**8**

votes

**1**answer

393 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

434 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

271 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

557 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

228 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

159 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

241 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

423 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

293 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

297 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

284 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

295 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

257 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

219 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

252 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

172 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

99 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

410 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 ...

**16**

votes

**11**answers

3k views

### Fastest way to factor integers < 2^60

I've been running a search for Mordell curves of rank >=8 for about 12 months and have identified approximately 280,000 curves in our archivable range, amongst many millions that aren't.
For this ...

**12**

votes

**2**answers

723 views

### Surjectivity of reduction maps of elliptic curves over Q

Let $E/\mathbf{Q}$ be an elliptic curve of rank $>0$. It is easy to see that there is a positive-density set of primes $p$ such that the reduction map $\mathrm{red}_p : E(\mathbf{Q}) \rightarrow ...

**1**

vote

**1**answer

411 views

### David Hilbert on Complex Multiplication [closed]

I have tried vainly to understand the significance of the following statement attributed to David Hilbert:
The theory of complex multiplication is not only the most beautiful part of mathematics ...

**1**

vote

**3**answers

359 views

### Express Weierstrass' g_2 and g_3 in terms of theta-functions of the periods

If E is a complex elliptic curve defined as the quotient of C over a lattice generated by w_1 and w_2, then it can be also written in Weierstrass form y^2=4*x^3-g_2*x-g_3. The coefficients g_2 and g_3 ...

**9**

votes

**3**answers

603 views

### What CASes say about the analytic rank of rank 8 elliptic curve '457532830151317a1'

For the rank $8$ elliptic curve with a-invariants $(0, 0, 1, -23737, 960366)$
sage 5.3 reports analytic rank $4$ in about 2.4 hours.
Almost sure this a bug, so I am interested what other CAS say on ...

**0**

votes

**0**answers

146 views

### elliptic curves in form $y^2=x^3+p^2x$ where p is prime with rank 0

We Know that from a conjecture by Goldfeld says that half of all elliptic curves have rank zero.
Are there any known infinite families of elliptic curves in form
$y^2=x^3+p^2x$ where p is prime with ...

**4**

votes

**0**answers

252 views

### Abelian cubic extensions of Q[i],

Recently I was considering cubic extensions $K/Q$ that have discriminant negative of a perfect square. Classifying these curves reduces to solving a Diophatine equation of the form $4a^3+27b^2=c^2$ ...

**3**

votes

**1**answer

243 views

### Hilbert scheme of 2 points on an elliptic curve

The Hilbert scheme of 2 points on an elliptic curve $C$, $Hilb^2(C)$, has a natural structure of ruled surface, given by the map $f:Hilb^2(C) \to C$ such that $f(P,Q)=P+Q$.
What can we say about the ...

**2**

votes

**0**answers

147 views

### Curve C of genus 2 whose equation satisfies equation in Igusa invariants, but where Jac(C) does not split

The background question is: Let $C$ be a curve of genus $2$ over a field $k$. When is there a degree $2$ morphism from $C$ to an elliptic curve (and therefore an isogeny from the Jacobian ...

**2**

votes

**0**answers

366 views

### Kernel of an \'etale isogeny of prime degree $\ell$ between elliptic curves

I recently try to read Vatsal's paper ``Multiplicative subgroups of $J_0(N)$ and applications to elliptic curves.'' He seemed to use the following fact freely:
Let $E$ be a semistable elliptic ...

**1**

vote

**0**answers

108 views

### Neighbours of division polynomials over finite field

Let $E$ be an elliptic curve over $\mathbb{F}_p$. The n-th division polynomial is $\psi_n$.
Given points $P=(x_P,y_P)$, $Q=k P$ (where $k$ is unknown) and $\psi_k(x_P,y_P)$, can one efficiently find ...

**6**

votes

**1**answer

247 views

### The existence of an elliptic curve with a specific Galois representation induced by a character

In Kevin Buzzard's survey article on potential modularity Buzzard writes:
Let us say that we have an elliptic
curve $E$ over a totally real ﬁeld $F$,
and we want to prove that $E$ is
...

**10**

votes

**0**answers

1k views

### Why doesn't functoriality immediately imply the modularity theorem?

Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. ...

**1**

vote

**0**answers

185 views

### Canonical forms for elliptic fibrations with Mordell-Weil group of rank 1 and zero torsion

Consider an elliptic fibration given by the following Weierstrass model:
$$
E: y^2 + a_1 x y + a_3 y =x^3 + a_2 x^2 + a_4 x + a_6,\quad a_6=a_2 a_4.
$$
( I work with characteristic zero).
With the ...

**21**

votes

**1**answer

910 views

### Do all curves have Néron models

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$.
Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$?
By a Néron model, I mean ...

**5**

votes

**1**answer

479 views

### Heegner Points and Binary Quadratic Forms

I've been trying to read Gross' paper on Heegner points on $X_0(N)$ and I am stuck on a few details. The definition he is working with is that a heegner points is a pair $y=(E,E')$, where $E$ and $E'$ ...

