An elliptic curve is an algebraic curve of genus one with some additional properties. Questions with this tag will often have the top-level tags nt.number-theory or ag.algebraic-geometry in addition; note also the tag arithmetic-geometry as well as some related tags such as rational-points, ...

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346 views

Hecke Characters

My question today concerns Hecke characters or Größencharaktere. I'm doing a project on complex multiplication of elliptic curves and need some help understanding Hecke characters. My main ...
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757 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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226 views

Abelian image of l-adic representation

For a number field F, let E/F be an elliptic curve with CM by a quadratic field K. Let $\rho_\ell: \text{Gal}_F \to \text{Aut}(T_{\ell}E)$ be the $\ell$-adic representation associated to E for some ...
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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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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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80 views

$E_2^*$ at elliptic curves over rational numbers having CM

Let $\tau_0$ be CM point. The values of $E_4,E_6$ at this point is algebraic multiple of the elliptic curve period power. The same is true about the modular completion of the second Eisenstein series ...
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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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125 views

Full $n$-torsion of elliptic curves and the cyclotomic field of order $n$

Hi, overflowers. I have a question concerning the torsion of elliptic curves over number fields. Let us consider an elliptic curve $E$ defined over ${\mathbb Q}$. From the Weil pairing one can ...
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3answers
321 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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167 views

What is the exact mathematical formulation of a claim

The motivation to this question can be found in Why are Galois Representations so important in Number theory ? My question is concerned with this sentence: where they relate the ranks of Selmer ...
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1answer
205 views

Weiestrass Form

How to convert this to weiestrass form? $x^{2}y^{2}-2\left( 1+2\rho \right) xy^{2}+y^{2}-x^{2}-2\left( 1+2\rho \right) x-1=0$
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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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1answer
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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284 views

Where do the product expansions of modular forms come from?

It is well-known that many modular forms can be expressed as infinite products. For instance, the most famous one is probably the expansion $$\Delta(q) = q \prod_{n=1}^\infty (1-q^n)^{24}$$ for the ...
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2answers
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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423 views

Best bounds toward Serre's uniformity conjecture

If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the Galois representations ...
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264 views

Order of torsion group

What can one say about the order of the torsion group of an elliptic curve defined over the compositum of all quadratic extensions of $\mathbb{Q}$ ?
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108 views

A nice rigid analytic model for local systems over an elliptic curve?

For $E$ an elliptic curve, let $LS(E)$ be the group of line bundles on $E$ with a flat connection. This is an $\mathbb{A}^1$-torsor over $E^\vee$. By Riemann-Hilbert (since the gauge group acts ...
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1answer
293 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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146 views

Can we find a set of elliptic curves over rationals associated with $f$?.

We know that for each elliptic curve over rationals, we can define the Dirichlet series of the Hasse–Weil $L$-function, i.e., the function associated with an elliptic curve over rationals. Then my ...
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1answer
255 views

Heegner Points on $X_0(N)$ when some primes dividing $N$ are inert in the imaginary quadratic field

If $K = \mathbb{Q}(\sqrt{-D})$ is a imaginary quadratic field with discriminant $-D$, then we get Heegner points on $X_0(N)$ as long as there exists $\mathfrak{n} \subset \mathcal{O}_K$ such that ...
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456 views

reduction types of elliptic curves

Let $E/K$ be an elliptic curve, where $K$ is a complete local field with residue field $k$ and char$(k) = p$. I'm trying to make sense of Kodaira symbols and Tate's algorithm. My current ...
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289 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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213 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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1answer
249 views

Inertia subgroup in the ordinary reduction case when $p=2$

Dear MO, Let $K/\mathbb{Q}_2$ be a finite extension, and let $E/K$ be an elliptic curve with good ordinary reduction, and such that $\mathbb{Q}_2(j(E))=K$. Let ...
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187 views

Isogenous elliptic curves have same conductor

Let $E/K, E'/K$ be elliptic curves defined over a number field $K$. Let $\phi: E \to E'$ be a non-constant isogeny defined over $K$. Why must the conductors be equal? I know that this is an ...
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3answers
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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2answers
140 views

Possible reasons why the image of 0 in the modular parametrization would always be O for a family of quadratic twists of elliptic curves

