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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4
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1answer
180 views

Derivatives of theta functions at zero

Let $L$ be a line bundle over complex elliptic curve, $\deg L = k>0$. Theta functions $$ \theta_s(z;\tau)_k=\sum_{r\in \mathbb{Z}} e^{\pi i [(\frac{s}{k} + r)^2 k \tau + 2kz(\frac{s}{k}+r)]}, ...
3
votes
2answers
260 views

Is it normal surface of general type to have infinitely many positive rank elliptic curves?

Cross-posted from MSE. I am not good at algebraic geometry and almost surely am misunderstanding something. Got an alleged argument against Bombieri-Lang conjecture and would like to know what the ...
2
votes
0answers
141 views

Help for reference of moduli stack of fake elliptic curve

I see everywhere says the following: Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, $\mathcal{O}_B$ be a maximal order, $N$ be an positive integer coprime to $D$. ...
0
votes
0answers
65 views

Polynomial identities for congruent numbers and Bunyakovsky's conjecture

Bunyakovsky's conjecture states that polynomial with integer coefficients takes infinitely many prime values unless there are obvious reasons not to. It appears to imply something about polynomial ...
4
votes
0answers
173 views

On $a+b+c= abc = n$, elliptic curves, and solvable Galois groups

Solving $a+b+c = abc = 6$ in the rationals entails solving, $$-24a+36a^2-12a^3+a^4=z^2\tag1$$ which is birationally equivalent to an elliptic curve. It can be shown that if $a$ is a solution, then ...
2
votes
0answers
67 views

Number of CM lifting of an ordinary elliptic curve

Before asking my questions I will start with an example: There are two CM elliptic curves over $\mathbb{Q}$ with CM field $\mathbb{Q}(\sqrt{-7})$, whose $j$-invariants are $-3^3.5^3$ and $3^3. 5^3. ...
7
votes
3answers
580 views

Ranks of elliptic curves depend only on the field?

Let $K/\mathbb{Q}$ be an algebraic extension, and let $E_1,E_2/\mathbb{Q}$ be elliptic curves. Is it possible that the Mordell-Weil rank of $E_1(K)$ is finite while that of $E_2(K)$ is infinite?
2
votes
1answer
205 views

Congruence Number of 197A1

It is reported in this paper by Zagier, as well as in Sage, that the elliptic curve $E=197A1$ has congruence number 10. (Since $E$ has prime conductor, a theorem of Ribet ensures that the congruence ...
1
vote
0answers
148 views

An elliptic curve trivial over any extension unramified outside 7 and infinity?

Is there an elliptic curve $E/\mathbb{Q}$ such that $E(K)$ is trivial for every finite extension $K/\mathbb{Q}$ with discriminant a power of $7$ ?
1
vote
0answers
114 views

Possible counterexample to a conjecture of Granville about automorphisms of twists of hyperelliptic curves

This might be a counterexample to a conjecture of Granville about automorphisms of twists of hyperelliptic curves. In this paper, the quadratic twist of $f(x)=y^2$ is denoted by $C_d : d y^2=f(x)$ ...
7
votes
0answers
199 views

Is the compositum of all quadratic extensions of the rationals an ample field?

Let $K$ be the compositum of all quadratic extensions of $\mathbb{Q}$, that is $$K = \mathbb{Q}(\sqrt{d} \ : \ d \in \mathbb{Q}).$$ Is there a (geometrically irreducible) smooth variety ...
0
votes
1answer
220 views

A particular interesting elliptic curve

Given the elliptic curve $E:y^2=x^3-4x+4$. (a) How to prove that the group of rational points $E(\mathbb{Q})$ is generated by $P=(2,2)$. (b) If we consider the piece of curve on the region ...
4
votes
1answer
487 views

Weierstrass form of genus one $y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0$

Related to the n-conjecture. We are looking for Weierstrass form and map from it of the genus one curve: $$ y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0 $$ It is ...
3
votes
4answers
458 views

Integral points on a particular family of curves

This is a follow-up to this question (and comments thereon). Namely, it follows from Felipe Voloch's comment that for any $n>2$ there is a finite set of integral $(x, y),$ such that $$ ...
8
votes
1answer
573 views

Remark 4.23.4 in Hartshorne

Crosspost from math.stackexchange, since it's quite possible I might not get a response there. Remark 4.23.4 in Chapter IV of Hartshorne's Algebraic Geometry references a paper by Elkies that ...
4
votes
1answer
172 views

Equidistribution of representations by a binary cubic form

Let $f(x,y)\in\mathbb{Z}[x,y]$ be a binary cubic form with nonzero discriminant, and for a positive integer $m$ consider the integral representations $f(x,y)=m$. Assume that the number of ...
14
votes
1answer
301 views

Why there are only finitely many $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves with CM by $\mathcal{O}$?

For someone who does not have a very extensive knowledge of number theory, what is a good intuitive explanation as to why there are only finitely many $\overline{\mathbb{Q}}$ isomorphism classes of ...
13
votes
0answers
253 views

Result of Deuring, intuitive way to see it's true/quickest way to prove?

