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, ...

learn more… | top users | synonyms

44
votes
0answers
2k 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 ...
41
votes
0answers
887 views

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, $$ \# ...
21
votes
0answers
2k views

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 ...
20
votes
0answers
825 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 ...
19
votes
0answers
212 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 ...
16
votes
0answers
282 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 ...
15
votes
0answers
417 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 ...
14
votes
0answers
295 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 ...
13
votes
0answers
343 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 ...
12
votes
0answers
568 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 ...
12
votes
0answers
1k views

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. ...
12
votes
0answers
575 views

Endomorphisms rings of elliptic curves and congruences of $j$

Let $p$ be a prime number, $K/\mathbf{Q}_p$ a finite extension, with integers $O_K$, valuation ideal $\mathfrak{p}$, and residue field $k_\mathfrak{p}$. Let $E$ be an elliptic curve over $K$ with good ...
11
votes
0answers
289 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 ...
11
votes
0answers
604 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
votes
0answers
396 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
votes
0answers
246 views

Preimage of torsion points under modular parametrizations

Let $E$ be an elliptic curve defined over $\mathbf{Q}$. It is known that $E$ admits a modular parametrization $\phi : X_0(N) \to E$, where $N$ is the conductor of $E$. We may normalize $\phi$ such ...
8
votes
0answers
139 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
votes
0answers
143 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 ...
8
votes
0answers
327 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 ...
8
votes
0answers
805 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 $(1 + \mathfrak{f} ) \cap \mathcal{O}_K^\times = \{1\}$. ...
7
votes
0answers
347 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 ...
7
votes
0answers
207 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
0answers
237 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 ...
7
votes
0answers
311 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 ...
7
votes
0answers
354 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 ...
7
votes
0answers
679 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 ...
7
votes
0answers
391 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 ...
6
votes
0answers
122 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
0answers
74 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
0answers
235 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. ...
6
votes
0answers
322 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
0answers
217 views

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 ...
6
votes
0answers
449 views

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
votes
0answers
688 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 ...
5
votes
0answers
78 views

Reference request: "effective'' semistable reduction

I am looking for the origin of the following idea: suppose $m$ and $n$ are relatively prime integers $\geq 3$. Let $E$ be an elliptic curve over a number field $K$. Let $L/K$ be a finite extension ...
5
votes
0answers
218 views

An example of a coarse moduli space of elliptic curves

Let $\mathcal{S}$ be the moduli stack (over $\text{Spec }\mathbb{Z}$) of elliptic curves endowed with a trivialisation of its Hodge bundle. Classically, we know that $\mathcal{S}\otimes ...
5
votes
0answers
204 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 ...
5
votes
0answers
174 views

When are Galois representations with open image attached to elliptic curves?

Let $K$ be a number field with absolute Galois group $G_K$. Let $\rho:G_K \rightarrow GL_2(\hat{\mathbb{Z}})$ be a Galois representation such that the image of $\rho$ is open in ...
5
votes
0answers
369 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 ...
5
votes
0answers
257 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 ...
5
votes
0answers
489 views

Elliptic curve with no points in a number field

The following is probably well-known (I'd appreciate a link): for a field $K$ that is a finite extension of the field of rational numbers, give a polynomial $f(x,y) ∈ Q[x,y]$ of the form $y^2 − x^3 ...
5
votes
0answers
422 views

Can the Galois representation on the $p$-adic Tate module of $E/\mathbf{Q}_p$ be recovered from the $p$-divisible group associated to the mod $p$ good reduction of $E$?

Let $p$ be a prime number, and $E/\mathbf{Q}_p$ and elliptic curve with good reduction $\bar E_p$. Can the Galois representation of $\mathbf{Q}_p$ given by the $p$-adic Tate module of $E$ be recovered ...
5
votes
0answers
501 views

Formal groups in the supersingular reduction case

Let $E/\mathbb{Q}$ be an elliptic curve with potential good supersingular reduction at $p$. Thus, there is a finite extension $K/\mathbb{Q}_p$ such that $E/K$ has good supersingular reduction. Let us ...
5
votes
0answers
724 views

Elliptic curves over finite fields and 2x2 matrices

Let $k$ be a finite field of order $p^a$ and characteristic $p$, and $\mathcal{C}$ a $k$ isogeny class of elliptic curves over $k$. Let $w$ be the (Galois conjugacy class of the) Weil $k$-number ...
5
votes
0answers
632 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 ...
4
votes
0answers
121 views

Ranks of elliptic curves over Q(t)

I have an elliptic curve $E/\mathbb{Q}(t)$, and I want to compute its rank. Does knowing the rank over $\mathbb{F}_p(t)$ for some prime of good reduction give a bound on the rank over ...
4
votes
0answers
93 views

Minimal discriminant of an elliptic curve in terms of its Galois representation

From the Galois representation of an elliptic curve $E$ we can read the conductor of $E$, and further some information about the minimal discriminant. So is there any more information about the ...
4
votes
0answers
225 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 ...
4
votes
0answers
328 views

The Modularity Theorem and Serre's/Faltings's Isogeny Theorem

Earlier this year I completed my Masters dissertation on Andrew Wiles's proof of The Modularity Theorem for semistable elliptic curves as a precursor to the accepted proof of Fermat's Last Theorem. ...
4
votes
0answers
106 views

minimal conductors among elliptic curves with a fixed CM type

Let $K$ be a quadratic imaginary field. To simplify my life, let us assume that $K$ has class number one. Consider the following infinite set: $S_1:=$ $\{$ $E\subseteq\mathbf{P}^2(\mathbf{C})$ is an ...