**3**

votes

**2**answers

170 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 ...

**4**

votes

**1**answer

131 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 ...

**3**

votes

**2**answers

312 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.
...

**4**

votes

**2**answers

217 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 $ ...

**2**

votes

**0**answers

147 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

**1**answer

245 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

148 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

116 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 ...

**2**

votes

**1**answer

232 views

### Elliptic units and Euler system

Maybe this question is quite obscure and ambiguous. I am really sorry for such ambiguity.
My question is, what is the good thing we get from defining elliptic units and Euler system? There are lots ...

**2**

votes

**1**answer

254 views

### Mordell-Weil and finiteness of rational points

Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order, that is, $\mathrm{End}_K(E)\cong \mathcal{O}_K$. Suppose the class number of $K$ is equal to $1$. Let ...

**3**

votes

**1**answer

150 views

### The rank of $y^2=x^3\pm i$

How Can I calculate the rank of curves $y^2=x^3\pm i$ over Q(i)?
Is there any soft function to do it?

**10**

votes

**0**answers

374 views

### divisibility of Tamagawa numbers

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$. Let $p\ge11$ be a prime of good ordinary reduction for $E$ and assume that $p$ does not divide the degree of a minimal modular parametrization ...

**1**

vote

**0**answers

121 views

### Cube-root of j-invariant [closed]

Cube-root of j-invariant is a modular function of level 3, does it have similar property as j-invariant? How about the its minimal polynomial？ In particular, are it's coefficients much smaller like ...

**1**

vote

**1**answer

84 views

### The relative sizes of coordinates of a point on projective genus 1 curve (second try)

Hopefully this is better than what I asked yesterday and Milton solved.
Let $ C : F(x,y,z)=0$ be a projective genus $1$ curve over $\mathbb{Q}$ with
no restriction on the degree.
Write a point $P = ...

**2**

votes

**1**answer

372 views

### $\lambda$-invariant is constant for isogenous elliptic curves

How to prove that the $\lambda$-invariant is constant for isogenous elliptic curves $?$

**5**

votes

**2**answers

428 views

### Mazur's torsion theorem on elliptic curves and its generalisations

I want to study Mazur's torsion theorem for elliptic curves over $Q$ and its generalizations for number fields, i.e., papers by Kamienny, Kenku & Momose, Filip Najman. So please suggest to me what ...

**6**

votes

**2**answers

179 views

### Rational points and torsion points of CM elliptic curve

Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order i.e., $\mathrm{End}_K(E)\cong \mathcal{O}=\mathcal{O}_K$. Let $\mathfrak{p}$ be a prime of $K$ such that ...

**1**

vote

**1**answer

96 views

### The relative sizes of coordinates of a point on projective genus 1 curve

Let $ C : F(x,y,z)=0$ be a projective genus $1$ curve over $\mathbb{Q}$ with
no restriction on the degree.
Write a point $P = (X , Y , Z)$ with the smallest coprime integers
$X,Y,Z$.
Is it true that ...

**1**

vote

**2**answers

291 views

### Is there an efficient algorithm to solve ECDLP over global field?

Let E be an elliptic curve over $\mathbb{Q}$. Is there an efficient algorithm which can solve an elliptic curve discrete logarithm in E?

**29**

votes

**0**answers

556 views

### The exponent of Ш of y^2 = x^3 + px, where p is a Fermat prime

For $d$ a non-zero integer, let $E_d$ be the elliptic curve
$$
E_d \colon y^2 = x^3+dx.
$$
When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD,
$$
\# ...

**1**

vote

**2**answers

255 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 ...

**13**

votes

**1**answer

453 views

### State of knowledge of $a^n+b^n=c^n+d^n$ vs. $a^n+b^n+c^n=d^n+e^n+f^n$

As far as I understand, both of the Diophantine equations
$$a^5 + b^5 = c^5 + d^5$$
and
$$a^6 + b^6 = c^6 + d^6$$
have no known nontrivial solutions, but
$$24^5 + 28^5 + 67^5 = 3^5+64^5+62^5$$
and
...

