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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Does the property (P) holds true for the derivatives of $L$?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. As $s$ takes on real negative values, there are trivial zeros ...
China-Hong Kong's user avatar
5 votes
0 answers
221 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]. In his paper Kihara considers the $1$-...
Jesper Petersen's user avatar
7 votes
0 answers
664 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 ...
Hugo Chapdelaine's user avatar
9 votes
2 answers
2k views

Supersingular elliptic curves over $\mathbb{Q}$

what are the examples of elliptic curves defined over $\mathbb{Q}$ with supersingular reduction at a prime $p$ and having a $p$-isogeny over $\mathbb{Q}$ ?
Suman's user avatar
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Visualizing singular points of real loci of elliptic curves

On one hand the real locus of a complex elliptic curve is the intersection of a plane with a torus (i.e. a torus embedded in $\mathbb{C}^2$ plus infinity). And an elliptic curve has no cusps or self-...
Colin McLarty's user avatar
7 votes
2 answers
862 views

BSD conjecture for X_0(17)

I use Magma to calculate the L-value, yields E:=EllipticCurve([1, -1, 1, -1, 0]); E; Evaluate(LSeries(E),1),RealPeriod(E),Evaluate(LSeries(E),1)/RealPeriod(E); Elliptic Curve defined by y^2 + x*y + y =...
Carl's user avatar
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1 vote
2 answers
461 views

Equations of elliptic curves

First part of question I have asked on mathoverflow already: https://math.stackexchange.com/questions/467088/explict-form-of-the-equation-of-elliptic-curve 1) Let $E(\mathbb{F}_{q^2})$ is elliptic ...
Alexey Milovanov's user avatar
4 votes
0 answers
242 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 $E_{\ell,2}(q)=E_{2}(q)-\...
DCT's user avatar
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2 votes
1 answer
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Question on x coordinates of Mordell Curves where $y^2=x^3+k$ and $k^2 = 1$ mod $24$

In my ongoing search for Mordell curves of rank 8 and above I have currently identified 144,499 curves of a type where $k$ is squarefree and $k^2 = 1$ mod $24$. In each case the x coordinates are ...
Kevin Acres's user avatar
10 votes
3 answers
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Why is the gcd so large in an identity related to the $abc$ conjecture?

Consider the identity $$ (x+z)^5+(y-z)^5 = (-3 x + 4 y)^2 (x + y)^3 + (x+y) f(x,y,z) $$ Where $f(x,y,z)=(-8*x^4 + 5*x^3*y + 24*x^2*y^2 - 9*x*y^3 - 15*y^4 + 5*x^3*z - 5*x^2*y*z + 5*x*y^2*z - 5*y^3*z + ...
joro's user avatar
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2 votes
2 answers
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What is the exact meaning of the real period in the $p$-adic formulation of BSD?

Let $E$ be an elliptic curve over $\mathbf{Q}$ which has split multiplicative reduction at $p$ (a prime). If one chooses a global Neron model of $E$ over $\mathbf{Z}$ (unique up to unique isomorphism ...
Hugo Chapdelaine's user avatar
3 votes
1 answer
569 views

Is it expected that every natural number is the rank of some elliptic curve over the rationals?

It is a well-known problem on the theory of elliptic curves that the rank of an elliptic curve (the number of generators of the free part of the Mordell group of the elliptic curve) cannot be ...
Stanley Yao Xiao's user avatar
6 votes
2 answers
346 views

Mordell-Weil of an elliptic surface after adjoining a nontorsion section: as small as possible?

Let $k$ be an algebraically closed field of characteristic $0$, let $C_{/k}$ be a nice (smooth, projective, geometrically integral curve), let $K = k(C)$, and let $\overline{K}$ be an algebraic ...
Pete L. Clark's user avatar
3 votes
0 answers
676 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). ...
BlackAdder's user avatar
3 votes
1 answer
406 views

Division Field of a nonCM elliptic curve

This might be a ridiculous question, but please bear with me. Let $E$ be an elliptic curve over a $p$-adic field $K$. Denote by $K(E_{p^∞}):=\bigcup_{n∈Z≥1} K(E[p^n])$ the field extension obtained by ...
Octobris's user avatar
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5 votes
1 answer
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Hecke $L$-series exercise in Silverman's Advanced Topics in Arithmetic of EC

This has been posted on SE, but I haven't gotten a reply, so I thought I'll try my luck here. I would like to refer you to 2.30 & 2.32 in Silverman's book Advanced Topics in the Arithmetic of ...
BlackAdder's user avatar
4 votes
1 answer
326 views

Elliptic Curves isogenous only over an extension?

