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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72
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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 $$ E_d : y^2 = x^3+dx. $$ When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD, $$ \# Ш(E_p)...
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3k views

Constructing non-torsion rational points (over Q) on elliptic curves of rank > 1

Consider an elliptic curve $E$ defined over $\mathbb Q$. Assume that the rank of $E(\mathbb Q)$ is $\geq2$. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank $=$ algebraic ...
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What are the possible singular fibers of an elliptic fibration over a higher dimensional base?

An elliptic fibration is a proper morphism $Y\rightarrow B$ between varieties such that the fiber over a general point of the base $B$ is a smooth curve of genus one. It is often required for the ...
22
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1k views

Fake CM elliptic curves

Suppose one has an elliptic curve $E$ over $\mathbb{Q}$ with conductor $N < k^3$ for some (large) positive $k$, with the property that its Fourier coefficients satisfy $$ a_p=0, \; \mbox{ for all }...
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442 views

Is every positive integer the rank of an elliptic curve over some number field?

For every positive integer $n$, is there some number field $K$ and elliptic curve $E/K$ such that $E(K)$ has rank $n$? It's easy to show that the set of such $n$ is unbounded. But can one show that ...
18
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622 views

Infinite extensions such that every elliptic curve has finite rank

The comments to this answer seem to make the following claim. Claim. Let $K$ be the maximal abelian extension of $\mathbf Q$ that is unramified away from $p$ (more generally, away from a finite set $S$...
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294 views

"High-concept" explanation for proof of a theorem of Ochanine?

See Akhil Mathew's notes on Ochanine's theorem for elliptic genera here and here. Let $\phi: \Omega_{SO} \to \Lambda$ be a genus. We might ask when $\phi$ satisfies the following multiplicative ...
17
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563 views

Trying to reconcile two facts about the Appell-Lerch sum learned from Polishchuk and Zwegers

One of the key characters in the thesis of Zwegers is the modular correction $\tilde\mu(u,v;\tau)=\mu(u,v;\tau)+\frac i2R(u-v;\tau)$ of the Lerch sum $\mu(u,v;\tau)=\frac{e^{\pi i u}}{\vartheta(v;\tau)...
17
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372 views

Elliptic $\infty$-line bundles over $B G$

Theorem 5.2 in Jacob Lurie's "Survey of Elliptic Cohomology" (pdf) states the equivalence of two maps $$ B G \longrightarrow B \mathrm{GL}_1(A) $$ for $A$ an $E_\infty$-ring carrying an oriented ...
17
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469 views

Cohomological characterization of CM curves

In his 1976 classical Annals paper on $p$-adic interpolation, N. Katz uses the fact that if $E_{/K}$ is an elliptic curve with complex multiplications in the quadratic field $F$, up to a suitable ...
16
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348 views

Quadratic non-residues in elliptic divisibility sequences

Let $E: y^2 = x^3 + ax + b$ be an elliptic curve over $\mathbb{Q}$ with $a,b \in \mathbb{Z}$. Recall that any rational point $P = (x,y)$ can be written uniquely as $P = (u/d^2, v/d^3)$ with $u,v,d \in ...
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534 views

For how many primes does an elliptic curve over a totally imaginary field have supersingular reduction?

An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a ...
14
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836 views

Recent developments in the proof of Fermat's last theorem

I posted on Mathematics Stack Exchange, but was encouraged to post on MathOverFlow instead. It has been 20 years since Fermat's last theorem was proved by Andrew Wiles. Has there been any ...
14
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Explicit example of elliptic curve of the kind needed for IUTT

At the nLab, we are currently trying to illustrate the definition of initial theta-data in Mochizuki's first IUTT paper by means of an explicit example. The exposition should end up at the following ...
14
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482 views

Am I missing something about this notion of Mirror Symmetry for abelian varieties?

This is a continuation of my recent question: Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s. In the comments of the question, I was directed to the paper http://arxiv.org/abs/...
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481 views

Case D=4l in Elkies' paper on Supersingular Primes of an Elliptic Curve over $\mathbb{Q}$

My question is regarding Elkies' paper on "The existence of infinitely many supersingular primes for every elliptic curve over $\mathbb{Q}$". In the section "Nuts and Bolts", Elkies has the ...
12
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299 views

Artin representations appearing in Mordell-Weil groups of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbf{Q}$, and let $K$ be a Galois number field. The Galois group $G=\mathrm{Gal}(K/\mathbf{Q})$ acts on the Mordell-Weil group $E(K)$ and thus on the ...
12
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683 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 ...
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Why doesn't functoriality immediately imply the modularity theorem?

Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. ...
11
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331 views

Record analytic rank for an elliptic curve?

