Questions tagged [elliptic-curves]
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. Note also the tag arithmetic-geometry as well as some related tags such as rational-points, abelian-varieties, heights. Please do not use this tag for questions related to ellipses; instead use conic-sections.
487
questions with no upvoted or accepted answers
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Rank 1 curves with prime conductor have trivial torsion. Why?
In the LMFDB database, there are 337912 elliptic curves over $\mathbb{Q}$ for which the rank is 1 and the conductor is a prime number.
All of these curves have trivial torsion group.
Is there a known ...
7
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0
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213
views
Counting elliptic curves over finite fields with a prescribed number of points
Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$, and let $\mathcal{O}$ be an order in $K$ of discriminant $D$ and class number $h(\mathcal{O})$. Then the Hurwitz-...
7
votes
0
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230
views
Magic hourglass of squares hyperelliptic equation
I have been looking into the problem of the magic square of squares, or more specifically, the magic hourglass of squares, like so:
$a^2$ $b^2$ $c^2$
$ $ $ $ $ $ $ $ $ $ $d^2$
$e^2$ $f^2$ $g^2$
...
7
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161
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Simultaneous reductions of elliptic curves: same number of points $|E(\Bbb F_p)| = |E'(\Bbb F_p)|$ for some prime $p$?
$
\newcommand{\End}{\mathrm{End}}
\newcommand{\Gal}{\mathrm{Gal}}
\newcommand{\kb}{\overline{k}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\Q}{\mathbb{Q}}
$
Let $E,E'$ be ...
7
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223
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Freys elliptic curves and Hilbert spaces?
Consider the Frey-Hellegouarch curve given $a,b$ positive rational numbers:
$$y^2= x\left(x-\frac{a}{\gcd(a,b)}\right)\left(x+\frac{b}{\gcd(a,b)}\right)$$
The j-invariant is given by:
$$j(a,b) = \frac{...
7
votes
1
answer
438
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Discriminant of elliptic curve (Frey-Hellegouarch), j-invariant and positive definite kernels, similarities?
Consider the Frey-Hellegouarch curve given $a,b$ natural numbers:
$$y^3= x(x-\frac{a}{\gcd(a,b)})(x+\frac{b}{\gcd(a,b)})$$
Then the discriminant is given by $\Delta = \Delta(a,b) = 16 \left(\frac{ab(a+...
7
votes
0
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175
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Meaning of Elliptic Irregular Primes
The Bernoulli numbers are defined by the equation
$$
\frac{t}{e^t-1}=\sum_k b_k \frac{t^k}{k!}.
$$
A prime number $p$ is irregular if it divides the numerator of one of the even Bernoulli numbers up ...
7
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451
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What is the right basis of solutions of the Picard-Fuchs equation of the Legendre family around 0?
I have been trying to reconstruct some elliptic curves theory computationally and have gotten stuck on some period computations.
Specifically, let $$E_\lambda:\ y^2=x(x-1)(x-\lambda)$$ be the ...
7
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0
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161
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$\lambda$-invariants in cyclotomic $\mathbb{Z}_p$ extensions
The idea that Selmer groups and class groups are related is not new. More recently, we understand that the growth patterns of fine Selmer groups are very similar to that of class groups in cyclotomic $...
7
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413
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A mysterious number related to Hasse-Weil L-function of elliptic curve
Let $E/K$ be a non-isotrivial elliptic curve over a function field $K$ of characteristic $p$, with field of constant $F_q$, with semistable reduction. Its Hasse-Weil L-function $L(s)$ is a polynomial ...
7
votes
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482
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Structure of elliptic curve $y^2 = x^3 - x$ over $\mathbb{F}_p$ with $p=(a+i)(a-i)$
I hope this question is good enough for this network.
I am trying to compute the group structure as the title says of $E:y^2=x^3 - x$ over $\mathbb{F}_p$ with $p\equiv 1\bmod 8$ and $p-1$ a square, ...
7
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0
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701
views
The role of Honda-Tate theory in (Scholze's refinement of) the Langlands-Kottwitz method?
I was wondering if somebody could give me a quick primer on Honda-Tate theory, specifically the number of isogeny classes of elliptic curves, and how specifically it's utilized in (Scholze's ...
7
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0
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434
views
Hilbert scheme of projectively normal elliptic curves
Consider the Hilbert scheme of degree $n$, genus $1$ curves in $\mathbb P^{n-1}$. It contains the locus of smooth curves embedded by the complete linear system of a degree $n$ divisor. Let $X_n$ be ...
7
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0
answers
722
views
On discriminants of elliptic curves
Let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$ and $\Delta_E$ denote the discriminant of $E$. We say an elliptic curve has entanglement fields if the intersection of the $m_1$ and $m_2$ ...
7
votes
0
answers
264
views
Invariant obstructions to gluing Galois representations on elliptic curves
Let $E$ be an elliptic curve over $\overline{\mathbb F_p}$, or another separably closed field of characteristic $p$. Let $K$ be the function field of $E$, and let $K_p$ be the local field at a point.
