All Questions
1,724 questions
3
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46
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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 ...
- 12.9k
1
vote
0
answers
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=\...
- 5,986
5
votes
1
answer
203
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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 $\ell$-...
CommunityBot
- 1
12
votes
2
answers
1k
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Ramanujan's tau function, $691$ congruence, and $\eta(z)^{12}$
Let $q = e^{2\pi i\,z}$.
I. 24th power
The Ramanujan tau function $\tau(n)$ is given by the expansion of the Dedekind eta function $\eta(z)$'s $\text{24th}$ power. Then
$$\begin{aligned}\eta(z)^{...
- 13.4k
7
votes
0
answers
122
views
Langlands correspondence of coverings of $\mathrm{SL}_2(\mathbb R)$ and modular forms with fractional weights
$\DeclareMathOperator\SL{SL}$Let $G \to \SL_2(\mathbb R)$ be a finite covering of degree $d \geq 2$. Then $G$ is a connected Lie group with semisimple Lie algebra $\mathfrak{g}=\mathfrak{sl}_2$ and ...
- 63.9k
1
vote
0
answers
113
views
The value of the Hauptmodul at CM point
Let $J$ be a classical normalized $j$-invariant (that is, J=j-744).
Then, it is a classical result that $J(\tau)$ is an algebraic integer if $\tau$ is an imaginary quadratic number (that is, $a\tau^2+...
- 13.4k
9
votes
1
answer
650
views
Sum of three squares as class numbers and Waldspurger's formula
It is known that the number of ways to express $n \in \mathbb{Z}_{\geq 0}$ as a sum of three squares (let's denote it as $r_3(n)$) can be expressed as Hurwitz-Kronecker class number (certain weighted ...
CommunityBot
- 1
2
votes
0
answers
114
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Whether or not the root number of GL$_3\times$GL$_2$ $L$-function $L(s, F \otimes g)$ contains the coefficients $\lambda_g(n)$ of $g$?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ and $q$ be two distinct primes. Let $$\Gamma_0(p)= \left\{ g\in \GL_3(\mathbb{Z}):g \equiv \left(\begin{matrix} \ast &\ast&\...
- 51
1
vote
2
answers
197
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What are the finite étale coverings of a quasi-hyperelliptic surface?
Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.
Question: Is there a finite étale covering $Y \rightarrow X$ such that
$Y$ is an abelian ...
- 103
2
votes
1
answer
413
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Motivation for $p$-stabilization in Hida theory
I'm currently reading Hida's paper "A $p$-adic measure attached to the zeta functions associated with two elliptic modular forms". The setup is the following: let $f$ be a weight $2$ newform ...
3
votes
0
answers
192
views
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
4
votes
1
answer
225
views
What is "the quotient of the universal ordinary Hecke algebra corresponding to an ordinary $\Lambda$-adic form"?
Let $\Lambda := \Bbb Z_p[[T]]$ be the usual Iwasawa algebra. In Jha and Sujatha - On the Hida deformations of fine Selmer groups on page 181, the authors refer to the quotient $\Bbb H^{\text{ord}}_{\...
3
votes
0
answers
64
views
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
13
votes
0
answers
626
views
On Ramanujan's pi formula $\frac 1\pi=\sum_{k=0}^\infty\frac {(4k)!}{k!^4}\frac {Ak+B}{396^{4k}}$ and the solvable quintic $z^5-5z-396 = 0$?
(Updated with new information.)
I. Five eta quotients and the Monster?
Given Dedekind eta function $\eta(\tau)$, define the five eta quotients which in fact are the McKay-Thompson series 1A, 2A, 3A, ...
- 12.6k
3
votes
0
answers
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 ...
- 12.9k
1
vote
1
answer
203
views
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/\...
- 12.3k
3
votes
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 $\mathcal H/\Gamma$ ...
- 63.9k
0
votes
0
answers
190
views
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 ...
- 4,713
2
votes
0
answers
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
14
votes
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
vote
0
answers
74
views
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
8
votes
1
answer
567
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 ...
- 424
1
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0
answers
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,
$$2^6\big(g_{58}^{12}+g_{58}^{-12}...
