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.

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2 votes
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70 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 ...
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-1 votes
0 answers
86 views

Solutions to an elliptic curve equation involving a prime [migrated]

Let $p$ be a prime and let $x$ be a positive integer. How do I prove that the equation $$y^2=px^3+p$$ has infinitely many integer solutions? I tried to tackle this problem using elliptic curves and ...
1 vote
1 answer
108 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 ...
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3 votes
0 answers
166 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
102 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
109 views

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(...
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3 votes
0 answers
115 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 ...
5 votes
2 answers
243 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 ...
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3 votes
1 answer
128 views

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^...
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-3 votes
0 answers
106 views

Number of points of $x^4+a y^4=y^2 z^2$ over finite fields?

Q1 What are bounds for the number of points of the genus $1$ projective curve $x^4+a y^4=y^2 z^2$,$ a \ne 0$ over $\mathbb{F}_p$? Q2 Why the Hasse-Weil bound fail for the following curves? ...
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1 vote
0 answers
79 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
86 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 $$0\to\mathfrak{a}^{-1}Λ\to\Bbb{C}\to\Ker(E^n\to E^m)\to Λ^n/A^tΛ^m\quad (1)$$ ...
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4 votes
0 answers
174 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
108 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{...
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5 votes
0 answers
119 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
136 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
77 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 $...
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3 votes
1 answer
241 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
136 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 ...
0 votes
1 answer
124 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 $...
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0 votes
1 answer
93 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 ...
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2 votes
1 answer
104 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
125 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 $ψ_E(I)$ ...
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7 votes
0 answers
110 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 ...
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1 vote
0 answers
150 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( \...
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9 votes
1 answer
450 views

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
99 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 ...
  • 365
4 votes
2 answers
249 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, ...
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0 votes
0 answers
116 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^...
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2 votes
0 answers
130 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
251 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 ...
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4 votes
0 answers
179 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
173 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 $ψ$, ...
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4 votes
1 answer
437 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!
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0 votes
0 answers
120 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 ...
2 votes
0 answers
195 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}...
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1 vote
0 answers
71 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
117 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 ...
3 votes
0 answers
77 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 ...
0 votes
0 answers
61 views

Modularity and Galois representations notation

Reading some literature concerning modularity of elliptic curves, more particularly the study of the corresponding Galois representations I sometimes see $$\rho_{E,p} \simeq \rho_{\mathfrak{f},p}$$ ...
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1 vote
0 answers
45 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 ...
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1 vote
0 answers
68 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 ...
4 votes
2 answers
261 views

Reference for universal elliptic curves

I've seen the following sentence come up in a few papers: Consider the modular curve $Y_1(N)$ and let $E$ be the universal elliptic curve over $Y_1(N)$. This comes up in Deligne's construction of ...
0 votes
0 answers
78 views

Counting points on Elliptic Curves with CM by $\mathbb{Q}[\sqrt{-d}]$, $d=1,3$ (CM ring with non-trivial units)

Consider an elliptic curve $E/H$ with CM by the entire ring of integers $\mathcal{O}_K$ of $K=\mathbb{Q}[\sqrt{-d}]$ (and such that $j(E)=j(\mathcal{O}_K)$) such that $H$ is the Hilbert class field of ...
0 votes
0 answers
31 views

Average rank of elliptic curves in an algebraic family parametrized by bihomogeneous forms

Let $\displaystyle q_i(u_1, u_2; v_1, v_2)$ be bihomogeneous polynomials of bidegree $(2,2)$ and $(3,3)$ respectively for $i = 2,3$ (i.e., each $q_i$ is a binary form of degree $i$ in each of the $u$ ...
1 vote
0 answers
106 views

Isomorphism of Brauer groups of curves

I asked this question a few days ago on math.stackexchange with no success, and it doesn't seem like there'll be any. So I thought I'll repost it here. A recent big result proved by $\mathrm{\check{C}}...
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3 votes
1 answer
202 views

Non-modular elliptic curves

Do we have any examples of non-modular elliptic curves over number fields $K \neq \mathbb{Q}$? In particular, I came across a paper by Freitas, Le Hung, and Siksek, "Elliptic curves over real ...
  • 347
2 votes
1 answer
215 views

Torsors over elliptic curves

Let $G$ be a finite abelian etale group scheme over a number field $k$. Let $E$ be an elliptic curve over $k$ and $C := E\backslash \{O\}$ its affine model of the same equation. Recall that for a ...
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1 vote
0 answers
128 views

Embedding of a genus 1 hyperbolic curve

Let $E$ be an elliptic curve over a number field $k$. We define the affine curve $C := E \backslash \{p_1,...,p_n\}$ by removing a finite number of points from $E$. Here, I would like to declare that ...
  • 573
2 votes
0 answers
144 views

Extending the analogy between cyclotomic units and elliptic units

There is a nice analogy between cyclotomic units and elliptic units given as follows: Cyclotomic units are related to special values of the Riemann Zeta function. This is because the logarithmic ...

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