**2**

votes

**2**answers

294 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

291 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

255 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

248 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

167 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

98 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

401 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

2k 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

719 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

406 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

351 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

602 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

249 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

241 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

362 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

107 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

184 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

887 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

476 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

243 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

276 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

399 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

152 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

355 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

180 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

512 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

373 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 ...

**0**

votes

**0**answers

332 views

### Linking L function dynamics with behavior close to s = 1 ?

A division, found on a sample set of semi-stable elliptic curves, calls for interpretation regarding the Birch and Swinnerton-Dyer conjecture and the dynamic behavior of the L functions involved.
...

**0**

votes

**1**answer

195 views

### A question regarding the j-function

Consider $\tau$ and $\tau'$ in the upper half plane such that $j(g \tau) = j(g \tau')$ for all $g \in GL_2^{+}(\mathbb{Q})$, where $j$ is the modular $j$-function and $GL_2^{+}(\mathbb{Q})$ acts as ...

**5**

votes

**1**answer

556 views

### Must the $j$-invariant of an elliptic curve with an isogeny be integral?

Let $K$ be a quadratic field, and $E/K$ a non-CM elliptic curve with a $K$-rational $p$-isogeny, for $p$ a prime. I would like to say the following:
For large enough $p$, the $j$-invariant $j(E)$ ...

**3**

votes

**3**answers

536 views

### elliptic curves with and without CM

Let $E/\mathbb C$ be an elliptic curve. It is known that if $E$ has CM, then $j(E)$ is an algebraic integer. My first question is: what about the converse? Is there a way to identify the subset of ...

**12**

votes

**2**answers

903 views

### Elliptic Curves with CM and Class Field Theory

Let $K$ be an imaginary quadratic field with Hilbert class field $H$, and let $E$ be an elliptic curve defined over $H$ with complex multiplication by the ring of integers $O_K$ of $K$. It is known ...

**6**

votes

**1**answer

265 views

### E an elliptic curve over Z[1/N], how many p such that E(Z/p^2) = (Z/p)^2?

For a fixed elliptic curve $E$ over $\mathbb{Z}[1/N]$, is there a non-trivial upper bound in terms of $x$ for the number of primes $p \leq x$ with $p \nmid N$ for which $E(\mathbb{Z}/p^2\mathbb{Z}) ...

**5**

votes

**1**answer

543 views

### Special value of $L$-function

Let $p$ be a prime number. Let $f$ be a newform of weight 2 on $Γ_0(p)$, and $E_f$ denote the associated newform quotient of $J_0 (N)$ over $\mathbb{Q}$. Is there a way to express the
algebraic part ...

**13**

votes

**3**answers

2k views

### Rational Points on $y^2=x^3-86069^5$

The analytic rank of the Mordell elliptic curve $y^2=x^3-86069^5$ indicates that it has rank 2. However, deriving a set of generators, and hence the regulator, is proving to be a little bit of an ...

**0**

votes

**1**answer

277 views

### Can two different elliptic curves have rational points in common

Can there be two different elliptic curve $E_{1}$ and $E_{2}$ and two different rational points $P_{1}$ and $P_{2}$ such that $P_{1}, P_{2} \in E_{1}$ and $P_{1}, P_{2} \in E_{2}$ but $P_{1} + P_{2}$ ...