# Tagged Questions

**2**

votes

**1**answer

221 views

### Does the property (P) holds true for the derivatives of $L$?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. As $s$ takes on real negative values, there are trivial zeros ...

**11**

votes

**2**answers

816 views

+300

### 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 ...

**4**

votes

**2**answers

311 views

### Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$

This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better.
I believe it is a theorem of Grauert that any holomorphic vector ...

**1**

vote

**0**answers

127 views

### Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$

According to a conjecture there are no three
consecutive powerful numbers.
Necessary condition for this is integer solution of
$$ z^3 y^2 = x(x-1)(x+1) \qquad (1) $$
What are integer solutions ...

**4**

votes

**3**answers

147 views

### Quadratic twist of an elliptic curve given by non-Weierstrass model

Suppose $f(x)$ is a polynomial of degree 4 with integer coefficients and nonzero discriminant. Let $C$ be the hyperelliptic curve of genus 1 defined by $y^2=f(x)$. If we assume that $C$ has a rational ...

**1**

vote

**1**answer

95 views

### Elliptic curves with square conductor

Is there a characterization of elliptic curves over $\mathbb Q$ whose conductor is a square? Does this property have a geometric meaning?

**-2**

votes

**0**answers

57 views

### What does this compute to $s=(3(16)^2+9)\cdot(2\cdot 5)^{-1}\bmod{23}$? [closed]

Sorry for asking such a n00b question but what does the following compute to?
$s=(3(16)^2+9)\cdot(2\cdot 5)^{-1}\bmod{23} = 11$
In an online resource, I saw this computes to $11$ but whenever I do ...

**8**

votes

**4**answers

4k views

### Is the ABC conjecture known to imply the Riemann hypothesis?

I once heard from a graduate student that the ABC conjecture implies the Riemann hypothesis. I can't find a reference for this, but given the department the student is from I tend to believe he might ...

**3**

votes

**1**answer

333 views

### $\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 2

Consider the elliptic curves -
$ E_{1}: y^{2}+y=x^{3}+x^{2}-769x-8470 $ $ [\text{Cremona}:19a2] $
$ E_{2}: y^{2}+xy+y=x^{3}-86x-2456 $ $ [\text{Cremona}:38a2] $
with both good ...

**-1**

votes

**0**answers

43 views

### Trace of Frobenius of elliptic curve is integer [migrated]

I recently started to read the book "Arithmetic of Elliptic Curves" by Silverman. And I can't solve an exercise 5.10. Let $E/\mathbb F_q$ be an elliptic curve and $\phi$ is Frobenius endomorphism, ...

**0**

votes

**1**answer

156 views

### Kernel of a 3-isogeny between two elliptic curves

Suppose $E_1$ and $E_2$ are two elliptic curves defined over $\mathbb{Q}$ and there exists a 3-isogeny $\varphi$: $E_1 \longrightarrow E_2$. If $E_1$ has no $\mathbb{Q}$-rational point of order 3, ...

**7**

votes

**5**answers

2k views

### Generalizations of Belyi's theorem

Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:
1) $X$ is defined over $\overline{\mathbb{Q}};$
2) There exists a meromorphic ...

**1**

vote

**0**answers

113 views

### Ratio of periods for elliptic curves in an isogeny class

Let $E \to E^\prime $ be an isogeny of elliptic curves defined over $\mathbb{Q}$. Then what is the definition of ratio of periods for elliptic curves in the isogeny class and how to calculate the ...

**1**

vote

**2**answers

121 views

### Anomalous elliptic curves over finite rings

I was wondering if it is possible to solve the discrete logarithm on an Elliptic Curve E(Z/nZ) (defined over the ring of integers modulo a composite n) with #E(Z/nZ)=n by applying a method analogous ...

**1**

vote

**1**answer

201 views

### On $x^3-y^2=1728 \text{ unit}$ in number fields

Consider solution of
$$x^3-y^2=1728 \text{ unit} \qquad (1)$$
in a number field.
This is related to the discriminant of elliptic curve
in terms of $c_4,c_6$.
Via elliptic curves it might have ...

**5**

votes

**1**answer

482 views

### Main conjecture for elliptic curves

Suppose that $E$ is an elliptic curve defined over $\mathbb{Q}$, and that $p$ is a prime where $E$ has good ordinary reduction. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ ...

**2**

votes

**1**answer

147 views

### Counting curves of degree 4 in $\mathbb{P}^{3}$

Let $p_1,...,p_8\in\mathbb{P}^{3}$ be points in linear general position. Then there exists a unique elliptic curve $C$ of degree $4$ passing through $p_1,...,p_8$. I am interested in what happens for ...

