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.
1,509
questions
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Reference request for recurrence relation of division polynomials
The recurrence relations for division polynomials of elliptic curves are well known:
$$\Psi_{2n} = \Psi_n \left( \Psi_{n+2} \Psi_{n-1}^2 - \Psi_{n-2} \Psi_{n+1}^2 \right) / \ 2y$$
$$\Psi_{2n+1} = \...
0
votes
0
answers
165
views
Why Lubin Tate character acts on torsion points of CM elliptic curve implies the group of torsion points is infinite?
Let $F$ be quadratic imaginary field, and $R_F$ be its ring of integers.
Let $E /\Bbb{Q} $ be an elliptic curve which has CM by $F$. Suppose $E$ has good reduction at $P$,which is prime ideal of $R_F$....
4
votes
0
answers
124
views
Weak version of (elliptic analog) Artin's primitive roots conjecture
Let $E/\mathbb{Q}$ be an elliptic curve, and $P\in E(\mathbb{Q})$ be any non-torsion point. Given any $\varepsilon>0,$ how often it is true that $\mathrm{ord}(P \pmod p)>p^{1-\varepsilon},~p~\...
1
vote
0
answers
86
views
Pure, residual, 2-dimensional, semisimple $G_{\mathbb{Q}}$ representations
Let $G_{\mathbb{Q}}$ be the absolute Galois group of the rationals. I want to know how many isomorphism classes of 2-dimensional, irreducible, pure Galois representations are there over a finite field ...
1
vote
0
answers
75
views
Why is the kernel cyclic if and only if the walk does not backtrack?
I'm reading Mathematics of Isogeny Based Cryptography by Luca De Feo. At some point (pg. 32), he says
"A walk of length $e_A$ in the $l_A$-isogeny graph corresponds to a kernel of size $l_A^{e_A};...
5
votes
1
answer
268
views
Transforming the Kondo quintic $5T2$ into the Lehmer quintic $5T1$?
I. Kondo-Brumer quintic
The deceptively simple solvable quintic,
$$x^5 + (a - 3)x^4 + (-a + b + 3)x^3 + (a^2 - a - 1 - 2b)x^2 + b x + a=0$$
is quite important for imaginary quadratic fields. For ...
1
vote
0
answers
121
views
Is there an analog of Weil pairing for modular forms?
Given a newform $f(z)$ (of weight $k$) and a prime $p,$ consider the classical Galois representation
$$\rho_{f,p}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}/p\mathbb{Z}).$...
4
votes
0
answers
202
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Effect of the surjectivity of Galois representation
Let $K$ be any arbitrary number field and $E$ be any elliptic curve over it. For any integer $m,$ consider the well-known Galois representation
$$\rho:\text{Gal}(\overline{K}/K)\to \text{GL}_2(\mathbb{...
2
votes
0
answers
121
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How to compute torsion subgroup $E[24]$ over $\overline{\mathbb{Q}}$
If I have an elliptic curve $E: y^2=x^3-15x+22$ over $\mathbb{Q}$ with CM from the imaginary quadratic field $\mathbb{Q}(\sqrt{-3})$ then how do I compute the $24$-torsion subgroup $E[24]$ over $\...
0
votes
0
answers
79
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Projection map $\pi:\left(\mathcal{O}/n\mathcal{O}\right)^\times \to\left(\mathcal{O}/\gcd(n,m)\mathcal{O}\right)^{\times}$ of a CM elliptic curve
In this paper the author has mentioned in page $693$ under section $2.2$ that for an Elliptic curve $E/\mathbb{Q}$ with CM by an order $\mathcal{O}$ of an imaginary quadratic field $K$ there is a ...
5
votes
2
answers
288
views
Generalization of $j(E) \in \overline { \Bbb{Z}}$ to abelian varieties of arbitrary dimension
Let $E/ \Bbb{C}$ be an elliptic curve which has complex multiplication over a number field $K$.
Then it is widely known that $j(E) \in \overline { \Bbb{Z}}$.
What is the known generalization of this ...
3
votes
0
answers
120
views
Isogeny of elliptic curve over positive characteristic $p$ which does not come from characteristic $0$
Let $K$ be quadratic imaginary field. Let $E$ be an elliptic curve which has CM over $R_K$
($R_K$ is ring of integers of $K$).