Let $E$ be an elliptic curve and $E_n$ be its quadratic twist by $n$. Let $\phi_n: X_0(N_n) \to E_n$ be the normalized modular parametrization of $E_n$ ($\infty \to O$) For some particular curves ...
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2answers
165 views

computing the order of the image of 0 under the modular parametrization map for an elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve of rank 0, with modular parametrization $\phi: X_0(N) \to E$. Let $\Omega_0$ be the least positive real period. A paper I'm reading (Yoshida, Some variants of ...
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3answers
445 views

Division by 3 on elliptic curve

Dear MO, There is a classical theorem (cf. Theorem 4.1 on Husemoller "Elliptic Curve" book) that states conditions on the coordinates of a point P in an elliptic curve to be twice another point. The ...
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2answers
453 views

About equivalent statements of the Birch and Swinnerton-Dyer Conjecture

The Birch and Swinnerton-Dyer Conjecture is well known in the current literature http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture My question is about the possible equivalent ...
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1answer
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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220 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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223 views

Analytic rank of an elliptic curve with algebraic rank 0

Let $E$ be an elliptic curve over $\mathbb{Q}$ with algebraic rank 0. Is there any way, one can argue that the analytic rank must be even? Of course this would follow from the standard conjectures, ...
5
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1answer
510 views

A Universal Elliptic Curve

I'm working through Deligne's "Formes modulaires et representations l-adiques" paper and I find one of his constructions particularily ambigious. I'm hoping someone can give me a bit of clarification ...
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2answers
259 views

Inequality relating rank and analytic rank

Let $E$ be an elliptic curve over $\mathbb{Q}$. It is known from Gross and Zagier that if $\textrm{rank}_{\textrm{an}}(E) \leq 1$, then $$\textrm{rank}(E) \geq \textrm{rank}_{\textrm{an}}(E).$$ ...
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1answer
232 views

The elliptic Lehmer problem for several independent algebraic points

The higher dimensional Lehmer problem asserts that if $\alpha_1,\ldots,\alpha_r$ are multiplicatively independent non-zero algebraic numbers generating an extension of $\mathbb{Q}$ of degree $d$, then ...
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1answer
167 views

Which level structures on elliptic curves are twist-invariant?

Let $N \geq 5$ be a prime, and $H$ a subgroup of $GL_2(\mathbb{F}_N)$. As shown in Chapter IV of [DeRap], there is a curve $X_H(N)$, defined over $K_N := \mathbb{Q}(\zeta_N)^{\det H}$, which is a ...
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1answer
264 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 $E\to \mathbb CP^2$ ...
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1answer
374 views

Best results regarding the Lang-Trotter conjecture

Let $E$ be an elliptic curve over $\mathbb Q$, which does not have complex multiplication over the algebraic closure of $\mathbb Q$. For $x>0$, let $P(x)$ be the number of primes $p < x$ such ...
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1answer
428 views

Like Diophantine equation

Dear all, I have posted this question on m.s.e. Unfortunately, no one responded to answer. I hope, this site and members of this site will answer my questions. The equation $x^n - ny^x-nxy$ = $0$ ...
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1answer
269 views

calculate function from its divizor

There is elliptic curve $C (y^2 = x^3 + Ax + B)$ over $GF(q)$. There is algebraic function f on C. We have div(f). How calculate f as rational function ( $f = (f_1(x) + yf_2(x)) / (g_1(x) + ...
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2answers
531 views

What is a(n algebro-geometric) family of modular forms?

We know that a family of elliptic curves is a morphism of schemes $f:X \to Y$ such that the fiber of every point of $Y$ is an elliptic curve (and we usually require the morphism to be smooth, proper, ...
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211 views

Elliptic curves with almost prime conductor

Let $E/\mathbb{Q}$ be an elliptic curve having a rational 3-torsion point. Then $E$ can be given an affine equation of the type $$y^{2} = x^{3} + (ax + b)^{2}$$ for $a, b, D \in \mathbb{Q}$. Has there ...
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154 views

First reference to the term “Weierstrass equation” in elliptic curves

I'm studying the theory of elliptic curves and in all the books I've read they use the term "Weierstrass equation" or a similar one. But so far I've failed to find out when that term was used for the ...
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234 views

Rank growth of elliptic curves after cubic extensions

Let $E/\mathbb{Q}$ be an elliptic curve and let $N_E(3,X)$ denote the number of cyclic cubic extensions $K/\mathbb{Q}$ of conductor no more than $X$ for which $rank~E(K)> ~rank~ E(\mathbb{Q})$. ...