All Questions
Tagged with moduli-spaces elliptic-curves
39 questions
27
votes
6
answers
4k
views
Does the moduli space of smooth curves of genus g contain an elliptic curve
Let $M_g$ be the moduli space of smooth projective geometrically connected curves over a field $k$ with $g\geq 2$. Note that $M_g$ is not complete.
Does $M_g$ contain an elliptic curve?
The answer ...
- 281
25
votes
3
answers
5k
views
Conceptual understanding of the Gross-Zagier theorem.
The Gross-Zagier paper "Heegner points and derivatives of $L$-series", is really computational and hard to plow through. It seems it is futile to read it as such and one must look for a more ...
- 1,417
21
votes
2
answers
2k
views
elliptic curves and group cohomology
Recently, I've been trying to understand Jacob Lurie's 2-equivariant elliptic cohomology a bit better than I had in the past.
From what I can tell, the fragment of the story that only deals with ...
- 43.2k
15
votes
3
answers
3k
views
Existence of fine moduli space for curves and elliptic curves
For the moduli problem of a curve of genus $g$ with $n$ marked points, how large an $n$ is needed to ensure the existence of a fine moduli space? For this question, terminology is that of Mumford's ...
- 7,442
12
votes
1
answer
1k
views
What's wrong with my understanding of the scheme $\text{Isom}(E_\lambda, E_{\lambda'})$?
Let $\mathcal{M}_{1,1}$ be the moduli stack of elliptic curves (over the complex numbers). Define
$$\begin{eqnarray*} X &:=& \Bbb{A}^1_{\lambda} - \{0,1\}\\
X' &=& \Bbb{A}^1_{\lambda'} ...
- 2,607
12
votes
1
answer
337
views
$\pi_1$ of the moduli of G-bundles on elliptic curves and the double affine braid group
For a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, the fundamental group of $\mathfrak{h}_\text{reg}/W$ (where $\mathfrak{h}$ is the Cartan subalgebra, $\mathfrak{h}_\text{reg}$ is the subset ...
- 6,123
10
votes
1
answer
535
views
examples of "exotic" moduli problems for elliptic curves?
Let $\textbf{Ell}$ be the category of elliptic curves over various base schemes, and where a morphism between $E\rightarrow S$ and $E'\rightarrow S'$ is a cartesian diagram with those two maps as ...
- 10.7k
10
votes
1
answer
581
views
Intuitive reasons for the existence of modular parametrizations
Whenever I encounter anything about modular parametrizations, I have a feeling it is something very unnatural: you have some kind of moduli space and all of a sudden it parametrizes an object ...
- 18.4k
9
votes
2
answers
839
views
$Pic$ of the stack of elliptic curves vs. $Pic$ of the coarse space
There's a natural map $f:\overline{\mathcal{M}}_{1,1}\to \overline{M}_{1,1}\cong \mathbb{P}^1$ from the stack of elliptic curves to the coarse space. Both spaces have $Pic=\mathbb{Z}$ hence $f^*:\...
- 3,779
8
votes
2
answers
2k
views
(nontrivial) isotrivial family of elliptic curves
I think it should be a standard procedure to construct such things, can anyone give a reference or give a hint? Can this be done over any base scheme?
- 1,503
8
votes
0
answers
333
views
Do automorphisms actually prevent the formation of fine moduli spaces?
I have found similar questions littered throughout this site and math.SE (for example [1], [2], [3],…), but I feel like like most of them usually just say that non-trivial automorphisms prevent the ...
- 821
8
votes
0
answers
416
views
Stacky proof of no elliptic curves over Z
It is a well known result that there are no Elliptic curves over the integers with every where good reduction. In fact this is even true for abelian varieties (and hence higher genus curves) but let ...
- 7,746
7
votes
1
answer
957
views
Modular curve X(2)
Let $\mathfrak{M}(2)$ be the algberaic stack over $\mathbb{Z}[1/2]$ which classifies the elliptic curves with the two level structure and let $X(2)$ be the coarse moduli space of $\mathfrak{M}(2)$ ($X(...
- 1,066
6
votes
1
answer
305
views
Definition of modular curve associated to $\Gamma(N)$
For a positive integer $N$, we define $$\Gamma(N)=\big \{ \begin{bmatrix} a & b \newline c & d\end{bmatrix}\in \operatorname{SL}_2(\mathbb{Z}): \begin{bmatrix} a & b \newline c & d\end{...