Let $l$ be a prime $\geq 5$. Does there exist a pair $E,E'$ of elliptic curves, both defined over the same number field $K$, which are not $l$-isogenous over $K$, but are $l$-isogenous over a ...
Barinder Banwait's user avatar
8 votes
0 answers
581 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 $\...
DCT's user avatar
  • 1,537
4 votes
1 answer
160 views

A certain property of elliptic curves in a paper by Rees

In the paper "On a problem of Zariski", David Rees presents a counterexample to the following problem of Zariski. Let $F/k$ be a f.g. field extension, $S$ a f.g. normal integral domain over $k$ ...
InvisiblePanda's user avatar
3 votes
1 answer
501 views

a question on CM elliptic curves

Let $E$ be an elliptic curve over $\overline{\mathbb{Q}}$ defined by an equation $y^2=4x^3-g_2x-g_3$ and let $\omega=\int_\gamma \frac{dx}{y}$ be the integral of the regular differential form $\...
user36475's user avatar
3 votes
1 answer
660 views

Hecke Character vs Grossencharakter

I would like to know if there is any difference between (1) an algebraic Hecke character (2) a Hecke character (3) a Grössencharakter All of the above in the setting of ellitpic curves with complex ...
BlackAdder's user avatar
7 votes
1 answer
716 views

Canonical differential on Tate curve

I am starting studying the theory of (algebraic) modular forms, and I have some trouble in understanding completely the construction of the Tate curve. My problem is the following: as far as I know ...
user36371's user avatar
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4 votes
1 answer
686 views

Rational points on $X_0(15)$

The modular curve $X_0(15)$ has a canonical model over $\mathbf{Q}$, and it has genus $1$. As the cusp $\infty$ is rational, it is an elliptic curve. Roughly, my question is whether we can find all ...
Bertie Wooster's user avatar
3 votes
1 answer
987 views

Hecke Characters

My question today concerns Hecke characters or Größencharaktere. I'm doing a project on complex multiplication of elliptic curves and need some help understanding Hecke characters. My main ...
BlackAdder's user avatar
19 votes
1 answer
882 views

What is $Aut(Ell)$?

Consider the stack $Ell$ (of groupoids) of elliptic curves. I'm interested in the autoequivalence 2-group of $Ell$, the objects of which consists of transformations $Ell \Rightarrow Ell: Ring \to Gpd$ ...
David Roberts's user avatar
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4 votes
1 answer
720 views

Abelian image of l-adic representation

For a number field F, let E/F be an elliptic curve with CM by a quadratic field K. Let $\rho_\ell: \text{Gal}_F \to \text{Aut}(T_{\ell}E)$ be the $\ell$-adic representation associated to E for some ...
abourdon's user avatar
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3 votes
0 answers
292 views

What does Hodge theory tell us about simply connected surfaces of general type

Let $X$ be a smooth complex projective variety. We know that $\Omega^1_X$ has a non-zero section if and only if the abelianization of the fundamental group of X is infinite. This follows from Hodge ...
Fabiano Rug's user avatar
2 votes
1 answer
443 views

Specialization of sections in an elliptic fibration

Let $\pi: X \rightarrow S$ be the Neron model of an elliptic curve over a dedekind domain (but probably any minimal elliptic fibration will suffice). Let $\eta$ be the generic point of $S$, $K = S(\...
None's user avatar
  • 33
0 votes
2 answers
958 views

A question on degeneration of elliptic curves with actions.

Let $E$ be an elliptic curve. I want to consider its degeneration to the union of two projective lines $C:=\mathbb{P}^1 \cup_{x,y} \mathbb{P}^1$ attaching at two points $x,y$. The involution $-1$ on $...
taryn's user avatar
  • 3
3 votes
1 answer
375 views

Affine neighborhood of an $S$-valued point

How can we understand an affine neighborhood of an $S$-valued point on a scheme, and when does it exist? I am looking at page 111 of Haruzo Hida's Geometric Modular Forms and Elliptic Curves, and he ...
NotFromBrazil's user avatar
2 votes
0 answers
246 views

Full $n$-torsion of elliptic curves and the cyclotomic field of order $n$

Hi, overflowers. I have a question concerning the torsion of elliptic curves over number fields. Let us consider an elliptic curve $E$ defined over ${\mathbb Q}$. From the Weil pairing one can ...
Chema Tornero's user avatar
6 votes
3 answers
2k views

Surfaces ruled over elliptic curves

Ground field $\Bbb{C}$. Algebraic category. Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve. Suppose $E$ is an elliptic curve ...
Heitor's user avatar
  • 105
1 vote
1 answer
243 views

Weiestrass Form

How to convert this to weiestrass form? $x^{2}y^{2}-2\left( 1+2\rho \right) xy^{2}+y^{2}-x^{2}-2\left( 1+2\rho \right) x-1=0$
James's user avatar
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0 answers
275 views

r-torsion points on elliptic curve on finite field

Let $E(\mathbb{F}_q)$ - elliptic curve, $r$ is prime, $|E(\mathbb{F}_q)[r]| > 1$. Let $r | q-1$. Is it true that $|E(\mathbb{F}_q)[r]| = r^2$?
Alexey Milovanov's user avatar
10 votes
1 answer
517 views

examples of "exotic" moduli problems for elliptic curves?