There is the following result of Deuring that goes as follows: Let $E/L$ be an elliptic curve defined over a number field $L$ with complex multiplication by an order $\mathcal{O}$ in an imaginary ...
6
votes
1answer
154 views

Question on paper of Stewart and Top about ranks of elliptic curves over Q(t)

I'm reading "On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms" by Stewart and Top, and struggling to understand the argument on pg 962 which shows that the rank of a ...
2
votes
1answer
141 views

isogeny clases of CM abelian varieties

Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$ and with complex multiplication by a CM field $K$. Looking at the action of $K$ on $H^0(A, \Omega^1_A)$ one gets a CM type of $K$, ...
0
votes
0answers
142 views

Conditions for splitting of short exact sequence?

Are there conditions under which the short exact sequence $$0\rightarrow E (K)/mE (K)\rightarrow H^1_{Sel}(K,E_m)\rightarrow \Sha(E|K)_m\rightarrow 0$$ splits? I assume $K $ to be a number field and ...
1
vote
0answers
101 views

References for modular curves over finite fields [closed]

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...
0
votes
2answers
139 views

Special type Diophantine equations with integer solutions

The following problem on Diophantine equation is still solved or not I don't know. However, I found few solutions by trail and error method. Problem: $X^2 - X = Y^5 - Y$ has integer solutions or not? ...
1
vote
1answer
126 views

From an eigenfom with $\mathbf{Q}$-coefficients to $j$-invariants

Given a cuspidal, classical eigenform $f\in S_2(\Gamma_0(N))$ of weight $2$ and with $\mathbf{Q}$-coefficients is there a way of describing the set $J_f$ of $j$-invariants of the elliptic curves lying ...
1
vote
0answers
57 views

What is the complexity of finding a generator for the cyclic elliptic curves?

Let $E$ be an elliptic curve which is defined over a finite field $\mathbb{F}_p$, where $p$ is a prime number. If we know that $E(\mathbb{F}_p)$ is cycyclic, is there an algorithm to find its ...
0
votes
0answers
72 views

Are the elliptic curve discrete log problem and the elliptic curve Diffie-Hellman problem equivalent?

Suppose that $G=\langle g\rangle$ is a general group of order $p$. Maurer has introduced an algorithm to reduce the discrete log problem to the Diffie-Hellman problem under a conjecture about smooth ...
4
votes
2answers
211 views

Secant varieties of curves in $\mathbb{P}^4$

My question is motivated by the following simple observations. By a standard dimensions count in $\mathbb{P}^4$ there should not exist neither an hypersurface of degree $3$ with multiplicity $2$ in ...
6
votes
0answers
67 views

Prime divisors of the norm of the first coefficient of an elliptic newform at width-1 cusps.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ and let $f$ be its newform. Suppose $p \geq 5$ is a prime such that $p^2 \mid N$. We assume $f$ is $p$-minimal, which is equivalent to that the ...
5
votes
1answer
160 views

On the number of 3-Selmer elements of rational elliptic curves

I am trying to understand a step in the proof of Theorem 39 in the recent work of Bhargava and Shankar, "Ternary Cubic Forms having bounded invariants, and the existence of a positive proportion of ...
13
votes
2answers
379 views

Elliptic curves and supercuspidal representations of conductor $p^2$

Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $p \geq 5$ be a prime of additive reduction for $E$. Let $f$ be the newform associated to $E$, and let $\pi$ be the irreducible admissible ...
4
votes
0answers
194 views

Example of a genus-1 degree-7 plane curve

I am wondering if anyone knows how to construct an explicit example of an irreducible plane curve of degree 7 with 14 double points. Such a curve would have genus 1. One can show that for a general ...
3
votes
2answers
279 views

n torsion groups of quadratic twists of elliptic curves

If $E$ is an elliptic curve over a number field $K$ and $E^{F}$ is a quadratic twist of $E$. Then it is stated in ``Ranks of twists of elliptic curves and Hilbert’s tenth problem" due to Mazur and ...
-4
votes
1answer
113 views

Elliptic Curve Multiplication [closed]

What would happen if I performed Elliptic Curve multiplication on some random point within the FiniteField that wasn't actually on the curve? I assume that I would get a point in return but would that ...
11
votes
0answers
433 views

Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system, $$x_1^2+x_2^2+\dots+x_n^2 = y^2$$ $$x_1^3+x_2^3+\dots+x_n^3 = z^3$$ and define the function, $$F(s_m) = x_1+x_2+\dots+x_n$$ For $n\geq3$, using an ...
13
votes
1answer
535 views

What are the strongest conjectured uniform versions of Serre's Open Image Theorem?

This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM ...
7
votes
1answer
244 views

how do automorphisms of elliptic curves act on the Tate module?