According to SIlverman's ''ADvanced topics in the arithmetic of elliptic ...
2
votes
1
answer
153
views
Decomposition of the Galois group of the $m$-th division field of an elliptic curve with CM into a direct product of Galois groups
Let $E/\mathbb{Q}$ be an elliptic curve with CM from an imaginary quadratic field $K$. Let $K(E[m])$ denote $m$-th division field (number field obtained by adjoining the coordinates of the $m$-torsion ...
3
votes
0
answers
231
views
Reverse engineering an elliptic curve from its modular form?
Does there exist an algorithm or something of the sort to reverse-engineer a curve from its modular form (weight two eigenform with complex coefficients)? I am aware that sometimes there isn’t a ...
2
votes
0
answers
132
views
Upper bound for the torsion subgroup of an elliptic curve over arbitrary number fields
Let $K$ be a finite extension of $\mathbb{Q}$ of degree $d$ and let $E(K)$ be an elliptic curve over the field $K$ with coefficients in $K$. Let us fix $d$ and vary over all the possible $K$, in turn ...
1
vote
0
answers
158
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How to link the rank of Elliptic Surfaces and Elliptic Curves over function fields?
I have been investigating certain elliptic surfaces for my research, and when giving a presentation, I was asked why when given an elliptic surface $E$, say given in Weierstrass form
$$E\colon y^2+a_1(...
4
votes
0
answers
140
views
Can you determine the least degree of a morphism between algebraic curves?
I have several questions regarding the degrees of morphisms between algebraic curves.
If we have algebraic curves $X$ and $Y$ defined over some perfect field $k$, can we determine the least degree of ...
6
votes
2
answers
282
views
Does the $p$-adic regulator depend on Weierstrass model?
I am a little confused on the $p$-adic regulator on elliptic curves and what happens when you switch to different Weierstrass models. Restrict to ell. curves over $\mathbb Q$ for simplicity.
From my ...
3
votes
1
answer
164
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Why an isogeny induces a surjection between points over maximal unramified extension?
Let $E$ and $E'$ be elliptic curves over $\mathbb Q$, and let $\phi\colon E\to E'$ be an isogeny defined over $\mathbb Q$. Let $p$ be a prime relatively prime to the degree of $\phi$. Let $\mathbb Q_p^...
1
vote
0
answers
93
views
An elliptic threefold and the Mordell–Weil lattices of its reductions
Let $T\!: y^2 = x^3 + a(t, s)x + b(t, s)$ be an elliptic threefold over a finite field $\mathbb{F}_q$ of characteristic $p > 3$. In other words, we have an elliptic curve over the function field $\...
0
votes
0
answers
98
views
Identity component of $\mathrm{Ker}(E^n→E^m)$ in the advanced topics in the arithmetic of elliptic curves
$\DeclareMathOperator\Ker{Ker}$Silverman's "Advanced topics in the arithmetic of elliptic curves", p.115 reads $$0\to\mathfrak{a}^{-1}Λ\to\Bbb{C}\to\Ker(E^n\to E^m)\to Λ^n/A^tΛ^m\quad (1)$$ ...
4
votes
0
answers
230
views
Lifting the connected-etale sequence of the $p$-torsion of an elliptic curve
Suppose that $R$ is a complete DVR with characteristic 0 fraction field $K$, maximal ideal generated by $p$ and characteristic $p>0$ residue field $k$ which is algebraically closed. Suppose that $\...
2
votes
0
answers
133
views
Lattice relations and isogenous elliptic curves
Consider two (primitive) elements $\pi_{i} \in \mathbb{C}$, such that $\pi_{1} = M \pi_{2}$ for $M \in \mathcal{S}_{m}$ with $$\mathcal{S}_{m}:=\Big\{\begin{pmatrix}
A & B \\
0 & D
\end{...
5
votes
0
answers
226
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$ ...
2
votes
1
answer
171
views
How can I calculate $\wp(αu), α\in \Bbb{C}$, $αL⊆L$
Let $\wp(u) = \frac{1}{u^2} + \sum\limits_{\omega \in L, \omega \neq 0} \left(\frac{1}{(u-\omega)^2} - \frac{1}{\omega^2}\right)$ be a Weierstrass pe function.
My question is, how can I calculate $\wp(...
2
votes
0
answers
97
views
Selmer ranks unbounded?