**20**

votes

**1**answer

767 views

### Is the Modularity Theorem (currently) effective?

The Modularity Theorem says every elliptic curve over $\mathbb{Q}$ can be gotten from the classic modular curve $X_0(N)$ by a rational map. Here $N$ is the conductor, easily calculable from a ...

**20**

votes

**3**answers

627 views

### Consecutive square values of cubic polynomials

Let $P(x)$ be a cubic polynomial with integer coefficients. Does there exist a constant $c$ such that at least one of the following values $P(0),P(1),...,P(c)$ is not a square?
It is known that the ...

**10**

votes

**1**answer

411 views

### Examples of elliptic curves over $\mathbb{Q}$

I need examples of two non-isogenous elliptic curves $E_{1}, E_{2}$ over $\mathbb{Q}$ having the following 2 properties -
1) $E_{1}, E_{2}$ have no rational torsion points.
2) $E_1[9] \cong E_2[9]$ ...

**1**

vote

**1**answer

107 views

### A special curve with points of order 3(or 6)

Could you please tell me what is the points of order 3 (or 6) Hon the elliptic curve $y^2=x^3+sx^2-x$ where
$$s = -\frac{1}{432}\frac{(81k^8-2592k^4-6912)}{k^6}$$ and $k$ is rational?

**5**

votes

**2**answers

311 views

### Salmon's proof that tangents to a cubic from a point on it have the same cross-ratio

In Higher plane curves, nr 167, Salmon proves that the cross-ration of the four tangents to a non-singular plane cubic, drawn from a point on the curve, is independent of the point.
A proof can be ...

**0**

votes

**1**answer

154 views

### Elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$

How to find out examples over elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$ $?$

**1**

vote

**0**answers

129 views

### Average rank of elliptic curves over function fields

De Jong showed in 2002 if the finite field $\mathbb{F}_q$ has characteristic not equal to 3, then the limsup of the average of 3-Selmer rank is bounded above, where the average is taken over the ...

**3**

votes

**2**answers

130 views

### Root number of a quadratic twist of an elliptic curve

Could someone provide a reference for the following fact which is stated without proof in section 4.3 of Alice Silverberg's survey "Open Questions in Arithmetic Algebraic Geometry"?
Let E be an ...

**4**

votes

**1**answer

373 views

### Point of order 5 over an elliptic curve

For this curve $y^2=x^3+b^2x^2-a^2b^2x$ where $a \neq b$ and $a,b$ are rational. I can prove that if $b^2+4a^2$ is square then torsion group of curve is $\mathbb Z2 \times \mathbb Z2$,
and when ...

**2**

votes

**3**answers

369 views

### transcendence of periods of CM elliptic curves

Let $E$ be an elliptic curve over $\overline{\mathbb{Q}}$ defined by a Weierstrass equation
$$
y^2=4x^3+g_2x+g_3.
$$ Then $H^1_{dR}(E/\overline{\mathbb{Q}})$ is spanned by the classes of the ...

**26**

votes

**4**answers

1k views

### Are most cubic plane curves over the rationals elliptic?

%This is a new version of the original question modified in the light of the answers and comments.
The word 'most' in the title is ambiguous. The following is one way of making it precise.
...

**3**

votes

**1**answer

58 views

### Algebraicity of isogenies as maps of lattices

Let $E_i\colon y^2=4x^3+A_ix+B_i$, for $i=1,2$ be two elliptic curves where $A_i,B_i \in \mathbb C$ are algebraic over $\mathbb Q$. For $i=1,2$ let $\Lambda_i\subseteq \mathbb C$ be the unique lattice ...

**14**

votes

**6**answers

2k views

### A non-technical account of the Birch—Swinnerton-Dyer Conjecture

I was wondering whether anyone knows of any good non-technical or even popular expositions of the Birch—Swinnerton-Dyer conjecture, for someone with minimal background in elliptic curves. I was ...

**7**

votes

**1**answer

216 views

### Algebraic equations for modular parameterizations

I was wondering if there some place where for some small $N$ I can find explicit modular parameterizations in an algebraic way.
One type of model for $X_0(N)$ is just given by a single algebraic ...

**5**

votes

**1**answer

322 views

### Explicit calculation of Weil Deligne representations

According to Grothendieck monodromy theorem, l-adic galois representations of a local field corresponds to Weil-Deligne representations.
However, given a galois representation, it is usually difficult ...

**7**

votes

**2**answers

308 views

### Is there a largest prime p such that J_0(p) completely splits into elliptic curves

The question in the title is related to a more general question. Namely does there exist an integer $N$ such that for all curves $C/\mathbb C$ of genus $> N$ one has that not all simple isogeny ...

**10**

votes

**2**answers

335 views

### Records for low-height points on elliptic curves over number fields

Elkies maintains a list of nontorsion points of low height on elliptic curves over Q; does anyone know of anything similar for curves over number fields?
Everest and Ward give examples of points of ...