**4**

votes

**2**answers

432 views

### Gross's paper on Heegner points

I try to read Gross's paper on Heegner points and it seems ambiguous for me on some points:
Gross (page 87) said that $Y=Y_{0}(N)$ is the open modular curve over $\mathbb{Q}$ which classifies ordered ...

**2**

votes

**0**answers

246 views

### Automorphisms of the L-function associated to an elliptic $\mathbb{Q}$-curve

Edited after Noam Elkies' comment: From what I understand (very few actually), there exist elliptic curves defined over some number fields $\mathbb{K}$ Galois over $\mathbb{Q}$ which are isogenous to ...

**3**

votes

**1**answer

195 views

### The existence of elliptic curves with prescribed supersingular primes

For a given infinite set of primes, not too big, eg, satisfying Lang-Trotter conjecture, can we always find an E.C. with supersingular reduction (at least) at these primes? How about E.C. without CM?
...

**0**

votes

**1**answer

94 views

### list of supersingular elliptic curves [closed]

Who can give me a table of supersingualr elliptic curves over F_p? At lsist for small p. If I make such a table using magma on my laptop, how long (as a function of p) will I use.

**3**

votes

**2**answers

310 views

### The existence of infinitely many supersingular primes for every elliptic curve over Q

Elkies proved The existence of infinitely many supersingular primes for every elliptic curve over Q. I read his paper, but found the supersingular primes he constructed are all 3(mod 4) type. So, how ...

**6**

votes

**1**answer

291 views

### Fundamental group of the moduli stack of ordinary generalized elliptic curves

Let $M$ be the moduli stack of ordinary but possibly nodal elliptic curves over the field $\overline{\mathbf{F}_p}$. Then $M$ has a $\mathbb{Z}_p^{\times}$-torsor over it, given by the moduli scheme ...

**4**

votes

**1**answer

257 views

### Isogeny classes and reduction types of elliptic curves at primes of bad reduction

Fix a conductor. Then
1) Do the elliptic curves in the same isogeny class have the same reduction type at a prime of bad reduction of the curve ?
2) Do the elliptic curves belonging to two ...

**1**

vote

**0**answers

238 views

### in the preface of “A first course in modular forms”

In the preface of "A first course in modular forms", the author considered the quadratic equation $Q: x^2=d, \ \ d \in \mathbb{Z}, d \not=0$, and for each prime $p$ define an integer $a_p(Q)=\#\tilde ...

**1**

vote

**0**answers

199 views

### Elliptic curve is to Tate module as genus one curve is to blank?

Let $E$ be an elliptic curve over $\mathbf{Q}$, and let $\rho:\mathrm{Gal}(\mathbf{Q}) \to \mathrm{GL}_2(\mathbf{Z}_\ell)$ be its Tate module.
Let $T$ be the set of isomorphism classes of ...

**14**

votes

**3**answers

480 views

### Average rank of elliptic curves, excluding those of low rank

It's conjectured that, asymptotically, half of elliptic curves have rank 0, half have rank 1, and elliptic curves of rank $\geq 2$ have density 0. But what if we disregard elliptic curves of rank 0 or ...

**2**

votes

**3**answers

406 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 ...

**7**

votes

**2**answers

500 views

### $Pic$ of the stack of elliptic curves vs. $Pic$ of the coarse space

There's a natural map $f:\overline{\mathcal{M}}_{1,1}\to \overline{M}_{1,1}\cong \mathbb{P}^1$ from the stack of elliptic curves to the coarse space. Both spaces have $Pic=\mathbb{Z}$ hence ...

**2**

votes

**1**answer

229 views

### Legendre relation for elliptic curves

Let $E$ be an elliptic curve over some subfield $k$ of $\mathbb{C}$, say given by an equation
$$
y^2=4x^3+ax+b.
$$ Then:
$E(\mathbb{C})$ is a complex torus, so $H_1(E(\mathbb{C}), \mathbb{Q})$ is ...

**0**

votes

**0**answers

152 views

### Can one find elliptic curve and a point of known order over $\mathbb{Z}/n\mathbb{Z}$?

Let $n$ be composite with unknown factorization.
Can one efficiently find elliptic curve $E$ over $\mathbb{Z}/n\mathbb{Z}$
and a point $P$ on $E$ of known order $m > 5$?
$P$ should be nontorsion ...

