6 - 2x^5 + x^4 -2x^3 + 6x^2 -4x + 1$ is $X_1(...
- 1,352
2
votes
0
answers
262
views
Moduli interpretation and Ogg's notation for the cusps on modular curves
In Ogg's paper "rational points on certain elliptic modular curves", the author says, using Ogg's notation for cusps,
that for fiexed $d$, if $(y, N) = d$, then for any $x$ satisfying $(x, y,...
- 1,352
7
votes
1
answer
719
views
Statement of classical Ramanujan-Petersson conjecture
I'm preparing for an expository talk on some topics in the representation theory of reductive p-adic groups, including tempered representations and Whittaker models, and as motivation I wanted to ...
- 509
1
vote
0
answers
116
views
Stabilizers of points in the upper half-plane
Suppose that $\Gamma$ is a group acting discontinuously on $\mathcal{H} = \{ z \in \mathbb{C} : \operatorname{Im}(z) > 1\}$. In order to keep things simple, suppose that $\Gamma \subseteq \...
- 163
1
vote
1
answer
168
views
Reference request for some fragments of Gauss with dubious origin
Gauss's results on the interconnection between the different values of the arithmetic-geometric mean of two complex numbers as recorded in his private notebooks led him to introduce foundational ...
- 1,869
6
votes
0
answers
169
views
Factorizing classical Eisenstein series
In the course of my research, I found some surprising (for me) factorizations
of Eisenstein series in levels $1$, $2$, $3$, and $4$. For instance, in level $1$
set with standard modular form notation
$...
- 11.7k
5
votes
1
answer
282
views
Modular forms and number of representations by binary quadratic forms
Let $Q(x,y)$ be a positive definite quadratic form of discriminant $d$. Let $r_Q(n)$ be the number of solutions of $Q(x,y)=n$. It is known that the function $f_Q(\tau)=\sum_{n=0}^{\infty}r_Q(n)q^n$ is ...
- 2,335
4
votes
0
answers
185
views
A formula in Ramanujan's lost notebook and its connection with Chudnovsky series for $1/\pi$
While studying Berndt's Ramanujan's Lost Notebook Vol. 2, page 369 (chapter on Springerlink), I found that Ramanujan gave values of a certain expression $$\frac{1}{\sqrt{Q_n}}\left(\sqrt {n} P_n-\frac{...
- 2,019
8
votes
2
answers
314
views
Do odd-weight cusp forms have analytic rank 0?
Let $f(z)=\sum_{n\ge 1}a_nq^n$ be a cusp form, where $q=e^{2\pi i z}$. Let $
L(s) = \sum_{n\ge 1} a_nn^{-s}
$ be its corresponding L-function. The completed L-function of $L(s)$, $\Lambda(s)$, should ...
- 9,421
4
votes
0
answers
150
views
Logarithmic vector-valued modular functions and quasimodular forms with misleading modular weights
I have a somewhat imprecise question about functions with reasonably nice modular transformations that don't seem to fit nicely into what I understand of the plain vanilla theory of modular and ...
- 390
5
votes
0
answers
89
views
Totally rational hypergeometric evaluations
This is a followup to the question at which rational points does the Hypergeometric function take rational values asked 10 years ago by
Eugene Starling, and is more a challenge than a question.
Let $F(...
- 11.7k
3
votes
0
answers
75
views
Continuation of $\sum \sigma_\nu(n) a(n) n^{-s}$ for $a(\cdot)$ coming from a half-integral weight form
In some of my work, I've run into a wall trying to understand whether a Dirichlet series has a meromorphic continuation or not. Let $f(z) = \sum_{n \geq 1} a(n) e(nz)$ be a half-integral weight ...
- 1,478
1
vote
0
answers
250
views
How can I compute the derivative of a modular form, for example an Eisenstein series?
Suppose that $f(z)$ is a modular form of weight $k$, therefore for any $z\in \mathbb{H}$, and any matrix in $SL_2(\mathbb{Z})$ we have:
$$f\left(\frac{az+b}{cz+d}\right) = (cz+d)^k f(z).$$ I do not ...
12
votes
1
answer
421
views
Two non constant meromorphic functions over a connected compact Riemann surface, could not be algebraically independent
Let $M$ be a connected compact Riemann surface. Let $f, g$ be two nonconstant meromorphic functions. Why is there a two-variable complex polynomial $F(x,y)$ that vanishes for $(x, y)=(f, g)$, (in ...
12
votes
1
answer
368
views
Chiral homology for the Virasoro algebra and/or affine Lie algebra
I want to understand what concrete analytical objects are found in chiral homology of higher degree of a vertex algera (-module) $M$. More precisely: I can obtain conformal blocks on a surface $\Sigma$...
- 1,746
4
votes
1
answer
388
views
What are the known number-theoretic functions, that are related to "the number of ideals of norm $n$, that belong to the class $[c]$"?