...
7
votes
0
answers
664
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 ...
7
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502
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$a^5+b^5=c^5+d^5$ and polynomial identities
No nontrivial integer solutions to $$ a^5+b^5=c^5+d^5 \qquad (1)$$ are known.
(1) has infinitely many solutions in an extension of $\mathbb{Z}$
(root of $9-15x+37x^2 $ ) resulting
from a genus 0 ...
7
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0
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264
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Hilbert space from the Tate pairing
Fix an elliptic curve $E$ over ${\Bbb Q}$ (or if you prefer, something more general over something more general). For each extension $F$ of ${\Bbb Q}$, the Néron-Tate height pairing gives
an inner ...
7
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503
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Constructing large rank elliptic curves by multiplying quadratic imaginaries by cubes so that all have same imaginary part
I've been thinking of the relation between elliptic curves of large rank and quadratic imaginary fields with large 3-rank class groups. There are quite a few papers constructing infinitely many ...
6
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195
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Ranks of elliptic curves over cubic fields
We are writing a paper on the ranks of elliptic curves over cubic fields. The curves of different torsion subgroups are created by the formulas in Jeon et al. and by our new parametrizations.
D. Jeon,...
6
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140
views
Fourier transform and Hodge-$*$ operator
Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says
$$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$
...
6
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0
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151
views
Explicit computations of Serre duality for elliptic curves
I have an elliptic curve $E$ defined over a ring $R$, I want to compute the pairing
$$
H^1(E,\mathcal{O}_E)\times H^0(E, \Omega_E^1){\rightarrow}R.
$$
Clearly we have that $H^0(E, \Omega_E^1)=R \...
6
votes
0
answers
523
views
Mistake in Silverman's book in proof of Neron-Ogg-Shafarevich criterion?
In Silverman's The arithmetic of elliptic curves, p. 201, theorem $7.1$ (Criterion of Neron-Ogg-Shafarevich),
he applies the theorem "When $K$ is complete with respect to it's discrete value, ...
6
votes
0
answers
226
views
Modularity switching for primes $p>7$
In Freitas, Le Hung, and Siksek's 2014 paper proving that elliptic curves over totally real quadratic fields are modular they prove (and use) the following result (Theorems 3 and 4): let $\bar \rho_{E,...
6
votes
0
answers
202
views
Concerning the omnipresence of hyper-elliptic curves in the construction of examples
Vague rambling: I hate asking these types of questions, but I feel that I would benefit immensely from hearing some discussion of the use of hyperelliptic curves in constructing certain examples. What ...
6
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answers
159
views
What are the genus 4 curves with Jacobians that are 4-th powers?
Consider the moduli space of all genus $4$ curves $\overline{\mathscr M_4}$ of dimension $3\times 4 - 3 = 9$. Under the Torelli map, there is a map to $\overline{\mathscr A_4}$ (which has dimension $...
6
votes
0
answers
224
views
Existence of elliptic curve with only one rational point over global fields
Let $K$ be a global field, does there always exist an elliptic curve $(E,e)$ over $K$, such that $E(K)=\{e\}$? (Is there an explicit way to find such a curve?)
6
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0
answers
92
views
Elliptic deformation of the second Chern class
Second Chern class
$$c_2 \in H^4(BGL,\mathbb{Q}(2))$$
admits a nice presentation using dilogarithm. The five term relation in this setting becomes a cocycle condition (details can be found here). ...
6
votes
0
answers
324
views
Geometric interpretation of j-invariants of supersingular elliptic curves
In the classical theory of Complex Multiplication, one considers elliptic curves with an endomorphism ring larger than the integers. In this theory, it's possible to determine the j-invariants of all ...
6
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328
views
How to decide whether the isogeny between Neron models is etale?
Let there be an isogeny $f:A_1 \rightarrow A_2$ between two abelian varieties over a $p$-adic field $F$ and assume $f$ has degree $p^n$. By the universal property we get a moprhism $f_0: \mathcal{A}_1 ...
6
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206
views
Can the base of an elliptically fibered Calabi-Yau threefold be an Enriques surface?
For this question, a Calabi-Yau manifold or variety of dimension $n$ is defined as a non-singular projective variety with trivial canonical bundle and $h^{i,0} = 0$ unless $i = 0$ or $i = n$.
If ...
6
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0
answers
153
views
Descent via an explicit isogeny (genus 2)
This question is related to a previous question posted by me here answered by Prof. M. Stoll. 5-Descent or ($\sqrt{5}$-Descent?) on certain genus 2 Jacobians.
Here I ask some technicalities of a ...
6
votes
0
answers
350
views
What happens to Neron-Ogg-Shfarevich when characteristic of the residue field equals the prime at which Tate module is considered?
Neron-Ogg-Shafarevich criterion states that an elliptic curve $E$ over a local field $K$ has a good reduction if and only if the Tate module $T_{\ell}(E)$ is unramified for some prime $\ell$ which ...