What is the current record (and reference) for the highest analytic rank of an elliptic curve over $\mathbb{Q}$? The highest algebraic rank is the Elkies curve with rank at least 28, but I cannot ...
11
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286 views

Non-vanishing modular forms

Prompted by this MO question, I have the following question about modular forms which do not vanish on the upper-half plane. Q1. Let $N \geq 1$ be an integer and let $\Gamma(N)$ be the principal ...
11
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906 views

Isogenies between supersingular elliptic curves

Suppose we are given two non-isomorphic supersingular elliptic curves $C$ and $C'$ (in characteristic $p$). Is there an isogeny $C\to C'$ of a given degree (say, power of a prime $l$ different from ...
10
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237 views

Integral points on elliptic curve and the Lee norm

This question is based on small experiments I have done in Sagemath and if it is not research level, I will move it to MSE: Let $E$ be an elliptic curve defined with coefficients in $\mathbb{Z}$. The ...
10
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173 views

Representation of the space of lattices in $\Bbb R^n$

The space of 2D lattices in $\Bbb R^2$ can be represented with the two Eisenstein series $G_4$ and $G_6$. Each lattice uniquely maps to a point in $\Bbb C^2$ using these two invariants, and the points ...
10
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430 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 $V/\mathbb{...
10
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460 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 ...
9
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411 views

Kihara-like Z/6Z elliptic curve families

Shoichi Kihara constructed a family of elliptic curves with Mordell–Weil group $\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}^3$ (generic rank at least 3) in 2006. Kihara's family produces a number of rank 8 ...
9
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236 views

Good reduction of finite etale covers of abelian varieties

Let $R$ be a dvr (whose residue characteristic is zero if it helps) with fraction field $K$. Let $A$ be an abelian variety over $K$ with good reduction over $R$. Let $X\to A$ be a finite etale ...
9
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601 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 ...
8
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425 views

Elkies' theorem on supersingular primes and inertness

Suppose $E_{/\mathbb{Q}}$ is an elliptic curve over $\mathbb{Q}$ without CM. By Elkies' theorem, there exist infinitely many primes $p$ for which $E$ has supersingular reduction at $p$. Question. Is ...
8
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372 views

Rational points of a "famous" elliptic curve

The following problem has already been discussed in mathoverflow, for example here It is essentially an elliptic curve of rank $1$, a generator of the Mordell-Weil group being the point $Q = (\frac{4}...
8
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140 views

Topological modular forms on cuspidal elliptic curves

If you take the moduli stack of smooth elliptic curves, there is a sheaf of $E_{\infty}$-rings on it whose global sections are isomorphic to $TMF$. This sheaf extends to the moduli stack of smooth/...
8
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0answers
426 views

What are the smallest positive $a,b,c$ for which $a/(b+c)+b/(a+c)+c/(a+b)$ is an integer $>2$?

The problem of finding the smallest positive $a,b,c$ for which $a/(b+c)+b/(a+c)+c/(a+b)=4$ turns out to be surprisingly difficult, and has made the rounds on the internet and social media, and Andrew ...
8
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0answers
144 views

What is a geometric construction corresponding to elliptic curve addition for Sharygin-isosceles triangles?

NB: this is a cross-posting from from MSE after two months with no progress (despite a bounty). It's totally elementary but I think it's cute. Consider the elliptic curve defined by the cubic: $$ a^...
8
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0answers
300 views

Elliptic curves and the $\ell$-adic image of the decomposition group

Let $E$ be an elliptic curve over $\mathbb{Q}_\ell$ and consider the image of the $\ell$-adic representation. Is there a description of this image similar to Serre's description of the image of the ...
8
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525 views

A property of supersingular $j$-invariants (reference request)

Edit 2: For those who understandably don't want to read such a long post, I think Voloch's suggestion reduces the problem to asking whether $j$-invariants of supersingular curves are 3rd powers in $\...
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520 views

Tameness criterion in the reducible case

Dear MO, This is a follow up to a previous question here in MO, but I will make this question self-contained for convenience. Those already familiar with the following paper [G] by Gross can safely ...
8
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947 views

Torsion points of CM elliptic curves

Let $K$ be an imaginary quadratic field, and $\mathfrak{f}$ an integral ideal of $K$ which is stable under complex conjugation. Assume that $(1 + \mathfrak{f} ) \cap \mathcal{O}_K^\times = \{1\}$. ...
8
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0answers
802 views

Semistable Elliptic Curves and irreducible Galois representations

I am interested in the set of number fields $K$ having the property that for \emph{any semistable} elliptic curve $E$ defined over $K$, there exists a constant $c(E,K)>0$ such that $$p>c(E,K)\...
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474 views

Binary quadratic forms attached to supersingular elliptic curves over F_p?

The question that I have is a more precise version of an earlier one (1), posted by myself on MO a little bit ago. Sorry for repeating myself. Let $p$ be prime number which for simplicity shall be ...
7
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0answers
141 views

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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0answers
199 views

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
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0answers
283 views

Stacky proof of no elliptic curves over Z

It is a well known result that there are no Elliptic curves over the integers with every where good reduction. In fact this is even true for abelian varieties (and hence higher genus curves) but let ...
7
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0answers
168 views

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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0answers
310 views

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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0answers
274 views

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
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0answers
140 views

Looking for an elliptic curve E st ${\large Ш}(\mathbb Q,E)$ cont. an element of order $p^2$ and certain other properties

I am looking for an elliptic curve $E$ with Weierstraß coefficients in $\mathbb{Q} $ so that for some prime $p$ the following conditions are satisfied: (1) ${\large Ш}_{p^{\infty}}(\mathbb{Q},E)$ ...
7
votes
0answers
313 views

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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0answers
642 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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0answers
331 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 ...

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