# Tagged Questions

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78 views

### Parabolic bundles on elliptic curves

as a warm up for his thesis I would like a student of mine to read something on parabolic bundles. He is reading the famous Atiyah paper on vector bundles on elliptic curves, so I think it would be ...

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**2**answers

321 views

### Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$

This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better.
I believe it is a theorem of Grauert that any holomorphic vector ...

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**1**answer

156 views

### Kernel of a 3-isogeny between two elliptic curves

Suppose $E_1$ and $E_2$ are two elliptic curves defined over $\mathbb{Q}$ and there exists a 3-isogeny $\varphi$: $E_1 \longrightarrow E_2$. If $E_1$ has no $\mathbb{Q}$-rational point of order 3, ...

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114 views

### Ratio of periods for elliptic curves in an isogeny class

Let $E \to E^\prime $ be an isogeny of elliptic curves defined over $\mathbb{Q}$. Then what is the definition of ratio of periods for elliptic curves in the isogeny class and how to calculate the ...

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**1**answer

485 views

### Main conjecture for elliptic curves

Suppose that $E$ is an elliptic curve defined over $\mathbb{Q}$, and that $p$ is a prime where $E$ has good ordinary reduction. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ ...

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**1**answer

148 views

### Counting curves of degree 4 in $\mathbb{P}^{3}$

Let $p_1,...,p_8\in\mathbb{P}^{3}$ be points in linear general position. Then there exists a unique elliptic curve $C$ of degree $4$ passing through $p_1,...,p_8$. I am interested in what happens for ...

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**1**answer

414 views

### Examples of elliptic curves over $\mathbb{Q}$

I need examples of two non-isogenous elliptic curves $E_{1}, E_{2}$ over $\mathbb{Q}$ having the following 2 properties -
1) $E_{1}, E_{2}$ have no rational torsion points.
2) $E_1[9] \cong E_2[9]$ ...

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312 views

### Salmon's proof that tangents to a cubic from a point on it have the same cross-ratio

In Higher plane curves, nr 167, Salmon proves that the cross-ration of the four tangents to a non-singular plane cubic, drawn from a point on the curve, is independent of the point.
A proof can be ...

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308 views

### Is there a largest prime p such that J_0(p) completely splits into elliptic curves

The question in the title is related to a more general question. Namely does there exist an integer $N$ such that for all curves $C/\mathbb C$ of genus $> N$ one has that not all simple isogeny ...

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133 views

### When are Galois representations with open image attached to elliptic curves?

Let $K$ be a number field with absolute Galois group $G_K$.
Let $\rho:G_K \rightarrow GL_2(\hat{\mathbb{Z}})$ be a Galois representation such that the image of $\rho$ is open in ...

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150 views

### Hesse pencil and Schrodinger representation of Heisenberg group

Let $E$ be a smooth elliptic curve over an algebraically closed field of characteristic zero. Let $\mathcal{L}$ be a line bundle of degree $3$. Heisenberg group $H_3$ acts on global sections of ...

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123 views

### Finite Heisenberg groups action on cohomology of line bundles

Let $E$ be a smooth elliptic curve over algebraically closed field $k$ of characteristic zero, $\mathcal{L}$ is a line bundle over $E$, $\operatorname{deg}(\mathcal{L})=n \geq 1$. Then I define the ...

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**1**answer

240 views

### Mordell-Weil and finiteness of rational points

Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order, that is, $\mathrm{End}_K(E)\cong \mathcal{O}_K$. Suppose the class number of $K$ is equal to $1$. Let ...

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348 views

### Mazur's torsion theorem on elliptic curves and its generalisations

I want to study Mazur's torsion theorem for elliptic curves over $Q$ and its generalizations for number fields, i.e., papers by Kamienny, Kenku & Momose, Filip Najman. So please suggest to me what ...

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160 views

### Rational points and torsion points of CM elliptic curve

Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order i.e., $\mathrm{End}_K(E)\cong \mathcal{O}=\mathcal{O}_K$. Let $\mathfrak{p}$ be a prime of $K$ such that ...