6
votes
0
answers
140
views
Modular forms for non-arithmetic subgroups
Modular forms are usually considered as complex-valued functions on the upper half-plane quite invariant by a discrete subgroup $\Gamma$ of isometries and satisfying smoothness and growth condition.
...
6
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0
answers
389
views
Kisin module for CM elliptic curve
Let $E$ be a CM elliptic curve with CM by the field $K$ and assume that $p$ is ramified in $K$ so that $\pi^2 = p \in \mathcal{O}_K$. In particular, then $E$ has supersingular reduction at $p$ and by ...
6
votes
0
answers
410
views
Field of definition of a point in $[p]^{-1}E(K)$
Let $E$ be an ordinary elliptic curve defined over a non-perfect field $K$ of characteristic $p$. If $P \in E(K)$ satisfies $P \not\in [p]E(K)$, is it true that its $p^m$-division points of $P$ are ...
6
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268
views
Is there a prime degree endomorphism on supersingular elliptic curves?
Let $E$ be a supersingular elliptic curve which is defined over field $\mathbb{F}_{p^2}$ and $l$ be a prime such that $gcd(l,p)=1$.
Is there an endomorphism $\phi\in End(E)$ such that $deg(\phi)=l$?
...
6
votes
0
answers
98
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 ...
6
votes
0
answers
635
views
Isogenous elliptic curves have same conductor
Let $E/K, E'/K$ be elliptic curves defined over a number field $K$. Let $\phi: E \to E'$ be a non-constant isogeny defined over $K$. Why must the conductors be equal?
I know that this is an ...
6
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0
answers
465
views
elliptic curves over function fields
Let $K$ be a number field, $E$ an elliptic curve over $K$, and $p$ an odd prime. If $v$ is a place of $K$, we know by the Kummer injection that $E(K_v)/pE(K_v) \hookrightarrow H^1 (K_v, E[p])$ for ...
6
votes
0
answers
919
views
Can you get Siegel's theorem "for free" from modularity and Mazur's Eisenstein Ideal paper?
There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction outside $S$ is ...
6
votes
0
answers
933
views
Curious propositon in "Les schemas de modules de courbes elliptiques"
Currently I am reading "Les schemas de modules de courbes elliptiques" by Deligne and Rapoport and I got myself seriously confused about the following proposition (in English translation):
(II ...
5
votes
0
answers
147
views
Generalization of Deuring's theorem
Deuring has the following important theorem on elliptic curves over a finite field: Let $p$ be a prime at least $5$, and $N=p+1-a$ an integer with $|a|\leq 2\sqrt{p}$, then the number of elliptic ...
5
votes
0
answers
223
views
The p^n torsion of a supersingular elliptic curve
Let k be an algebraically closed field of characteristic $p$ and $E/k$ a supersingular elliptic curve.
It is well known that $E[p]$ is the unique autodual local group $I_{1,1}$ of Lie dimension $1$ ...
5
votes
0
answers
259
views
Torsion points of an elliptic curve over number fields. Another proof of Silverman AEC theorem
I am studying the following theorem from Silverman's AEC:
I am wondering whether there exists another proof that doesn't make use of formal groups and is still valid for a number field $K$. Could you ...
5
votes
0
answers
203
views
Elliptic curves and localizations at various primes
Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime at which $E$ has good reduction. Let $D=D_{E,p}$ be the $p$-torsion in the cokernel of the map $E(\mathbb{Q})\otimes\mathbb{Z}_p\...
5
votes
0
answers
206
views
Making Virasoro uniformization explicit for elliptic curves
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Lie{Lie}\DeclareMathOperator\Der{Der}\newcommand\el{\mathrm{ell}}$Let $\mathcal{M}_{g,1^{\infty}}$ denote the space ...
5
votes
0
answers
290
views
2-descent on elliptic curves, and units modulo squares of units
Setup: Let $p$ be a prime, let $f(x) \in \mathbb{Q}_p[x]$ be a separable monic cubic polynomial cutting out the maximal order $\mathcal{O}_{K_f}$ in the etale algebra $K_f := \mathbb{Q}_p[x]/(f(x))$, ...
5
votes
0
answers
247
views
Goldfeld resolution of the quadratic class number problem
Goldfeld proved the following result. Let $E$ be an elliptic curve (with conductor $N$) over $\mathbb{Q}$ whose Hasse-Weil L-function has a zero at $s = 1$ with multiplicity $g$ then for sufficiently ...
5
votes
0
answers
184
views
Given a point $P$ on a genus-$1$ curve over $\mathbb{Q}_p$, is there an $R$ such that $2 \mid [P - R]$ and $x(\overline{P}) \neq x(\overline{R})$?
Let $p \in \mathbb{Z}$ be prime, and let $f \in \mathbb{Z}_p[x]$ be a quartic polynomial with nonzero discriminant. Let $C/\mathbb{Q}_p$ be the genus-$1$ curve with affine equation $y^2 = f(x)$. Let $\...