All Questions
1,724 questions
1
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15
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p-torsion in the Tate-Shafarevich group of supersingular elliptic curves
Let E be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the p-torsion of the Tate-Shafarevich group in this case? In particular, I would like to know if (or if known ...
- 83
1
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0
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89
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Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 5,966
7
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0
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122
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Langlands correspondence of coverings of $\mathrm{SL}_2(\mathbb R)$ and modular forms with fractional weights
$\DeclareMathOperator\SL{SL}$Let $G \to \SL_2(\mathbb R)$ be a finite covering of degree $d \geq 2$. Then $G$ is a connected Lie group with semisimple Lie algebra $\mathfrak{g}=\mathfrak{sl}_2$ and ...
- 6,622
1
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0
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113
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The value of the Hauptmodul at CM point
Let $J$ be a classical normalized $j$-invariant (that is, J=j-744).
Then, it is a classical result that $J(\tau)$ is an algebraic integer if $\tau$ is an imaginary quadratic number (that is, $a\tau^2+...
- 111
2
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0
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114
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Whether or not the root number of GL$_3\times$GL$_2$ $L$-function $L(s, F \otimes g)$ contains the coefficients $\lambda_g(n)$ of $g$?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ and $q$ be two distinct primes. Let $$\Gamma_0(p)= \left\{ g\in \GL_3(\mathbb{Z}):g \equiv \left(\begin{matrix} \ast &\ast&\...
- 51
3
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192
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Voronoi formula on $\mathrm{GL}_4$ in the level aspect with ramification
$\DeclareMathOperator\GL{GL}$Let $f$ be an automorphic form on $\GL_3$ for $\Gamma_0(p)$ with $p$ being a prime (see Bump or Goldfeld's books for definitions). Recall that, in this paper-"The ...
- 51
3
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0
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64
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Congruences regarding $4n$-dimensional lattices
A sequence of integers $(a_n)_{n\geq 1}$ satisfies Gauss congruence if
$$\sum_{d\mid n}\mu(d)a_{n/d}\equiv 0\pmod{n}$$
for every $n\geq 1$. Such sequences are also called Dold sequences, Newton ...
- 111
2
votes
0
answers
128
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Solving the quintic using the eta quotients $\frac{\eta(\tau)}{\eta(2\tau)},\,\frac{\eta(\tau)}{\eta(3\tau)},\,\frac{\eta(\tau)}{\eta(4\tau)},$ etc?
I. Reduced quintics
The general quintic can be reduced to the one-parameter forms,
$$x^5+5x+\alpha=0\\[5pt]
x^5+5\alpha x^2-\alpha=0$$
for some generic alpha. The first is the Bring form and there are ...
- 12.6k
3
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0
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91
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Reference request: étale local system on $\Gamma\backslash\mathcal H$ for $\Gamma$ non-congruence
Suppose $\mathcal H$ is the upper half plane, and $\Gamma\subset \operatorname{PSL}_2(\mathbb Z)$ is a finite index subgroup. Then by Belyi's theorem we know that $\Gamma\backslash\mathcal H$ is an ...
- 785
3
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1
answer
169
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Express fundamental group of $\mathcal H/\Gamma$ by $\Gamma$
Suppose $\mathcal H$ is the upper half plane, and $\Gamma$ is an arithmetic subgroup of $\operatorname{PSL}_2(\mathbb Z)$, I want to ask can we interpret the fundamental group of $\mathcal H/\Gamma$ ...
- 785
1
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1
answer
203
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Is $\mathcal H/\Gamma$ defined over a number field when $\Gamma$ is not congruent?
Let $\mathcal H$ denote the upper half plane of complex numbers, and $\Gamma$ an arithmetic subgroup of $\operatorname{SL}_2(\mathbb Z )$ (not necessarily congruent). I wonder whether $\mathcal H/\...
- 785
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0
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190
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About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
- 348
2
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0
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95
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Using Ramanujan's cubic continued fraction $C(q)$ and $x^3+y^3=1$ to solve the Bring quintic?