- 821
6
votes
1
answer
815
views
Is there an elliptic surface over $Y(1)$?
Actually I have a few related questions.
Here, by $Y(1)$ I mean the affine $j$-line $\text{SL}_2(\mathbb{Z})\backslash\mathcal{H}$.
I know $Y(1)$ is only a coarse moduli space, so there isn't a ...
- 10.7k
6
votes
1
answer
433
views
Choosing tau for elliptic curves over the rational numbers with prescribed ramification data
Let $r>2$ and let $b_1,b_2,\ldots,b_r$ be in $\mathbf{P}^1(\mathbf{Q})$. Let $B$ be the divisor $$B:= \sum [b_i].$$ We consider this data to be fixed. For $d>1$, we define $\textrm{Ell}(b_1,b_2,\...
- 9,497
6
votes
0
answers
164
views
What are the genus 4 curves with Jacobians that are 4-th powers?
Consider the moduli space of all genus $4$ curves $\overline{\mathscr M_4}$ of dimension $3\times 4 - 3 = 9$. Under the Torelli map, there is a map to $\overline{\mathscr A_4}$ (which has dimension $...
- 7,746
6
votes
0
answers
971
views
Curious propositon in "Les schemas de modules de courbes elliptiques"
Currently I am reading "Les schemas de modules de courbes elliptiques" by Deligne and Rapoport and I got myself seriously confused about the following proposition (in English translation):
(II ...
- 839
5
votes
1
answer
752
views
Some help in digesting a paragraph in the introduction of Deligne/Rapoport's "Les Schemas de Modules de Courbes Elliptique"
http://www.springerlink.com/content/04x54gr171v556m4/fulltext.pdf
On page 149 (DeRa-7), in the middle of the page, I can translate the middle paragraph that starts "3. La surface de Riemann ..." as ...
- 10.7k
5
votes
1
answer
333
views
Maps to the universal punctured elliptic curve
I have just started reading Hain's paper On the Universal Elliptic KZB Connection. I am a bit confused about a comment made there about base points on orbifolds. I am still very new to the idea of ...
- 661
5
votes
1
answer
412
views
Modular curve parametrizing two cyclic subgroups of an elliptic curve
The aim of this question is to better understand the following moduli space/modular curve, for which I propose (temporarily) the name $Y_0(M,N)$. We define $Y_0(M,N)$ as the moduli space parametrizing ...
4
votes
1
answer
372
views
Unirationality of moduli spaces of marked elliptic curves
Let us consider the moduli space of genus one curves with an effective divisor of degree d; this space is birational to the quotient of the usual moduli space of genus one curves with $d$ marked ...
- 2,300
4
votes
2
answers
661
views
What does this quotient of the upper half plane parametrize?
Let $G(N)$ be the congruence subgroup
$\big\{ \begin{pmatrix} a&b \\ c&d \end{pmatrix} \in SL_2(\mathbb{Z}) \ \ | \ \ a \equiv d \mod N \textrm{ and } b \equiv c\equiv 0 \mod N \big\}$.
$G(...
- 1,905
4
votes
0
answers
163
views
Regularity of the modular curves $Y(N)$, $Y_1(N)$
I'm reading the five chapter of the book of Katz-Mazur, Arithmetic moduli of elliptic curves, concerning regularity of the moduli problems of $\Gamma(N)$-structures, $\Gamma_1(N)$-structures and $\...
- 133
4
votes
0
answers
412
views
A website which explains Mazur's torsion point theorem
I'm about to read Mazur's paper "Modular curves and the Eisenstein ideal".
It's so long and difficult for me, but I found a website which shows the Mazur's theorem.
This is very short and very very ...
- 1,364
3
votes
1
answer
441
views
Moduli space of genus 1 curves with a degree n divisors
I am sure this is well known, but I don't know what to search for:
Consider $M_{1,n}$, the moduli space of genus 1 curves with $n$ marked points. The symmetric group on $n$ letters acts on this space ...
- 7,746
3
votes
0
answers
159
views
The Weil pairing on a generalized elliptic curve
Now I'm trying the section 6 (and 3.20) of chapter IV of Deligne-Rapoport's "Les schemas de module de courbes elliptiques".
