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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208 views

Point halving on elliptic curves over $\mathbb{Q}$

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $E(\mathbb{Q})[2]=\{o,T_1,T_2,T_3\}$. Let $P=2R$ be a point in $2E(\mathbb{Q})$, using $2$-division polynomial, we can compute $1/2P$, but it gives ...
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219 views

Other elliptic curves for $x^4+y^4+z^4 = 1$

Given, $$a^4+b^4+c^4 = d^4\tag{0}$$ we have the identity, $$(-11980 + 1673 u + 54u^2)^4 + (36 - 2321 u + 3u^2)^4 + t^4 = (24677 + 203 u + 71u^2)^4$$ where, $$591800025 + 20030510 u + 1671327 u^2 ...
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243 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 ...
3
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1answer
192 views

relation between Faltings height and periods

Let $E$ be an elliptic curve defined by an equation $y^2=4x^3+ax+b$ where $a$ and $b$ are algebraic numbers. What is the relation between the Faltings height $h_F(E)$ and the periods $$ \int_{\gamma} ...
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1answer
351 views

Determining $\mu$-invariant of elliptic curves over $\mathbb{Q}$

From Pollack's table on his homepage, I have the values of mu invariant of elliptic curves 38B1 & 38B2 (labeled as in Cremona table). But I need to know the values of mu invariants of 38A1, 38A2, ...
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249 views

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 ...
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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 ...
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296 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 ...
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2answers
694 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 $\mathbb{Q}$ ?
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194 views

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 ...
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573 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 + ...
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2answers
388 views

Equations of elliptic curves

First part of question I have asked on mathoverflow already: http://math.stackexchange.com/questions/467088/explict-form-of-the-equation-of-elliptic-curve 1) Let $E(\mathbb{F}_{q^2})$ is elliptic ...
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188 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 ...
2
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1answer
351 views

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 ...
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3answers
729 views

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 + ...
2
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2answers
195 views

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 ...
3
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0answers
202 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 ...
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223 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 ...
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284 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
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1answer
140 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 ...
3
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1answer
129 views

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 ...
4
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1answer
202 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 ...
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287 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 ...
4
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1answer
100 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 $k$ ...
3
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1answer
286 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 ...
1
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1answer
109 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 ...
5
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1answer
247 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 ...
4
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1answer
238 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 ...
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1answer
375 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 ...
19
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1answer
781 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 $Ell \Rightarrow Ell: Ring \to Gpd$ ...
4
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1answer
231 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 ...
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0answers
138 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 ...
2
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1answer
157 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 = ...
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2answers
294 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 ...
3
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1answer
184 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 ...
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0answers
132 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 ...
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3answers
372 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 ...
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1answer
207 views

Weiestrass Form

How to convert this to weiestrass form? $x^{2}y^{2}-2\left( 1+2\rho \right) xy^{2}+y^{2}-x^{2}-2\left( 1+2\rho \right) x-1=0$
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0answers
135 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 $|E(\mathbb{F}_q)[r]| = r^2$?
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1answer
333 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 ...
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1answer
316 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 ...
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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 ...
5
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1answer
304 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 ...
5
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2answers
281 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 ...
9
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2answers
519 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 ...
6
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1answer
270 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 $\mathbb{Q}$ ?
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118 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 ...
4
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1answer
457 views

equivalence between katz and classical modular forms

$\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\PP}{\mathbb{P}}$ $\newcommand{\QQ}{\mathbb{Q}}$ $\newcommand{\hH}{\mathcal{H}}$ $\newcommand{\eE}{\mathcal{E}}$ ...
0
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1answer
148 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 ...
3
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1answer
294 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 ...