Let $\textbf{Ell}$ be the category of elliptic curves over various base schemes, and where a morphism between $E\rightarrow S$ and $E'\rightarrow S'$ is a cartesian diagram with those two maps as ...
Will Chen's user avatar
  • 10k
3 votes
1 answer
547 views

Shafarevich's theorem for elliptic curves defined over function field of algebraic curve over algebraically closed field

Let $K$ be a number field and $S$ a finite set of places of $K$. Then Shafarevich's theorem states that there are only finitely many isomorphism classes of elliptic curves $E$ over $K$ with good ...
Adam Harris's user avatar
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3 votes
0 answers
182 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 $\Gamma(...
ghost's user avatar
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9 votes
1 answer
1k views

Where do the product expansions of modular forms come from?

It is well-known that many modular forms can be expressed as infinite products. For instance, the most famous one is probably the expansion $$\Delta(q) = q \prod_{n=1}^\infty (1-q^n)^{24}$$ for the ...
Bruno Joyal's user avatar
  • 3,860
6 votes
2 answers
988 views

elliptic curve with a degree 2 isogeny to itself?

I've come across the following question, which I think must be easy for experts: is there a complex elliptic curve $E$ with an isogeny of degree 2 to itself? Of course one can ask the same question ...
rita's user avatar
  • 6,213
13 votes
2 answers
1k views

Best bounds toward Serre's uniformity conjecture

If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the Galois representations $\...
Joël's user avatar
  • 25.7k
14 votes
3 answers
778 views

Order of torsion group

What can one say about the order of the torsion group of an elliptic curve defined over the compositum of all quadratic extensions of $\mathbb{Q}$ ?
Suman's user avatar
  • 1,211
2 votes
0 answers
168 views

A nice rigid analytic model for local systems over an elliptic curve?

For $E$ an elliptic curve, let $LS(E)$ be the group of line bundles on $E$ with a flat connection. This is an $\mathbb{A}^1$-torsor over $E^\vee$. By Riemann-Hilbert (since the gauge group acts ...
Dmitry Vaintrob's user avatar
5 votes
1 answer
1k views

equivalence between katz and classical modular forms

$\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\PP}{\mathbb{P}}$ $\newcommand{\QQ}{\mathbb{Q}}$ $\newcommand{\hH}{\mathcal{H}}$ $\newcommand{\eE}{\mathcal{E}}$ $\newcommand{...
Will Chen's user avatar
  • 10k
0 votes
1 answer
163 views

Can we find a set of elliptic curves over rationals associated with $f$?.

We know that for each elliptic curve over rationals, we can define the Dirichlet series of the Hasse–Weil $L$-function, i.e., the function associated with an elliptic curve over rationals. Then my ...
China-Hong Kong's user avatar
3 votes
1 answer
616 views

Heegner Points on $X_0(N)$ when some primes dividing $N$ are inert in the imaginary quadratic field

If $K = \mathbb{Q}(\sqrt{-D})$ is a imaginary quadratic field with discriminant $-D$, then we get Heegner points on $X_0(N)$ as long as there exists $\mathfrak{n} \subset \mathcal{O}_K$ such that $\...
stl's user avatar
  • 585
4 votes
3 answers
3k views

reduction types of elliptic curves

Let $E/K$ be an elliptic curve, where $K$ is a complete local field with residue field $k$ and char$(k) = p$. I'm trying to make sense of Kodaira symbols and Tate's algorithm. My current ...
stl's user avatar
  • 585
6 votes
1 answer
787 views

Is there an elliptic surface over $Y(1)$?

Actually I have a few related questions. Here, by $Y(1)$ I mean the affine $j$-line $\text{SL}_2(\mathbb{Z})\backslash\mathcal{H}$. I know $Y(1)$ is only a coarse moduli space, so there isn't a ...
Will Chen's user avatar
  • 10k
1 vote
0 answers
244 views

lifts of maps to $\mathcal{M}_{1,1}$

Hi, here's there's a construction about elliptic curves that I do not completely understand. Suppose I consider the two following families of elliptic curves over $\mathbb{C}^*$. The first, which I ...
IMeasy's user avatar
  • 3,717
3 votes
0 answers
145 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 ...
George's user avatar
  • 31
5 votes
1 answer
337 views

Inertia subgroup in the ordinary reduction case when $p=2$

Dear MO, Let $K/\mathbb{Q}_2$ be a finite extension, and let $E/K$ be an elliptic curve with good ordinary reduction, and such that $\mathbb{Q}_2(j(E))=K$. Let $\rho:\operatorname{Gal}(\overline{K}/K)...
Álvaro Lozano-Robledo's user avatar

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