Is it known if the Selmer ranks of quadratic twist families are unbounded?
Suppose that $E/K$ is an elliptic curve defined over a number field. For each quadratic extension $F/K$ I can form the twist $...
3
votes
1
answer
288
views
Hecke operators on universal elliptic curves
Suppose that $\tau \in \mathbf{H}$ belongs to the complex upper half plane. The quotient $\mathbf{C}/(\mathbf{Z}+\mathbf{Z}\tau)$ gives an elliptic curve over $\mathbf{C}$. Write this elliptic curve ...
1
vote
0
answers
143
views
Elliptic curves whose $2,3,5$-parts of Sha are large
Let $E$ be an elliptic curve, and $\text{Sha}(E)$ its Shafarevich-Tate group which measures the failure of the local-to-global principle for the curve. It is conjectured that $\text{Sha}(E)$ is a ...
1
vote
1
answer
141
views
Power series corresponding to $[a]\in \operatorname{End}(E)$ ($a \in R_K$) can be expressed as $[a](t)=at+\text{(term higher than degree $2$)}$?
Let $K$ be an imaginary quadratic field and $E/K$ be an elliptic curve which has complex multiplication on $K$.
Let $R_K$ be ring of integers of $K$.
Let $ \hat{E}$ be its formal group of $E$.
Take $...
0
votes
1
answer
124
views
Why does $[I](P)=0$ ($P\in E$) imply $[\psi(I)](P)=0$ ? ($\psi$ is Hecke character of elliptic curve)
Let $K$ be a imaginary quadratic field, $R_K$ be ring of integers of $K$, and $E/K$ be elliptic curve which has CM over $K$.
Let $\psi_E$ be Hecke (Grössencharakter) character of $E/K$.
Let fix prime ...
2
votes
1
answer
180
views
Image of Kummer map for CM Elliptic curves
Let $K$ be an imaginary quadratic field and let $F$ be a finite extension of $K$. Let $E$ be an elliptic curve over $F$ with CM by $K$. Suppose that $p$ is a prime that splits as $p=\pi\pi^*$ in $K$. ...
0
votes
0
answers
138
views
Why is image of prime ideal under Hecke (Grossencharacter) character is prime element of the local field?
Let $K$ be a imaginary quadratic field, and $E/K$ be elliptic curve which has CM over $K$.
Let $ψ_E$ be Hecke(Grossencharacter) character of $E/K$.
Let fix prime ideal $I$ of $K$.
Then, why $ψ_E(I)$ ...
7
votes
0
answers
117
views
Upper bound on $\#\{p \leq B :\#E(\mathbb{F}_p) \equiv a \mod b\}$ for large $b$
Let $E$ be a fixed elliptic curve over $\mathbb{Q}$. Is there a good upper bound on $\#\{p \leq B :\#E(\mathbb{F}_p) \equiv a \mod b\}$ when $b$ is large (maybe around $\sqrt{B}$)? I don't mind ...
1
vote
0
answers
171
views
Homomorphism of formal group of elliptic curve corresponding to its endomorphism
Let $E$ be an elliptic curve and $ \hat{E}$ be its formal group.
Rubin's lemma $3.7$ in 'Elliptic curves with complex multiplication' reads
For arbitrary $φ∈End(E)$, there exists unique $φ(t)∈End( \...
10
votes
1
answer
1k
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Universal elliptic curve and the Tate curve
I've seen the following sentence come up a few times in papers:
Let $E$ be the universal elliptic curve over the modular curve $Y_1(N)$. Then the localization of $E$ at any choice of cusp is ...
0
votes
0
answers
117
views
How extension $\Bbb{Q}_p(\hat{E}[p])/\Bbb{Q}_p$ looks like?
Let $E/ \Bbb{Q}_p$ be an elliptic curve over $ \Bbb{Q}_p$. $\hat{E}$ denote the corresponding formal group of $E$.
I want to know what
$\Bbb{Q}_p(\hat{E}[p])/\Bbb{Q}_p$ is .
At first I tried to prove ...
5
votes
2
answers
304
views
Additivity of Elliptic Curve Rank over Compositum of Fields
Assume that BSD holds for number fields. Let $E/\mathbf{Q}$ be an elliptic curve. For simplicity, let's assume it has Mordell-Weil rank zero. Let $F_1/\mathbf{Q}$ and $F_2/\mathbf{Q}$ be finite, ...