**11**

votes

**1**answer

315 views

### $S$-Tate-Shafarevich groups of elliptic curves

Let $S$ be a finite set of places of a number field $k$ and let $E$ be an elliptic curve over $k$. Define the ''$S$-Tate-Shafarevich group" of $E$ to be
$$Ш(E,S) = \ker\left(H^1(k,E) \to \prod_{v ...

**3**

votes

**2**answers

184 views

### $\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 1

1) What are the examples of elliptic curves over $\mathbb{Q}$ with good reduction and $\mu$-invariant $\geq 2$ at $p = 3$ and how to find them $?$
2) Let $\Lambda = \mathbb{Z}_{p}[[T]] $ and $ ...

**5**

votes

**0**answers

133 views

### When are Galois representations with open image attached to elliptic curves?

Let $K$ be a number field with absolute Galois group $G_K$.
Let $\rho:G_K \rightarrow GL_2(\hat{\mathbb{Z}})$ be a Galois representation such that the image of $\rho$ is open in ...

**5**

votes

**1**answer

381 views

### What did Shimura say about $y^2 + y = x^3 - x$?

From the introduction of Ribet-Stein:
Shimura showed that if we start with the elliptic curve $E$ defined by the equation $y^2 +y = x^3 −x^2$ then for “most” $n$ the image of $\rho$ is all of ...

**3**

votes

**2**answers

149 views

### Hesse pencil and Schrodinger representation of Heisenberg group

Let $E$ be a smooth elliptic curve over an algebraically closed field of characteristic zero. Let $\mathcal{L}$ be a line bundle of degree $3$. Heisenberg group $H_3$ acts on global sections of ...

**3**

votes

**1**answer

110 views

### What is the complexity of finding a point with a given height on elliptic curve?

It is well known that there exists a canonical height function $\hat{h}:E(\mathbb{Q})\longrightarrow\mathbb{R}$. My question is: I have a real number $h$ and elliptic curve $E$. Is there a feasible ...

**6**

votes

**1**answer

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 $\mathbb{Q}$ ?

**7**

votes

**2**answers

608 views

### Supersingular elliptic curves over $\mathbb{Q}$

what are the examples of elliptic curves defined over $\mathbb{Q}$ with supersingular reduction at a prime $p$ and having a $p$-isogeny over $\mathbb{Q}$ ?

**1**

vote

**1**answer

280 views

### Determining $\mu$-invariants of elliptic curves over $\mathbb{Q}$

From Pollack's table on his homepage, I have the values of mu invariant of elliptic curves 38B1 & 38B2 (labeled as in Cremona table). But I need to know the values of mu invariants of 38A1, 38A2, ...

**2**

votes

**1**answer

306 views

### Invariants for isogenous elliptic curves

How to prove that for elliptic curves, the $\lambda$-invariant is always unchanged by an isogeny ?

**1**

vote

**2**answers

207 views

### Isogeny classes and elliptic curves over finite fields

Fix a conductor and a prime $p$. Then
1) Do the elliptic curves in the same isogeny class after reduction modulo $p$ have the same number of points over the finite field $\mathbb{F}_{p} ?$
2) Do the ...

**2**

votes

**0**answers

122 views

### Finite Heisenberg groups action on cohomology of line bundles

Let $E$ be a smooth elliptic curve over algebraically closed field $k$ of characteristic zero, $\mathcal{L}$ is a line bundle over $E$, $\operatorname{deg}(\mathcal{L})=n \geq 1$. Then I define the ...

**3**

votes

**2**answers

183 views

### Example of elliptic curve with CM (complex multiplication) by \sqrt{-7}

Can someone give me an example of elliptic curve with CM by sqrt(-7) with the action.
I've found a list of examples in the following link but not the action.
...

**3**

votes

**1**answer

226 views

### Are elliptic Kummer extensions big?

Loosely speaking, are elliptic Kummer extensions big? More concretely:
Let $E$ be an elliptic curve over $\mathbb{Q}$, let $p$ be a prime, and
let $F$ be a subfield of $\overline{\mathbb{Q}}$ ...

**0**

votes

**0**answers

120 views

### Inflection points on elliptic curves over a field of characteristic 2

I'm looking at the elliptic curve $C:={\cal Z}(XY^2+ZX^2+YZ^2)$ in the field $k:=\overline{\mathbb{F}_2}$. I want to prove that this curve has 9 inflection points. Since the characteristic of $k$ is ...

**0**

votes

**1**answer

110 views

### Does the modified Szpiro conjecture require minimal model?

The modified Szpiro conjecture is described in
Wikipedia
and here and here.
The modified Szpiro conjecture states that: given $\varepsilon > 0$, there exists a constant $C(\varepsilon)$ such ...