**1**

vote

**1**answer

247 views

### What is the reduction of this hyperelliptic curve

Let $K$ be a number field and $E/K$ an elliptic curve with equation $Y^2Z = X^3 +AXZ^2+BZ^3$ in $\mathbf{P}^2_K$, where $A,B\in K$.
Let $S$ be non-empty finite set of finite places of $K$ and suppose ...

**4**

votes

**0**answers

278 views

### What is the status of the equidistribution root numbers of elliptic curves' L-functions

In Section 7 of Alice Silverberg's Rank "Cheat Sheet", Silverberg stated
The Bhargava Conjecture: For each $n >
> 1$ the average size of
$S_{n}(E/\mathbb{Q})$ is
...

**6**

votes

**3**answers

324 views

### Example of a diophantine application of an open image theorem

I'm an applied model theorist, and open image theorems are important in the mathematical structures I study (they limit the number of types of elements being realised, and therefore keep things model ...

**3**

votes

**3**answers

432 views

### Another question related to the isogeny theorem for elliptic curves

I was reading the following question: About isogeny theorem for elliptic curves and was interested in the following statement at the end of Torsten Ekedahl's answer:
"Note also that the situation is ...

**3**

votes

**1**answer

373 views

### Imaginary quadratic field contained in Hecke orbit field?

Let $\tau$ in the upper half plane lie in an imaginary quadratic field $K$.
Then is $K \subset \mathbb{Q}(\{j(g \tau) \ | \ g \in GL_2^+(\mathbb{Q}) \})$?
(here $j$ is the modular $j$-function, and ...

**9**

votes

**5**answers

1k views

### The significance of modularity for all Galois representations

On pg. 1 of the slides of a talk, Henri Darmon wrote:
Question: What is an interesting Diophantine equation?
A “working definition”. A Diophantine equation is interesting
if it reveals or ...

**18**

votes

**3**answers

1k views

### Over which fields does the Mordell-Weil theorem hold?

According to a well-known theorem of Mordell, the group of rational points $E(\mathbf{Q})$ of an elliptic curve $E/\mathbf{Q}$ is finitely generated. Weil generalized this theorem to abelian varieties ...

**18**

votes

**1**answer

2k views

### Modern proof of Serre's open image theorem?

Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic ...

**4**

votes

**2**answers

401 views

### What does this quotient of the upper half plane parametrize?

Let $G(N)$ be the congruence subgroup
$\big\{ \begin{pmatrix} a&b \\ c&d \end{pmatrix} \in SL_2(\mathbb{Z}) \ \ | \ \ a \equiv d \mod N \textrm{ and } b \equiv c\equiv 0 \mod N \big\}$.
...

**0**

votes

**1**answer

153 views

### Efficiency in deriving differences of divisor pairs

I have a computational problem where I need to derive the differences in divisor pairs in as few cpu cycles as possible.
In particular I am interested in divisors of numbers of the form ...

**4**

votes

**2**answers

356 views

### Intersection of Hilbert class fields of imaginary quadratic fields

In this question Hilbert class field of Quadratic fields it is mentioned that if $d\equiv 1 \mod 4$ then the Hilbert class field of $\mathbb{Q}(\sqrt{-d})$ contains $\mathbb{Q}(i,\sqrt{d})$.
Could ...

**2**

votes

**1**answer

185 views

### $K$-groups and dual graphs of special fibers

Let $p$ be a prime number, let $E$ be an elliptic curve defined over $\mathbb{Q}_p$. Let $\mathcal{E}_p$ be the special fiber of the Néron model of $E$ over $\mathbb{Z}_p$ and let ...

**6**

votes

**2**answers

1k views

### Isogeny classes of elliptic curves

Let $E \subset \mathbb{P}_\mathbb{C}^2$ be an elliptic curve. If $E$ has complex multiplication (by anything) then the theory of complex multiplication in particular tells us that if $\sigma \in ...

**5**

votes

**2**answers

519 views

### 12 descent scripts for pari/gp

I'm looking around for scripts to facilitate 12 descent on Mordell curves, preferably in Pari/gp.
I understand that Magma implements this feature, but unfortunately this software isn't available to ...

**2**

votes

**1**answer

384 views

### reduction of elliptic curves

Let $X$ be an elliptic curve over a complete local field.
The definition of semi-abelian reduction is: "the Neron model of $X$ is a semi-abelian scheme". On the other hand, the definition of ...

**0**

votes

**0**answers

143 views

### $T^2$-fibered K3 surface with involution

Let $S$ be a K3 surface and $f:S\rightarrow \mathbb{P}^1$ a $T^2$-fibration (not necessarily holomorphic, I have a special Langrangian fibration in mind). Assume there is a $k$-section, then a fiber ...