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400 views

### Gross's paper on Heegner points

I try to read Gross's paper on Heegner points and it seems ambiguous for me on some points:
Gross (page 87) said that $Y=Y_{0}(N)$ is the open modular curve over $\mathbb{Q}$ which classifies ordered ...

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**1**answer

268 views

### Fundamental group of the moduli stack of ordinary generalized elliptic curves

Let $M$ be the moduli stack of ordinary but possibly nodal elliptic curves over the field $\overline{\mathbf{F}_p}$. Then $M$ has a $\mathbb{Z}_p^{\times}$-torsor over it, given by the moduli scheme ...

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369 views

### transcendence of periods of CM elliptic curves

Let $E$ be an elliptic curve over $\overline{\mathbb{Q}}$ defined by a Weierstrass equation
$$
y^2=4x^3+g_2x+g_3.
$$ Then $H^1_{dR}(E/\overline{\mathbb{Q}})$ is spanned by the classes of the ...

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471 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 ...

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**1**answer

131 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 ...

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**1**answer

209 views

### Example of non-modular elliptic surface?

In "On elliptic modular surfaces", Shioda proves some interesting theorems on smooth elliptic surfaces (admitting a section); he then focuses on "modular elliptic surfaces" and proves some more ...

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**1**answer

119 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 $\mathbb{E}$ ...

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**1**answer

174 views

### relation between Faltings height and periods

Let $E$ be an elliptic curve defined by an equation $y^2=4x^3+ax+b$ where $a$ and $b$ are algebraic numbers. What is the relation between the Faltings height $h_F(E)$ and the periods
$$
\int_{\gamma} ...

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106 views

### Elliptic curves and quasi-self-reciprocal polynomials

I am reading Shoichi Kihara's On the rank of the elliptic curve $y^2=x^3+k, II$ [Proc. Japan Acad. Ser. A Math. Sci. Volume 72, Number 10 (1996), 228-229] which is available here ...

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252 views

### Explicit family of generalized elliptic curves with level n structure

Let $\pi:\mathcal{E}\rightarrow U$ be a family of elliptic curves with level $n$ structure (in the sense of Deligne-Rapoport) where $U\subseteq C$ is some (non-empty) Zariski open set of a smooth ...

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**1**answer

174 views

### Visualizing singular points of real loci of elliptic curves

On one hand the real locus of a complex elliptic curve is the intersection of a plane with a torus (i.e. a torus embedded in $\mathbb{C}^2$ plus infinity). And an elliptic curve has no cusps or ...

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376 views

### Equations of elliptic curves

First part of question I have asked on mathoverflow already: http://math.stackexchange.com/questions/467088/explict-form-of-the-equation-of-elliptic-curve
1) Let $E(\mathbb{F}_{q^2})$ is elliptic ...

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182 views

### Is it expected that every natural number is the rank of some elliptic curve over the rationals?

It is a well-known problem on the theory of elliptic curves that the rank of an elliptic curve (the number of generators of the free part of the Mordell group of the elliptic curve) cannot be ...

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80 views

### Picard-Fuchs equation of elliptic curves with level $N$ structure.

Let $X_N$ be the moduli space of elliptic curves with level $N$ structure. Is the Piard Fuchs equation of the universal family over $X_N$ (take some covering if $X_N$ is not a fine moduli space)?
In ...

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97 views

### A certain property of elliptic curves in a paper by Rees

In the paper "On a problem of Zariski", David Rees presents a counterexample to the following problem of Zariski.
Let $F/k$ be a f.g. field extension, $S$ a f.g. normal integral domain over $k$ ...

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227 views

### Canonical differential on Tate curve

I am starting studying the theory of (algebraic) modular forms, and I have some trouble in understanding completely the construction of the Tate curve. My problem is the following: as far as I know ...

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758 views

### What is $Aut(Ell)$?

Consider the stack $Ell$ (of groupoids) of elliptic curves. I'm interested in the autoequivalence 2-group of $Ell$, the objects of which consists of transformations $Ell \Rightarrow Ell: Ring \to Gpd$ ...