0
votes
0
answers
129
views
Proof of $[p](x)≡x^p\operatorname{mod}p \Bbb{Z}_p$ for formal group of elliptic curve
Let $E$ be an elliptic curve over $\Bbb{Q}_p$.
Let $ \hat{E}$ be formal group of $E$.
Let $[p](x)=x+_\hat{E}+・・・+_\hat{E}x$ (add by formal group law $p$ times).
I want to know the proof of $[p](x)≡x^...
2
votes
0
answers
175
views
Relation between division point of elliptic curve and formal group of elliptic curve, $\Bbb{Q}_p(E[p])=\Bbb{Q}_p(\hat{E}[p])$
Let $E/ \Bbb{Q}_p$ be an elliptic curve over $ \Bbb{Q}_p$. $\hat{E}$ denote the corresponding formal group of $E$.
I want to prove
$\Bbb{Q}_p(E[p])=\Bbb{Q}_p(\hat{E}[p])$.
$ \hat{E}[p]$ denotes $p$ ...
1
vote
1
answer
286
views
What's a right parameter space of abelian varieties over a non algebraically closed fields?
Let $k$ be a field of characteristic not 2 or 3. Then the set of elliptic curves over $k$ can be parametrized by the affine variety $S=D(4a^3+27b^2)\subset\mathbb{A}^2_k$ via the family $E\to S$ where ...
4
votes
0
answers
265
views
modularity of elliptic curves over function fields in positive characteristic
Let $F$ be a global function field over a finite field, and $E$ be an elliptic curve over $F$. Much like the number field case, it is natural to study the Galois representation on the Tate module of $...
-2
votes
1
answer
219
views
Special value of Hecke $L$ function
Let $E:y^2=x^3-x/ \Bbb{Q}(i)$ be elliptic curve and $L(E,1)$ be a special value of $L$ function of $E$ at $1$.
Let $L(ψ,1)$ be value at $1$ of Hecke $L$ function with respect to Hecke character $ψ$, ...
4
votes
1
answer
521
views
Are Frey elliptic curves semi-stable?
Are all Frey elliptic curves semi-stable? If so, where exactly is this needed in the modularity approach, now that we know modularity for all rational elliptic curves?
Thank you!
0
votes
0
answers
124
views
Is there a kind of uniqueness of Poincaré duality? [duplicate]
There is a related question. I would like to know if there is a more intrinsical way to show this. I want to know if we can get this through the uniqueness of Poincaré duality or the comparison ...
3
votes
0
answers
216
views
Proof of $L(E,1)/Ω(E)=1/8$ for elliptic curve $E:y^2=x^3-x/ \Bbb{Q}$?
Let
$E:y^2=x^3-x$ be an elliptic curve over $ \Bbb{Q}$ and
$ω_E=dx/2y=dx/2\sqrt{x^3-x}$.
Then
$$
\begin{split}
\Omega(E)&=\int_{E(\Bbb{R})} ω_E\\
\\
&=2\int\limits_1^{+\infty} dx/\sqrt{x^3-x}...
1
vote
0
answers
74
views
How to construct explicitly defining polynomials of an morphism between smooth irreducible curves?
Let $\phi\!: C_1 \to C_2$ be a separable morphism of smooth irreducible curves embedded as projectively normal models by invertible sheaves $\mathcal{L}_1$ and $\mathcal{L}_2$ respectively. Theorem 4....
7
votes
0
answers
174
views
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 ...
4
votes
0
answers
99
views
Reconstructing coefficients of an elliptic curve L-series from the modular form divisor
Let $E$ be an unknown elliptic curve over $\mathbb{Q}$.
Let $L(E, s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ be the L-function of $E$ and write $f(q) = \sum_{n=1}^{\infty} a_n q^n$.
I'm in a setting ...
1
vote
0
answers
53
views
Parametrizations of elliptic curves that "fixes" torsion points
I am not entirely sure this question is research-level question, but I have tried stack exchange and have received no response there.
During a conversation with a professor, I was informed of the ...
1
vote
0
answers
90
views
Iterated integrals on higher dimensional Calabi-Yau manifolds?
I recently read about the construction of closed quasi-periodic differential forms on elliptic curves (1-dim Calabi-Yaus) via the Kronecker-Eisenstein series. I now wonder if similar constructions are ...