- 12.6k
6
votes
0
answers
179
views
Modularity from cubic reciprocity: does it generalize?
Background. Let $a$ be a cubefree integer, $N=3\prod_{p\mid a}p$ and
$$
a_p=\#\{\text{solutions to}\,\, x^3\equiv a\,\,(\text{mod}\,\, p)\}-1.
$$ Let $\zeta$ be a primitive cube root of unity and $A=\...
- 171
0
votes
2
answers
223
views
What is the definition of Tr in the context of Hilbert modular forms?
I am currently reading Garrett's book "Holomorphic Hilbert Modular Forms". But I meet trouble at the starting line. Let $F = \mathbb{Q}(\sqrt D)$ be a real quadratic field, $u= a + b\sqrt{D}\...
- 185
3
votes
0
answers
84
views
Eigenvalues of Hecke operators for Siegel eigenforms are algebraic
Cross-posted from MSE (sorry about that, I now think it is more likely to get answer here).
Let $F$ be a Siegel modular form for $\text{Sp}_4(\mathbb{Z})$ of genus two. Let it also be an eigenform for ...
- 367
2
votes
0
answers
167
views
Bounds of modular functions on the Ford circles
Assume a holomorphic function from a product of two upper half planes $Z: \mathbb{H}_+\times \mathbb{H}_+\rightarrow \mathbb{C}$ with an expansion of the form
$$
Z(\tau,\tau') = \sum_{(h,h')\in S} a_{...
- 21
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-...
CommunityBot
- 1
14
votes
1
answer
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
1
vote
0
answers
82
views
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 $U_\chi(\mathfrak g)$ ...
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9
votes
1
answer
1k
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Coderivations of $S(V)$ correspond to linear maps $S(V) \to V.$ Only over characteristic $0$?
Definition. Let $k$ be a commutative ring. Let $V$ be a $k$-module. We turn the symmetric algebra $\mathrm{S}\left(V\right)$ of $V$ into a graded Hopf algebra by defining the comultiplication
\begin{...
- 33.8k
51
votes
7
answers
11k
views
How is representation theory used in modular/automorphic forms?
There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and ...
- 23k
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 ...
- 23k
2
votes
0
answers
93
views
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
votes
0
answers
101
views
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
2
answers
664
views
A coincidence concerning Fermat primes, binomial sums, and eta quotients?
Let $w_k$ be a primitive k th root of unity, where k is a power of 2. In response to a question, Robert Israel gave the solution,
$$\sum_{n=0}^\infty \frac{(-1)^n}{\binom{kn}{kn/2}} = \frac{2^k}{2^...
- 12.6k
4
votes
0
answers
323
views
Monstrous moonshine, Dedekind eta function, and the hypergeometric function
I. Monstrous Moonshine
Let $q = e^{2\pi i\tau}$ and $\tau = \sqrt{-d}$ or $\tau = \frac{1+\sqrt{-d}}2$ for positive integer $d$. Given the Dedekind eta function $\eta(\tau)$, consider the known ...
- 12.6k
1
vote
0
answers
164
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On the Jacobi theta functions and the Borweins' cubic theta functions
The post has been divided into sections to show some patterns, as well as possible evaluations of,
$$_2F_1\big(s,1-s,1,z\big)$$
with $s = \frac12, \frac13, \frac14, \frac16$ for infinitely many ...
- 12.6k
3
votes
1
answer
228
views
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 ...
- 149k
2
votes
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 $...
- 37k
3
votes
0
answers
125
views
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
4
votes
0
answers
103
views
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 $\...
- 63.9k
5
votes
0
answers
126
views
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 ...
- 12.9k
10
votes
2
answers
286
views
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 $...
- 20.4k
12
votes
1
answer
583
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{...
- 2,259
1
vote
0
answers
85
views
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
votes
0
answers
241
views
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
9
votes
1
answer
2k
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How did Gauss characterize the metrical relations in the uniform (4 4 4) tiling of the hyperbolic unit disk?
My purpose is to verify an historical hypothesis I have on Gauss's tesselation of the unit disk as described in John Stilwell "Mathematics and its history". Looking at the relevant pages in ...
- 2,099
4
votes
0
answers
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 $\...
- 63.9k