The octic Ramanujan-Selberg continued fraction $S(q)$ and $x^8+y^8=1$ can solve the Bring quintic. So can the Rogers-Ramanujan continued fraction $R(q)$ and $x^5+y^5=1.\,$ It turns out Ramanujan's ...
- 12.6k
1
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74
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Does Hermite's approach to the Bring quintic yield pairs of methods?
In a previous post, we mentioned $4+2=6$ methods to solve the Bring quintic. The first four uses the same quartic to find the elliptic modulus $k$ and the last two uses the same octic. For balance, we ...
- 12.6k
1
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0
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114
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Solving the Bring quintic using the Ramanujan $g$- and $G$-functions?
Ramanujan defined two functions now called the Ramanujan g- and G-functions. One of the more well-known values is,
$$g_{58} = \sqrt{\tfrac{5+\sqrt{29}}2}$$
If we let,
$$2^6\big(g_{58}^{12}+g_{58}^{-12}...
- 12.6k
6
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0
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179
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Modularity from cubic reciprocity: does it generalize?
Background. Let $a$ be a cubefree integer, $N=3\prod_{p\mid a}p$ and
$$
a_p=\#\{\text{solutions to}\,\, x^3\equiv a\,\,(\text{mod}\,\, p)\}-1.
$$ Let $\zeta$ be a primitive cube root of unity and $A=\...
- 171
0
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2
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223
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What is the definition of Tr in the context of Hilbert modular forms?
I am currently reading Garrett's book "Holomorphic Hilbert Modular Forms". But I meet trouble at the starting line. Let $F = \mathbb{Q}(\sqrt D)$ be a real quadratic field, $u= a + b\sqrt{D}\...
- 185
3
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0
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84
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Eigenvalues of Hecke operators for Siegel eigenforms are algebraic
Cross-posted from MSE (sorry about that, I now think it is more likely to get answer here).
Let $F$ be a Siegel modular form for $\text{Sp}_4(\mathbb{Z})$ of genus two. Let it also be an eigenform for ...
- 367
2
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0
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167
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Bounds of modular functions on the Ford circles
Assume a holomorphic function from a product of two upper half planes $Z: \mathbb{H}_+\times \mathbb{H}_+\rightarrow \mathbb{C}$ with an expansion of the form
$$
Z(\tau,\tau') = \sum_{(h,h')\in S} a_{...
- 21
14
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2
answers
749
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Solving the Bring quintic using the Monster?
I. Method
Hermite's method to solve the Bring quintic by functions that obey $x^8+y^8=1$ implicitly uses octahedral symmetry, while Emil Jann Fiedler's solution by the Rogers-Ramanujan continued ...
- 12.6k
1
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0
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82
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Behavior of translation functors in characteristic $p$
Let $G$ be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic $p>0$, and let $\mathfrak g$ be the Lie algebra of $G$. Let $U_\chi(\mathfrak g)$ ...
- 2,974
14
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1
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457
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Who solved the Bring quintic using the Rogers-Ramanujan continued fraction $R(q)$ and how to find all five roots?
I. The octahedral group
Given the nome $q=e^{\pi i \tau}$, then the elliptic lambda function $\lambda(\tau)$ shown below with its Ramanujan–Selberg continued fraction,
\begin{align}
\big(\lambda(\tau)...
- 12.6k
10
votes
2
answers
404
views
Impact of the squarefreeness of the level for modular forms
I often notice papers and results that assume that the level is squarefree in the setting of modular forms, but have a hard time figuring how where this impacts or simplifies the argument. Is there in ...
- 249
2
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93
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Quillen bundles and 2D CFTs
Roughly speaking, a (mathematical) genus-$0$ conformal field theory (CFT) is a projective symmetric
monoidal functor $Z$ from $C$ to $GrVec$ [1], where $GrVec$ is the category of graded complex vector ...
- 5,230
0
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0
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101
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Identities for Prime Coefficients of Certain Cusp Forms
While working with Fourier expansions of cusp forms of congruence subgroups of the modular group, I observed the following patterns in their prime coefficients.
Let $a(n)$ be the Fourier coefficients ...