I can't understand what $e_n$ (of 6.5.(d)) is.
It seems to be the ...
- 1,364
3
votes
0
answers
278
views
Rationality of Eisenstein series g2 and g3 for elliptic curves defined over numberfields
Let $K$ be a number field and let $E/K$ be an elliptic curve. (Fix an embedding of $K$ into the complex numbers $\mathbb{C}$). Let $\eta$ be the invariant differential of $E/K$. Let $\omega_1$ and $\...
- 139
3
votes
0
answers
323
views
Lifting a real quadratic twist of an Elliptic Curve to the modular curve
Let $E$ be an elliptic curve of conductor $N\cdot p^2$ over $\mathbb{Q}$, defined by the equation
$$y^2=x^3+p^2b\cdot x + p^3\cdot c$$
and parametrized by a map
$$X_{0}(N\cdot {p}^{2})\rightarrow E$$
...
- 1,805
2
votes
2
answers
334
views
For which composite $N$ does $X_0(N)$ possess a non-cuspidal rational point?
According the the introduction to Mazur's Rational Isogenies of Prime Degree the following question was open in 1978:
Let $N$ be one of the integers 39, 65, 91, 125, or 169. Does the modular ...
- 3,268
2
votes
1
answer
312
views
Sheaf of elliptic curves up to isogeny
For a scheme $X$, denote by $\mathcal{Ell}_X[\text{isog}^{-1}]$ the category of elliptic curves on $X$ localized at isogenies. Consider the functor
$$
\mathcal{Ell}^{isog}:Sch/S^{op}\rightarrow \text{...
- 4,418
2
votes
1
answer
258
views
picard group of moduli of elliptic r-prym curves
Let $\overline{\mathcal{M}}_{1,1}$ be the DM compactification of the moduli stack of elliptic curves. Its Picard group is $\mathbb{Z}$. Let us now consider stack of $r$-prym curves $\overline{\mathcal{...
- 3,779
2
votes
0
answers
94
views
Complex Structure Moduli of Elliptic Fibrations
Given an elliptically fibered Calabi-Yau threefold in Weierstrass form I want to compute the number of complex structure moduli of the fibration.
I know how it is done for the generic Weierstrass ...
2
votes
0
answers
309
views
Moduli space of points of fixed order N on elliptic curves
Let us consider the moduli space $Y_1(N)$, parametrizing an elliptic curve, together with a choice of a point of N-torsion on it. Over the complex numbers, it is easy to see that this moduli space is ...
- 63
1
vote
2
answers
582
views
Modular interpretation of an action of the linear group SL_2 on the cohomology of an elliptic curve
Let $E$ be an elliptic curve and $x,y \in H^1(E, \mathbb{Q})$ be a basis for the first rational cohomology group of $E$. There is an action of the linear group $SL_2(\mathbb{Q})$ on $H^*(E,\mathbb{Q})$...
- 690
1
vote
1
answer
186
views
Hodge bundle on F-curves
Let $\mathbb{E}\rightarrow\overline{M}_{g,n}$ be the Hodge bundle. Let us cosider an $F$-curve of type $\overline{M}_{1,1}\subseteq\overline{M}_{g,n}$. Is the degree of the restriction of $\mathbb{E}$ ...
- 8,998
1
vote
0
answers
115
views
Compactifications of product of universal elliptic curves
Let $\mathcal{E}$ be the universal elliptic curve over the moduli stack $\mathcal{M}$ of elliptic curves. As $\mathcal{E}$ is an abelian group scheme over $\mathcal{M}$, we obtain a product-preserving ...
- 12.1k
1
vote
0
answers
248
views
lifts of maps to $\mathcal{M}_{1,1}$
Hi,
here's there's a construction about elliptic curves that I do not completely understand. Suppose I consider the two following families of elliptic curves over $\mathbb{C}^*$.
The first, which I ...
- 3,779
0
votes
0
answers
130
views
Two elliptic curves with the same j-invariants
This is an interesting observation of mine when exploring moduli of elliptic curves.
Let's fix a base field $k$ and assume that we have two elliptic curves $E$, $E'$ which are isomorphic over $\bar k$...
- 1