**4**

votes

**1**answer

171 views

### realizing uniform boundedness of Galois representations associated to elliptic curves

This is less of a direct question and more of an argument that I've been worried about for a while and want to check (apologies for the length and if my writing is unclear).
Suppose I have an ...

**18**

votes

**2**answers

354 views

### Deep/precise relationship between two approaches to FLT for polynomials, $n = 3$

David Speyer commented the following here.
I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials ...

**6**

votes

**1**answer

261 views

### Analytic continuation for $L$-functions of elliptic curves

Let $E$ be an elliptic curve over a number field.
When $E$ has no CM and is a $\mathbb Q$-curve (i.e. it is $\overline{\mathbb Q}$-isogenous to all of its conjugates), it is nowadays known that $E$ ...

**10**

votes

**2**answers

356 views

### BSD and congruent numbers

Let $n$ be a positive integer, and let $E_n$ denote the elliptic curve $y^2=x^3-n^2x$. By work of Tunnell, it's known that if $E_n$ satisfies the BSD conjecture, then there is an algorithm to decide ...

**0**

votes

**1**answer

75 views

### Is there an algorithm to find a linear dependence between points on elliptic curves?

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_p$ of characteristic $p$. Let $P,Q\in E(\mathbb{F}_q)$, such that $Q=mP+n\tau(P)$, where $\tau$ is the p-th power of frobenious map and $m$ ...

**3**

votes

**2**answers

222 views

### Congruent numbers and elliptic curves

Who first explicitly stated the link between $N$ being a congruent number and the existence of rational points of infinite order on $y^2=x(x^2-N^2)$?

**1**

vote

**0**answers

87 views

### Elliptic curves with potential good reduction over a prescribed extension

Notation: Let $K/\mathbb{Q}$ be a quadratic number field; let $p\geq 3$ be a rational prime and let $\mathfrak{p}$ denote a prime lying above $p$; let $K_{\mathfrak{p}}$ denote the completion of $K$ ...

**4**

votes

**1**answer

193 views

### Checking whether two rational points of infinite order are generating the torsion free part of an elliptic curve

Let an elliptic curve be given.
As the title says I want to know if we can show that two independent points $P$ and $Q$ are generators of the torsion free part of $E$.
For instance let ...

**1**

vote

**1**answer

131 views

### Is an Isomorphism from an Abelian variety to a Shimura variety always defined over a solvable extension?

Are there counterexamples to the following:
Given two varieties $A$, $\tilde{A}$, both defined over $\mathbb{Q}$, one of which, say $A$, is a Shimura variety. Then, every isomorphism defined over ...

**8**

votes

**1**answer

221 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 ...

**11**

votes

**1**answer

465 views

### Reference to a Don Zagier Result and the Congruent Number Problem

I was looking for a reference/explanation as to how Don Zagier managed to find the side lengths of a rational right triangle with area 157. There have been many literature references to the fact that ...

**7**

votes

**2**answers

416 views

### Galois cohomologies of an elliptic curve

I asked this question at math stackexchange but did not get any answer and I was suggested to post the question here.
I am studying basic theory of elliptic curves. I encountered Galois cohomology. ...

**5**

votes

**1**answer

554 views

### Why are integer points on elliptic curves interesting and useful?

I read some papers which dealed with integer points on elliptic curves. One of these papers are
http://projecteuclid.org/euclid.rmjm/1214947612.
My question is: Why are integer points on elliptic ...

**0**

votes

**2**answers

324 views

### For what integer $n$ are there infinitely many $-a+nb+c = -d+ne+f$ where $a^6+b^6+c^6 = d^6+e^6+f^6$?

(Much revised for clarity.) I was considering the system of equations,
$$-a+nb+c = -d+ne+f\tag1$$
$$a+b+c = d+e+f\tag2$$
$$a^2+b^2+c^2 = d^2+e^2+f^2\tag3$$
$$a^6+b^6+c^6 = d^6+e^6+f^6\tag4$$
...

**4**

votes

**0**answers

78 views

### Minimal discriminant of an elliptic curve in terms of its Galois representation

From the Galois representation of an elliptic curve $E$ we can read the conductor of $E$, and further some information about the minimal discriminant. So is there any more information about the ...

**4**

votes

**2**answers

200 views

### Explicit $2$-descent on elliptic curves

Let $k$ be a field of characteristic $0$ and let
$$E: y^2 = f(x)$$
be an elliptic curve over $k$, with $\mathrm{deg}(f) = 3$. Kummer theory yields a map
$$\varphi:\mathrm{H}^1(k, E[2]) \to ...

**6**

votes

**1**answer

207 views

### Integral points on elliptic curves of the form $y^2=x^3+px$

As the title says. Can we determine all the integral points on elliptic curves of the form
$$y^2=x^3+px$$
for a prime $p$? If yes, can someone explain me how? A good reference would also be ...

