An elliptic curve is an algebraic curve of genus one with some additional properties. Questions with this tag will often have the top-level tags nt.number-theory or ag.algebraic-geometry in addition; note also the tag arithmetic-geometry as well as some related tags such as rational-points, ...

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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
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1answer
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
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2answers
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
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2answers
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
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0answers
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
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1answer
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 $\overline{\mathbb{Q}}$ ...
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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
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1answer
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
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1answer
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
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1answer
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
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1answer
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?
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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 ...
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0answers
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
1answer
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
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1answer
372 views

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

How to prove that the $\lambda$-invariant is constant for isogenous elliptic curves $?$
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2answers
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
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2answers
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
1answer
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
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2answers
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
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0answers
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
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2answers
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
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1answer
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
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2answers
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
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0answers
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
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1answer
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
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1answer
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
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2answers
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
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1answer
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
1answer
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 ...
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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 ...
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0answers
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
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3answers
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
3answers
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
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2answers
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
1answer
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
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0answers
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
1answer
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
1answer
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 ...
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1answer
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
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2answers
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
1answer
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
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2answers
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 ...
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1answer
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 ...
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0answers
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
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0answers
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
1answer
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
1answer
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, ...
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0answers
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
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0answers
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
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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 ...