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, ...

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2k views

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

Consider an elliptic curve E defined over Q. Assume that the rank of E(Q) is >=2. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank = algebraic rank.) How do you construct ...
29
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556 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 $$ E_d \colon y^2 = x^3+dx. $$ When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD, $$ \# ...
19
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1k 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 ...
16
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732 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 ...
15
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218 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
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396 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 ...
12
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255 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
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531 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
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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. ...
10
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374 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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233 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
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286 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
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0answers
472 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 ...
8
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749 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
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241 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
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309 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
578 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
325 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
211 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
294 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
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0answers
213 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
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0answers
426 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
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0answers
635 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
153 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
205 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
469 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
379 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
414 views

Formal groups in the supersingular reduction case

Dear MO, 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 ...
5
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638 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 ...
4
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221 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
85 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 ...
4
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0answers
288 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 ...
4
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0answers
187 views

Evaluation of $E_{\ell,2}$ on supersingular curves over $\mathbb{F}_{p^2}$

As mentioned in an answer to Modularity of $E_2$ on congruence subgroups, there exist modular forms $E_{\ell,2}$ of level $\Gamma_{0}(\ell)$ and weight 2, with $q$-expansion ...
4
votes
0answers
254 views

Abelian cubic extensions of Q[i],

Recently I was considering cubic extensions $K/Q$ that have discriminant negative of a perfect square. Classifying these curves reduces to solving a Diophatine equation of the form $4a^3+27b^2=c^2$ ...
4
votes
0answers
279 views

What is the status of the equidistribution root numbers of elliptic curves' L-functions

In Section 7 of Alice Silverberg's Rank "Cheat Sheet", Silverberg stated The Bhargava Conjecture: For each $n > > 1$ the average size of $S_{n}(E/\mathbb{Q})$ is ...
4
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0answers
221 views

A natural mod-n representation of the automorphisms of an elliptic curve

Let $E$ be an elliptic curve over a field of characteristic $p$, and let $n$ be an integer coprime to $p$. Then $E[n]$, the kernel of multiplication by $n$ on $E$, is (etale-locally) isomorphic to ...
4
votes
0answers
578 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 ...
3
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0answers
148 views

Fourier expansions of newforms at width-1 cusps

Let $f_E$ be the newform attached to the Elliptic Curve $E$ with cremona label $\textbf{100a1}$ and let $\alpha = \left[\begin{matrix} 1&0 \\ 10&1 \end{matrix}\right] \in SL_2(\mathbb{Z})$. ...
3
votes
0answers
89 views

Lang's height conjecture over $\mathbb{F}_q(T)$?

Is the canonical height of a non-torsion $\mathbb{F}_q(T)$-rational point of an elliptic curve over $\mathbb{F}_q(T)$ known or supposed to be bounded from below by an absolute positive number (or ...
3
votes
0answers
118 views

Elliptic curves and quasi-self-reciprocal polynomials

I am reading Shoichi Kihara's On the rank of the elliptic curve $y^2=x^3+k, II$ [Proc. Japan Acad. Ser. A Math. Sci. Volume 72, Number 10 (1996), 228-229] which is available here ...
3
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0answers
281 views

Birch/Swinnerton-Dyer “Notes on Elliptic Curves II”

I would like to know if any of you know if there is a more general treatment to what Birch and Swinnerton-Dyer did in "Notes on Elliptic Curves II" (http://www.ams.org/mathscinet-getitem?mr=179168). ...
3
votes
0answers
139 views

Is Hasse-witt map isomorphism?

Fix a level $N \geq 3$ and denote by $\Gamma(N)\subset SL(2,\mathbb{Z})$ the subgroup of matrices which are congruent to the identity modulo $N$. The open modular curve $Y(N)$ corresponding to ...
3
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0answers
119 views

P-adic Weierstrass Lemma for several variables

The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in the ring of power ...
3
votes
0answers
284 views

Rational points and Tesla cards

I'm rapidly approaching 300,000 curves in my ongoing search for Mordell curves of rank >=8. Currently I'm finding that I have a bottleneck in the code that locates rational points on these curves. ...
3
votes
0answers
99 views

Extending cohomology classes to compactifications of Kuga varieties

I am trying to understand the proof of lemma 3 in the paper "Algebraic cycles and the Hodge structure of a Kuga fiber variety" by B. Brent Gordon, available at ...
3
votes
0answers
317 views

Homomorphism ring of two elliptic curves with the same supersingular reduction.

$\textbf{Motivation:}$ I am studying the article "The existence of infinitely many supersingular primes for every elliptic curve over $\mathbb{Q}$", by Noam Elkies, and there is a part that I do not ...
3
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0answers
172 views

Computing Elliptic Curves of Conductor Divisible by a Large Prime Factor

A little while ago, I came across a paper (or slides from a talk or something) that seemed to suggest that the modular symbol method for computing elliptic curves over $\mathbf{Q}$ of prescribed ...
3
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0answers
337 views

Cyclotomic fields and singular moduli

Let $\mu$ be the roots of unity and $S$ be the image under the modular $j$-function of all imaginary quadratic $\tau$. Then what is $\mathbb{Q}(\mu)\cap\mathbb{Q}(S)$?
3
votes
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202 views

Find the canonical subgroup of a CM curve with ordinary reduction!

Let $p$ be a prime and $K$ a quadratic imaginary field in which $p$ splits. Let $\mathcal{O}$ be an order in $K$ and $A$ an elliptic curve with CM by $\mathcal{O}$. Then $A$ can be defined over the ...
3
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0answers
275 views

Rank of Subgroup of Elliptic Curve

I'm currently looking at two rational points $ p, q $ on an elliptic curve $E$ over $ \mathbb{Q} $. SAGE tells me that $E$ has rank 5 and no torsion, and that $p$ and $q$ both have infinite order. ...