**2**

votes

**1**answer

194 views

### Conductor of a CM elliptic curve and its Grössencharacter

For a CM elliptic curve $E$ and its Grössencharakter, their conductors are both supported on bad primes of $E$. Moreover, by comparing their functional equation, there should be some obvious ...

**11**

votes

**1**answer

216 views

### What is the chromatic number of the “conic hypergraph” on a non-singular plane cubic?

Can we color the points of a complex non-singular plane cubic curve with finitely many colors so that no conic intersects the curve at 6 distinct points of the same color?
If so, what is the smallest ...

**6**

votes

**2**answers

473 views

### confounding riddle about fine moduli schemes and twists of elliptic curves

I've encountered a strange situation while thinking about modular curves... Consider the modular curve $Y(3)$ parametrizing elliptic curves with a symplectic basis for their 3-torsion. This curve has ...

**8**

votes

**1**answer

149 views

### Is an elliptic curve that is isomorphic to its Frobenius conjugate defined over $\mathbb{F}_p$?

Let $p$ be prime and $q = p^n$. Let $E$ be an elliptic curve over $\mathbb{F}_q$, and let $E^{(p)}$ be the pullback of $E$ by the $p$-power Frobenius of $\mathbb{F}_q$. If $E$ is isomorphic (over ...

**11**

votes

**1**answer

198 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 ...

**4**

votes

**2**answers

240 views

### What is the fastest algorithm for counting points in elliptic curves mod n?

I need an algorithm for getting the order of the group in random elliptic curves mod n, being n a composite module. As far as I know, usual algorithms like Schoof's algorithm only works with prime ...

**2**

votes

**2**answers

175 views

### Finite orbits on an elliptic curve with two generic involutions

Let $C$ be a (very) general genus 1 curve embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a (2,2)-divisor.
Each projection defines $C$ as a double cover of $\mathbb{CP}^1$ and induces an ...

**3**

votes

**1**answer

175 views

### Veronese embeddings of elliptic curves in weighted projective space

Let $E$ be an elliptic curve and $D_k=kp$ a divisor on $E$, where $p\in E$, for $k\in\mathbb{N}$.
Then we can reconstruct $E$ from the graded ring $R(D_k)=\bigoplus_{n\geqslant0}\mathcal{L}({nD_k})$: ...

**6**

votes

**2**answers

292 views

### Order of reduction of infinite order rational point on an Elliptic Curve

Let $E/$ℚ be an elliptic curve and $P$ ∈ $E($ℚ$)$ a rational point of infinite order. Does the reduction of $P$ mod $p$ generate a maximal cyclic subgroup of $E(\mathbb{F}$$p$$)$ for ...

**6**

votes

**3**answers

665 views

### Do there exist elliptic curves over schemes which have all primes as residue characteristics?

It's well known that there are no elliptic curves over Spec $\mathbb{Z}$, but it's unclear (to me at least) if the proof generalizes.
My question is: If $S$ is a connected scheme such that has every ...

**7**

votes

**1**answer

205 views

### Ramification of the map from the stack of elliptic curves to the $j$-line

Let $\mathcal{M}_{1, 1}$ be the stack of elliptic curves. Its coarse moduli space is $\mathbb{A}^1_{\mathbb{Z}}$ with the map $\mathcal{M}_{1, 1} \rightarrow \mathbb{A}^1_{\mathbb{Z}}$ given by the ...

**7**

votes

**0**answers

175 views

### On discriminants of elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$ and $\Delta_E$ denote the discriminant of $E$. We say an elliptic curve has entanglement fields if the intersection of the $m_1$ and $m_2$ ...

**10**

votes

**1**answer

422 views

### Converse to Modularity I: weight 2 newforms

Since 2008 we have the following remarkable correspondence:
Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1
newforms
note: all Galois representations in this question are ment ...

**7**

votes

**1**answer

309 views

### Serre's surjective theorem importance

I'm studying Serre's paper in wich he shows the following theorem:
Let K be a number field, $E$ an elliptic curve over K without CM. Then the representation $$\rho_{\ell}:\mathrm{Gal}(\bar ...

**2**

votes

**1**answer

110 views

### The $p$-th power of the invariant derivative on an elliptic curve in characteristic $p$

I am not an expert in elliptic curves at all, so my question may naive and/or obvious. Let $E$ be an (affine) elliptic curve defined over a finite (or perfect) field of characteristic $p$. Since its ...

**5**

votes

**1**answer

291 views

### Derivatives of theta functions at zero

Let $L$ be a line bundle over complex elliptic curve, $\deg L = k>0$. Theta functions
$$
\theta_s(z;\tau)_k=\sum_{r\in \mathbb{Z}} e^{\pi i [(\frac{s}{k} + r)^2 k \tau + 2kz(\frac{s}{k}+r)]}, ...

**3**

votes

**2**answers

272 views

### Is it normal surface of general type to have infinitely many positive rank elliptic curves?

Cross-posted from MSE.
I am not good at algebraic geometry and almost surely am
misunderstanding something.
Got an alleged argument against Bombieri-Lang conjecture and
would like to know what the ...