Let $L$ be a number field, $\mathcal{O}_L$ its ring of integers, and $\mathcal{Cl(O}_L)$ its ideal class group. Let's fix an arbitrary class $[c] \in \mathcal{Cl(O}_L)$. By $r(n)=r([c], n)$, I mean ...
11
votes
1
answer
600
views
Modularity of higher genus curves
The modularity conjecture for elliptic curves over number fields is well known, and indeed, is a theorem for all elliptic curves over $\mathbb{Q}$, and at least potentially, over any CM field.
What ...
- 1,267
9
votes
1
answer
491
views
How can I transform $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^k\sin(\pi rn)}$ into a modular form?
Let
$$f_k(z)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^k\sin(\pi zn)}$$
be a family of holomorphic functions on the upper-half plane $\mathbb{H}=\{a+bi|b>0\}$ for each odd natural number $k$. These ...
- 2,817
1
vote
0
answers
123
views
Siegel's formula for generalized theta series with characteristics?
Siegel's formula(Siegel-Weil) directly relates the weighted sum of theta functions to Eisenstein series. (Or equivalently, the weighted sum of the cusp form is zero). I wonder if there is a ...
- 11
14
votes
1
answer
404
views
Bounding the fourier coefficient field
Let $f = \sum_n a_n q^n \in S_2(\Gamma_0(N))$ be a normalized, non-CM, newform of weight $N \geq 1$ and level $2$. Let $K_f := {\mathbb Q}(\{a_n\}) \subset {\mathbb C}$ be the number field generated ...
- 1,190
9
votes
1
answer
691
views
Spectral decomposition of product of modular functions
The eigenfunctions of the Laplacian on $SL(2,\mathbb Z)\backslash \mathbb H$ are known to be given by three types: the constant function, the real analytic Eisenstein series (which come in a ...
2
votes
0
answers
69
views
Hecke convergence factor
I was reading a paper here. There the author define an infinite series
$$\sum_{ad-cb=1}(cz+d)^{-(k-j)}(az+b)^{-j}$$
where $k$ is an even integer bigger than 2 and $2\leqslant j\leqslant k-2$. Then ...
- 235
6
votes
2
answers
1k
views
What are the applications of modular forms in number theory?
I am new to the topic, so I'm trying to get an overview. I am aware of the relation between modular forms and $L$-series (but don't know what that does) and FLT.
Are there other applications of ...
3
votes
0
answers
152
views
Finiteness of points over the cyclotomic extension for modular forms
Let $\rho(f):G_\mathbb{Q} \rightarrow GL_2(K_f)$ be the Galois representation attached to some cuspidal modular form $f$ where $K_f$ is a finite extension of $\mathbb{Q}_p$.
Let $V_f$ be the vector ...
- 313
26
votes
4
answers
1k
views
Equations defining hyperbolic geodesics in $\mathbb C \setminus\{0,1\}$
Let $X=\mathbb C\setminus\{0,1\}$, equipped with the hyperbolic structure it inherits from Klein's modular $\lambda$ function $\lambda:\mathbb H \to X$. In each (non-peripheral and nontrivial) free-...
- 261
2
votes
1
answer
203
views
Global section of vertical differential 1 forms on universal elliptic curve
Let $B$ be a modular curve (of some level) over a number field $K$ (here, we implicitly assume that $K$ is large enough to make sense the phrase "$B$ is a $K$-variety"). Let $E\to B$ the ...
- 1,378
16
votes
0
answers
264
views
Why should an abelian variety with few places of bad reduction and a lot of endomorphisms not have many points?
In the paper "Points of Order 13 on Elliptic Curves" by Mazur-Tate, they say in the introduction:
It seemed ... that if such an abelian variety $J$, which has bad reduction at only one ...
- 7,646
2
votes
1
answer
154
views
Theta series analogues for higher degree forms
It is simple to see that the following series converges absolutely and uniformly on $\mathcal{H}$ for all k positive:
$F_{2k}(z) = \sum_{n \in \mathbb{Z}} q^{n^{2k}}$
And this series being a ...
- 73
1
vote
0
answers
134
views
Automorphic representation of weight 3 eigenforms
Let $f$ be a weight 3 eigenform with rational Fourier coefficients. As shown by Elkies and Schutt, $f$ is associated to a singular K3 surface over $\mathbb{Q}$. A construction of Shioda and Inose ...
- 167
3
votes
0
answers
93
views
Principal series representation of $SL(2,\mathbb{R})$ restricted to principal congruence subgroup
Given a principal congruence subgroup $\Gamma(N)$ of $SL(2,\mathbb{R})$, since $\Gamma(N)$ is free, consider a probability distribution $\mu_1$ of a simple random walk on $\Gamma(N)$ and consider its ...
- 131
4
votes
0
answers
228
views
Lower bound on symmetric square L-function
In a paper of Soundararajan, equation (16c) states that for any Hecke eigenform $f$ of weight $k$, the symmetric square L-function at $1$ satisfies the bound
$$(\log k)^{-2}\ll L(1, \mathrm{sym}^2f)\...
- 1,831