8
votes
1
answer
356
views
Average bounds on Rankin-Selberg coefficients for modular forms
Let $f$ be a cuspidal Hecke newform of weight $k$ and level $N$, and denote by $a_f(n)$ its $n$-th Fourier coefficient. The newform $f$ is normalized so that $a_f(1) = 1$. As a consequence of Rankin-...
- 5,424
4
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0
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323
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Monstrous moonshine, Dedekind eta function, and the hypergeometric function
I. Monstrous Moonshine
Let $q = e^{2\pi i\tau}$ and $\tau = \sqrt{-d}$ or $\tau = \frac{1+\sqrt{-d}}2$ for positive integer $d$. Given the Dedekind eta function $\eta(\tau)$, consider the known ...
- 12.6k
3
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1
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228
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On the local factor of Rankin-Selberg L-functions
I have a puzzle on the local factors of Rankin-Selberg $L$-functions. Consider two newforms on $\text{GL}_2$. Let $f$ be a newform of square-free level $N$, and $g$ a newform of trivial level. As ...
- 353
2
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1
answer
147
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Finiteness and bounds for elliptic curves realizing a given galois representation
Let $\rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}_p)$ be a continuous, irreducible Galois representation. Consider the set $\mathcal{L}_\rho$ of all elliptic curves $...
3
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0
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125
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Why the hyperbolic Laplacian?
In the theory of automorphic forms there is the weight $k\in\mathbb{Z}$ Laplacian
\begin{align*}
\Delta_k:=-y^2 \left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+iky\frac{\...
- 177
1
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0
answers
164
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On the Jacobi theta functions and the Borweins' cubic theta functions
The post has been divided into sections to show some patterns, as well as possible evaluations of,
$$_2F_1\big(s,1-s,1,z\big)$$
with $s = \frac12, \frac13, \frac14, \frac16$ for infinitely many ...
- 12.6k
4
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0
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103
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Automorphic forms on $\mathrm{GL}_{2}$, $\mathrm{SL}_{2}$, and $\mathrm{Mp}_{2}$ — classical counterparts
I asked exact same question on MSE but haven't got answer yet, so asking here, too. I may erase the original one once I got an answer here.
--
I'm confusing about automorphic representations of $\...
- 2,215
5
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0
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126
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Using Lang–Trotter to get bounds on averages of Fourier coefficients
Let $E$ be an elliptic curve over $\mathbf{Q}$ and let $(a_n)$ be the sequence of Fourier coefficients for the weight two newform attached to $E$. The coefficients $a_p$ are the Frobenius traces given ...
- 265
10
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2
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286
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Finding a model for $X_G$ with $G\subseteq \mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$
I'm studying modular curves as part of my doctorate work and would like to understand how one gets from a subgroup $G$ of $\operatorname{GL}_2(\mathbb{Z}/N\mathbb{Z})$ to an equation for $X_G$. By $...
- 195
1
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85
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Action of Atkin--Lehner involution on CM points
In their first paper on Heegner points and derivatives of $L$-series, Gross and Zagier describe the action of Atkin--Lehner involutions on certain CM $\mathbf{C}$-points of the modular curve $X_0(N)$. ...
- 265
3
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0
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241
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Generating algebraic points on elliptic curves
Let $E$ be an elliptic curve over $\mathbf{Q}$. One has the modular parameterisation
\begin{align*}
\mathbf{H} \to X_0(N)(\mathbf{C}) \to E(\mathbf{C})
\end{align*}
where $X_0(N)$ is the modular curve ...
- 265
4
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0
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124
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A coefficient in Dirichlet series associated with a cofinite subgroup of $\mathrm{SL}(2,\mathbb R)$
Let $\Gamma$ be a discrete subgroup of $\operatorname{SL}(2,\mathbb R)$, acting on the upper half-plane $\mathbb H$. Suppose that $\Gamma\backslash \mathbb H$ is non-compact and its compactification $\...
- 7,193
1
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0
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58
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Asymptotics of Jacobi form
What are the large $x\in\mathbb R$ asymptotics of $f(x)=\theta_3(c_1+c_2 x^3,e^{-x^2})$ where $c_1,c_2$ are a pair of complex numbers (say, $\Re(c_2)>0$ and $\Im(c_2)<0$), and $\theta_3(a,b)=\...