**2**

votes

**1**answer

149 views

### picard group of moduli of elliptic r-prym curves

Let $\overline{\mathcal{M}}_{1,1}$ be the DM compactification of the moduli stack of elliptic curves. Its Picard group is $\mathbb{Z}$. Let us now consider stack of $r$-prym curves ...

**0**

votes

**1**answer

106 views

### One parameter families of elliptic curves over rings of integers of number fields

Let $A(n), B(n) \in \mathbb{Z}[n]$ be polynomials, not both constant, such that $4A^3(n) + 27B^2(n)$ is not the zero polynomial and the polynomial (in variables $x, y$) $$y^2 - x^3 - A(n)x - B(n) \in ...

**1**

vote

**1**answer

214 views

### Example of non-modular elliptic surface?

In "On elliptic modular surfaces", Shioda proves some interesting theorems on smooth elliptic surfaces (admitting a section); he then focuses on "modular elliptic surfaces" and proves some more ...

**0**

votes

**2**answers

378 views

### Rational points or a Weierstrass model for degree 8 elliptic curve

Related to rationally derived polynomials.
Neither Maple nor Magma online couldn't solve it.
Choose $s,h \in \mathbb{Q}$.
I am looking for rational points (possibly of finite order)
on
this ...

**1**

vote

**1**answer

132 views

### Hodge bundle on F-curves

Let $\mathbb{E}\rightarrow\overline{M}_{g,n}$ be the Hodge bundle. Let us cosider an $F$-curve of type $\overline{M}_{1,1}\subseteq\overline{M}_{g,n}$. Is the degree of the restriction of $\mathbb{E}$ ...

**0**

votes

**2**answers

270 views

### what is complexity of finding a non-torsion point on elliptic curve

Let E be an elliptic curve over $Q$ with positive rank $r$. I am looking for algorithms which find a rational point on $E$. I think the algorithms find points with the lowest height. But when I use ...

**0**

votes

**1**answer

196 views

### Point halving on elliptic curves over $\mathbb{Q}$

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $E(\mathbb{Q})[2]=\{o,T_1,T_2,T_3\}$. Let $P=2R$ be a point in $2E(\mathbb{Q})$, using $2$-division polynomial, we can compute $1/2P$, but it gives ...

**2**

votes

**0**answers

216 views

### Other elliptic curves for $x^4+y^4+z^4 = 1$

Given,
$$a^4+b^4+c^4 = d^4\tag{0}$$
we have the identity,
$$(-11980 + 1673 u + 54u^2)^4 + (36 - 2321 u + 3u^2)^4 + t^4 = (24677 + 203 u + 71u^2)^4$$
where,
$$591800025 + 20030510 u + 1671327 u^2 ...

**7**

votes

**0**answers

241 views

### Modular interpretation of Ramanujan theta operator?

I'm a beginner to the theory of modular forms trying to understand a certain construction from the point of view of elliptic curves. Let $f(q) = \sum a_n q^n$ be a formal power series. Define $\theta ...

**3**

votes

**1**answer

189 views

### relation between Faltings height and periods

Let $E$ be an elliptic curve defined by an equation $y^2=4x^3+ax+b$ where $a$ and $b$ are algebraic numbers. What is the relation between the Faltings height $h_F(E)$ and the periods
$$
\int_{\gamma} ...

**1**

vote

**1**answer

351 views

### Determining $\mu$-invariant 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, ...

**1**

vote

**0**answers

249 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 ...

**3**

votes

**0**answers

118 views

### Elliptic curves and quasi-self-reciprocal polynomials

I am reading Shoichi Kihara's On the rank of the elliptic curve $y^2=x^3+k, II$ [Proc. Japan Acad. Ser. A Math. Sci. Volume 72, Number 10 (1996), 228-229] which is available here ...

**4**

votes

**0**answers

288 views

### Explicit family of generalized elliptic curves with level n structure

Let $\pi:\mathcal{E}\rightarrow U$ be a family of elliptic curves with level $n$ structure (in the sense of Deligne-Rapoport) where $U\subseteq C$ is some (non-empty) Zariski open set of a smooth ...