**2**

votes

**0**answers

145 views

### Help for reference of moduli stack of fake elliptic curve

I see everywhere says the following:
Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, $\mathcal{O}_B$ be a maximal order, $N$ be an positive integer coprime to $D$. ...

**0**

votes

**0**answers

89 views

### Polynomial identities for congruent numbers and Bunyakovsky's conjecture

Bunyakovsky's conjecture states that polynomial with integer coefficients
takes infinitely many prime values unless there are obvious reasons not
to.
It appears to imply something about polynomial ...

**5**

votes

**0**answers

191 views

### On $a+b+c= abc = n$, elliptic curves, and solvable Galois groups

Solving $a+b+c = abc = 6$ in the rationals entails solving,
$$-24a+36a^2-12a^3+a^4=z^2\tag1$$
which is birationally equivalent to an elliptic curve. It can be shown that if $a$ is a solution, then ...

**2**

votes

**0**answers

85 views

### Number of CM lifting of an ordinary elliptic curve

Before asking my questions I will start with an example: There are two CM elliptic curves over $\mathbb{Q}$ with CM field $\mathbb{Q}(\sqrt{-7})$, whose $j$-invariants are $-3^3.5^3$ and $3^3. 5^3. ...

**7**

votes

**3**answers

630 views

### Ranks of elliptic curves depend only on the field?

Let $K/\mathbb{Q}$ be an algebraic extension, and let $E_1,E_2/\mathbb{Q}$ be elliptic curves. Is it possible that the Mordell-Weil rank of $E_1(K)$ is finite while that of $E_2(K)$ is infinite?

**2**

votes

**1**answer

212 views

### Congruence Number of 197A1

It is reported in this paper by Zagier, as well as in Sage, that the elliptic curve $E=197A1$ has congruence number 10. (Since $E$ has prime conductor, a theorem of Ribet ensures that the congruence ...

**2**

votes

**0**answers

151 views

### An elliptic curve trivial over any extension unramified outside 7 and infinity?

Is there an elliptic curve $E/\mathbb{Q}$ such that $E(K)$ is trivial for every finite extension $K/\mathbb{Q}$ with discriminant a power of $7$ ?

**1**

vote

**0**answers

122 views

### Possible counterexample to a conjecture of Granville about automorphisms of twists of hyperelliptic curves

This might be a counterexample to a conjecture of Granville
about automorphisms of twists of hyperelliptic curves.
In this paper,
the quadratic twist of $f(x)=y^2$ is denoted by
$C_d : d y^2=f(x)$ ...

**7**

votes

**0**answers

215 views

### Is the compositum of all quadratic extensions of the rationals an ample field?

Let $K$ be the compositum of all quadratic extensions of $\mathbb{Q}$, that is $$K = \mathbb{Q}(\sqrt{d} \ : \ d \in \mathbb{Q}).$$
Is there a (geometrically irreducible) smooth variety ...

**0**

votes

**1**answer

238 views

### A particular interesting elliptic curve

Given the elliptic curve $E:y^2=x^3-4x+4$.
(a) How to prove that the group of rational points $E(\mathbb{Q})$ is generated by $P=(2,2)$.
(b) If we consider the piece of curve on the region ...

**4**

votes

**1**answer

496 views

### Weierstrass form of genus one $y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0$

Related to the n-conjecture.
We are looking for Weierstrass form and map from it of the genus one curve:
$$ y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0 $$
It is ...

**3**

votes

**4**answers

486 views

### Integral points on a particular family of curves

This is a follow-up to this question (and comments thereon). Namely, it follows from Felipe Voloch's comment that for any $n>2$ there is a finite set of integral $(x, y),$ such that
$$
...

**9**

votes

**1**answer

581 views

### Remark 4.23.4 in Hartshorne

Crosspost from math.stackexchange, since it's quite possible I might not get a response there.
Remark 4.23.4 in Chapter IV of Hartshorne's Algebraic Geometry references a paper by Elkies that ...

**4**

votes

**1**answer

175 views

### Equidistribution of representations by a binary cubic form

Let $f(x,y)\in\mathbb{Z}[x,y]$ be a binary cubic form with nonzero discriminant, and for a positive integer $m$ consider the integral representations $f(x,y)=m$. Assume that the number of ...

**14**

votes

**1**answer

317 views

### Why there are only finitely many $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves with CM by $\mathcal{O}$?

For someone who does not have a very extensive knowledge of number theory, what is a good intuitive explanation as to why there are only finitely many $\overline{\mathbb{Q}}$ isomorphism classes of ...

**14**

votes

**0**answers

276 views

### Result of Deuring, intuitive way to see it's true/quickest way to prove?

There is the following result of Deuring that goes as follows:
Let $E/L$ be an elliptic curve defined over a number field $L$ with complex multiplication by an order $\mathcal{O}$ in an imaginary ...