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119 views

### What does Hodge theory tell us about simply connected surfaces of general type

Let $X$ be a smooth complex projective variety. We know that $\Omega^1_X$ has a non-zero section if and only if the abelianization of the fundamental group of X is infinite. This follows from Hodge ...

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136 views

### Specialization of sections in an elliptic fibration

Let $\pi: X \rightarrow S$ be the Neron model of an elliptic curve over a dedekind domain (but probably any minimal elliptic fibration will suffice).
Let $\eta$ be the generic point of $S$, $K = ...

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234 views

### A question on degeneration of elliptic curves with actions.

Let $E$ be an elliptic curve. I want to consider its degeneration to the union of two projective lines $C:=\mathbb{P}^1 \cup_{x,y} \mathbb{P}^1$ attaching at two points $x,y$. The involution $-1$ on ...

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175 views

### Affine neighborhood of an $S$-valued point

How can we understand an affine neighborhood of an $S$-valued point on a scheme, and when does it exist?
I am looking at page 111 of Haruzo Hida's Geometric Modular Forms and Elliptic Curves, and he ...

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323 views

### Surfaces ruled over elliptic curves

Ground field $\Bbb{C}$. Algebraic category. Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve.
Suppose $E$ is an elliptic curve ...

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121 views

### r-torsion points on elliptic curve on finite field

Let $E(\mathbb{F}_q)$ - elliptic curve, $r$ is prime, $|E(\mathbb{F}_q)[r]| > 1$.
Let $r | q-1$. Is it true that $|E(\mathbb{F}_q)[r]| = r^2$?

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308 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 ...

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289 views

### Shafarevich's theorem for elliptic curves defined over function field of algebraic curve over algebraically closed field

Let $K$ be a number field and $S$ a finite set of places of $K$. Then Shafarevich's theorem states that there are only finitely many isomorphism classes of elliptic curves $E$ over $K$ with good ...

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131 views

### Is Hasse-witt map isomorphism?

Fix a level $N \geq 3$ and denote by $\Gamma(N)\subset SL(2,\mathbb{Z})$ the subgroup of matrices which are congruent to the identity modulo $N$. The open modular curve $Y(N)$ corresponding to ...

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223 views

### elliptic curve with a degree 2 isogeny to itself?

I've come across the following question, which I think must be easy for experts: is there a complex elliptic curve $E$ with an isogeny of degree 2 to itself?
Of course one can ask the same question ...

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**1**answer

297 views

### equivalence between katz and classical modular forms

$\newcommand{\CC}{\mathbb{C}}$
$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\PP}{\mathbb{P}}$
$\newcommand{\QQ}{\mathbb{Q}}$
$\newcommand{\hH}{\mathcal{H}}$
$\newcommand{\eE}{\mathcal{E}}$
...

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**1**answer

291 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 ...

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214 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 ...

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116 views

### P-adic Weierstrass Lemma for several variables

The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in the ring of power ...

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608 views

### Elliptic curve over a scheme is a group scheme?

In Katz's article p-adic properties of modular schemes and modular forms in the Antwerp proceedings, the following definition of an elliptic curve over a base scheme $S$ is given:
By an elliptic ...

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**1**answer

672 views

### Pairing on elliptic curve

Let $E(\mathbb{F_q})$ - elliptic curve.
$G_1 = E(\mathbb{F_q})[r]$. $|G_1| = r$.
$k$ is minimal such $r | q^k - 1$.
$\pi_q$ - $q$-power Frobenius endomorphism.
$G_2 = E(\mathbb{F_{q^k}})[r] \cap ...

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221 views

### On the expilicite example of Parshin Construction.

In the Proof of Mordell Conjecture by Gerd Faltings, it is famous that Parshin constructed curve C_P for each Q-rational point P on the given curve C over Q such that genus of C satisfies g(C) > 0.
...

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**1**answer

265 views

### Injective morphism from an elliptic curve to $\mathbb CP^2$.

Let $E$ be the elliptic curve $x^3+y^3+z^3=0$.
Question. Are there injective morphisms $E\to \mathbb CP^2$ of arbitrary high degree?
Comments. 1) There are injective morphisms $E\to \mathbb CP^2$ ...