**1**

vote

**1**answer

208 views

### Weiestrass Form

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

**0**

votes

**0**answers

135 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 $|E(\mathbb{F}_q)[r]| = r^2$?

**9**

votes

**1**answer

336 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 ...

**3**

votes

**1**answer

320 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 ...

**3**

votes

**0**answers

140 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 ...

**5**

votes

**1**answer

308 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 ...

**5**

votes

**2**answers

288 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 ...

**9**

votes

**2**answers

523 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 ...

**6**

votes

**1**answer

271 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 $\mathbb{Q}$ ?

**2**

votes

**0**answers

119 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 ...

**4**

votes

**1**answer

467 views

### equivalence between katz and classical modular forms

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

**0**

votes

**1**answer

148 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 ...

**3**

votes

**1**answer

302 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 ...

**3**

votes

**3**answers

680 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 ...

**3**

votes

**1**answer

346 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 ...

**1**

vote

**0**answers

216 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**

votes

**0**answers

122 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 ...

**5**

votes

**1**answer

261 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 ...

**5**

votes

**0**answers

215 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 ...

**15**

votes

**3**answers

739 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 ...

**2**

votes

**2**answers

151 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 ...

**4**

votes

**2**answers

180 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 ...

**4**

votes

**3**answers

525 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 ...

**3**

votes

**2**answers

498 views

### About equivalent statements of the Birch and Swinnerton-Dyer Conjecture [closed]

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 ...

**0**

votes

**1**answer

688 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 ...

**2**

votes

**0**answers

232 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.
...

**0**

votes

**0**answers

232 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**

votes

**1**answer

553 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 ...

**3**

votes

**2**answers

276 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
$$\textrm{rank}(E) \geq \textrm{rank}_{\textrm{an}}(E).$$
...

**3**

votes

**1**answer

256 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 ...

**8**

votes

**1**answer

185 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 ...

**8**

votes

**1**answer

286 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 $E\to \mathbb CP^2$ ...

**8**

votes

**1**answer

411 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 ...

**0**

votes

**1**answer

438 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 $x^n - ny^x-nxy$ = $0$ ...

**1**

vote

**1**answer

273 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) + ...

**9**

votes

**2**answers

568 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, ...

**1**

vote

**0**answers

236 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 ...

**1**

vote

**0**answers

161 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 ...

**5**

votes

**1**answer

244 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 $rank~E(K)> ~rank~ E(\mathbb{Q})$. ...

**4**

votes

**2**answers

447 views

### Are there Heronian triangles that can be decomposed into three smaller ones?

Is there anything known about the existence of Heronian triangles ABC (i.e. with rational side lengths and rational area) that can be decomposed into three Heronian triangles ABD, BCD, CAD? ...

**6**

votes

**0**answers

295 views

### elliptic curves over function fields

Let $K$ be a number field, $E$ an elliptic curve over $K$, and $p$ an odd prime. If $v$ is a place of $K$, we know by the Kummer injection that $E(K_v)/pE(K_v) \hookrightarrow H^1 (K_v, E[p])$ for ...

**2**

votes

**2**answers

306 views

### Supersingular Elliptic Curves with rational isogeny?

Let $E/K$ be an elliptic curve over a number field, and $\mathfrak{p}$ a prime of good supersingular reduction. Let $p$ be the prime below $\mathfrak{p}$. I believe that the following is true, but I ...

**3**

votes

**0**answers

286 views

### Rational points and Tesla cards

I'm rapidly approaching 300,000 curves in my ongoing search for Mordell curves of rank >=8.
Currently I'm finding that I have a bottleneck in the code that locates rational points on these curves.
...

**0**

votes

**2**answers

299 views

### The origin of the root number $w(C/ℚ)=±1$ (the sign of the functional equation)

The motivation for this question is the same as in my previous question in MO: http://mathoverflow.net/questions/115179/real-root-1-of-the-hasse-weil-l-function-of-c-over
I am just curious to know ...

**1**

vote

**1**answer

269 views

### complex multiplication

For an abelian variety $A$, it is said to be have $complex \ multiplication$ if $\mathrm{End}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$ contains a number filed $F$ of degree $2 \cdot \mathrm{dim} (A)$. ...

**3**

votes

**1**answer

223 views

### Torsion of elliptic curves is finite

Let $S$ be an integral 1-dimensional scheme with function field $K$.
Let $E$ be an elliptic curve over $K$. The torsion of $E$ over $K$ is not necessarily finite. As an example consider an elliptic ...

**12**

votes

**0**answers

277 views

### For how many primes does an elliptic curve over a totally imaginary field have supersingular reduction?

An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a ...

**3**

votes

**1**answer

183 views

### Kernel of powers of Frobenius on supersingular elliptic curves

I am trying to understand some things related to elliptic curves and finite flat group schemes but I am a little bit confused.
Let $A$ be a supersingular elliptic curve over an algebraically closed ...

**3**

votes

**0**answers

102 views

### Extending cohomology classes to compactifications of Kuga varieties

I am trying to understand the proof of lemma 3 in the paper "Algebraic cycles and the Hodge structure of a Kuga fiber variety" by B. Brent Gordon,
available at ...

**11**

votes

**3**answers

421 views

### The boundedness of the rank of twists of a fixed curve.

It is conjectured that there are elliptic curves over $\mathbb Q$ of arbitrarily high rank. I was wondering wether someone made a similar conjecture if one restricts to a fixed $j$-invariant. If there ...