Let $E/k$ be an elliptic curve over some algebraically closed field $k$ of characteristic $p\ge 0$. It's known that $Aut(E)$ acts faithfully on the Tate module $T_\ell(E)$ ($\ell\ne p$) with ...
4
votes
0answers
147 views

Models for the moduli space $\overline{M}_{1,n}$

Let $\overline{M}_{1,n}$ denote the coarse moduli space of $n$-pointed elliptic curves. Is there an explicit description of these spaces (a la Kapranov's construction) for low $n$? Apparently this ...
3
votes
1answer
131 views

Interpreting Frobenius pullback as an invariant differential in the case of an elliptic curve

Let $S$ be an affine scheme of characteristic $p > 0$, let $E \rightarrow S$ be an elliptic curve over $S$, and let $F$ denote the absolute Frobenius. Since $E$ is its own $\mathrm{Pic}^0$ there is ...
0
votes
1answer
134 views

homological invariant of the “universal elliptic curve” over the punctured $j$-line

My question considers the curve $E$ over the affine $j$-line $S$ given by $$Y^2 - (j-1728)XY = X^3 - 36(j-1728)^3X - (j-1728)^5$$ This curve has the property that it's $j$-invariant is $j$ (see ...
3
votes
1answer
209 views

Addition law for elliptic curves of the form $x^2y^2+a(x+y)+b=0$

Did anybody consider addition law for elliptic curves of the form $$x^2y^2+a(x+y)+b=0\,?$$ Does this form have any specific name?
0
votes
1answer
62 views

The group of real points on quadratic twist of elliptic curve has one connected component

I am trying to understand the proof of assertion (i) in Proposition 3.10 (page 14) of this paper http://arxiv.org/pdf/1312.3884v3.pdf $M$ stands for a square free integer which is prime to $7$, $A$ ...
0
votes
0answers
106 views

Ward's formula for elliptic divisibility sequences

M. Ward in his Memoir on elliptic divisibility sequences proved that the sequence $\{a_n\}$ defined by recurrence $$a_{n+2}a_{n-2}=a_2^2a_{n+1}a_{n-1}-a_1a_3a_n^2$$ and initial conditions ...
2
votes
1answer
141 views

Frey's Formula and utilisation of the Hasse Invariant in “Links between Stable elliptic curves and Diophantine equations.”

In the paper "Links between Stable elliptic curves and Diophantine equations" for an elliptic curve $E$ with normal Weierstrass form $$y^2 = x^3 -g_2x -g_3$$ with $g_i \in \mathbb{Z}$ w.l.o.g. Then ...
8
votes
2answers
593 views

Sum of consecutive cubes

I'm investigating when the sum of $n$ consecutive cubes equals a cube, i.e., for which $n$ does $$\sum_{i=0}^{n-1} (k+i)^3 = k^3 + (k+1)^3 + \cdots + (k+n-1)^3 = Y^3 $$ have nontrivial solutions ...
2
votes
0answers
257 views

When an elliptic curve is a quotient of $\mathbb{G}_a$?

I want to know when an elliptic curve $E \rightarrow S$ is a quotient of $\mathbb{G}_a$. When $S$ is an analytic space, there is an exact sequence $$0 \rightarrow R^1 \mathbb{Z}^{\vee} \rightarrow ...
8
votes
1answer
316 views

A reference for $\mathbb{A}^1_R$ being a coarse moduli space of the stack of elliptic curves

Let $R$ be a ring and let $\mathcal{M}_{1, R}$ be the algebraic $R$-stack of elliptic curves (over $R$-schemes as bases). One knows that the coarse moduli space of $\mathcal{M}_{1, R}$ is supposed to ...
1
vote
0answers
50 views

If $E(K)=E(L)$ for an elliptic curve $E$ and an algebraic extension $L/K$, what can we say about $Sel(E/K), Sel(E/L), L/K$?

More generally, when $E(K_m)$ is stable as $m$ increases for an extension equence $K_0<K_1<K_2<\cdots<K_m<\cdots$ ? In the case, is $\mathrm{Sel}(E/K_m)$ stable as $m\rightarrow ...
1
vote
1answer
106 views

Q(\sqrt{-l_0}) satisfies Heegner hypothesis for an Elliptic curve of conductor C implies C is a square

Trying to understand the proof of Corollary 2.3 in the following paper, http://arxiv.org/pdf/1312.3884.pdf I would like to be able to justify that the root number of the quadratic twist ...
0
votes
1answer
180 views

Hyperelliptic curve of genus 2 over R

I know that the points of an elliptic curve over $\mathbb{Q}$, $\mathbb{R}$ or other field $K$ form a group, particularly the most common example to explain the naive way is with this curve ...
2
votes
1answer
145 views

How can one parametrize a real elliptic normal curve such that four points are coplanar iff their parameters sum to zero?

Let $E \subset \mathbb{P}^3_{\mathbb{R}}$ be a real elliptic normal curve with two non-null-homotopic connected components. Is there a parametrization $$ \chi: (\mathbb{R}/\mathbb{Z})\times ...