- 11
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0
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81
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Computing elliptic periods from modular form
How are the periods of a modular elliptic curve computed as path integrals of its associated normalized weight 2 cusp form on the modular curve? Please provide specific paths for both periods and cite ...
- 2,907
1
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0
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125
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Where have you encountered the following arithmetic function?
The following arithmetic function is studied by Zagier in connection with values at odd negative integers of zeta functions of real quadratic fields:
$$e_r (n)=\sum_{\underset{|x|\leq n}{x^{2}\equiv n\...
- 347
1
vote
0
answers
71
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Relation between the field and $\mathbb{Z}$-algebra generated by eigenvalues of modular form
Cross-posted from MSE: https://math.stackexchange.com/questions/4944262/relation-between-the-field-and-mathbbz-algebra-generated-by-eigenvalues-of
Let $f$ be a cusp form of weight $k\in\mathbb{Z}$ for ...
- 367
3
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0
answers
192
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How can I prove this stronger version of Fedder's Criterion?
I was reading Fedder's original paper which proved what is now known as "Fedder's criterion". I noticed that the abstract stated something which is a priori stronger than what is proved in ...
- 317
0
votes
0
answers
100
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Algebraic degrees of $j\left(\sqrt{-n}\right)$ and $j\left(\frac{1 + \sqrt{-n}}{2}\right)$ and class numbers of $Q(\sqrt{-n})$
Let $n\in\mathbb N$ be squarefree. Denote by $h(n)$ the class number of $Q(\sqrt{-n})$ and by $d_1(n)$ and $d_2(n)$ the degrees of the algebraic numbers $j\left(\sqrt{-n}\right)$ and $j\left(\frac{1 + ...
- 13.4k
5
votes
0
answers
141
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Compute weight of modular form from its Fourier coefficients
It is known that Hecke eigenform $f \in S_{k}(\Gamma_0(N), \chi)$ is uniquely determined by first $C_{k,N}$ many Fourier coefficients, where $C_{k,N}$ is a constant only depends on $k$ and $N$. For ...
- 2,215
5
votes
1
answer
203
views
Isogenous elliptic curves and canonical modular polynomials
Let $\ell$ and $p$ be two primes. We are looking for a method for checking whether two supersingular elliptic curves over the finite field $F_p$, given through their $j$-invariants, are $\ell$-...
- 81
2
votes
0
answers
125
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Is it true that all algebraic values of the $j$-invariant have a real Galois conjugate?
This is a follow-up of my recent question on real values of the $j$-invariant. It has had a partial response so far, establishing that any "reality root" $n$ is indeed rational. Now it ...
- 13.4k
3
votes
1
answer
272
views
Does there exist a polynomial that extracts the highest digit of an integer in base p?
Given an odd prime $p$, a positive integer $1 \lt n$, and an integer $x \in \mathbb{Z}/p^n\mathbb{Z}$, does there exist an an integer-coefficient polynomial that extracts the highest digit of $x$?
The ...
- 31
3
votes
0
answers
347
views
Modern integral $p$-adic Hodge theory and modularity lifting and Fontaine-Mazur
As a follow-up to a comment on this answer, I'm wondering if there are expected to be applications of the new point of view on integral $p$-adic Hodge theory, à la Bhatt-Morrow-Scholze and others, to ...
- 15.4k
0
votes
0
answers
64
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modular properties of macmahon function?
How does the MacMahon function for counting plane partitions
$M(q) = \frac{1}{(1-q^n)^n}$
behave under modular transformations?
For instance for $q= e^{2 \pi i \tau}$ where $\tau \rightarrow -1/\tau$.
- 11
3
votes
1
answer
212
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Arguments where the $j$-invariant has non-trivially real values — and a conjectured duality
The $j$-function (a.k.a. $j$-invariant up to a factor $1728$) is most often only considered for arguments where it yields real values.
It is well known that for $a,b,n\in\mathbb N$, $j\left(\dfrac {a+...
- 13.4k