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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Isogeny classes and elliptic curves over finite fields

Fix a conductor and a prime $p$. Then 1) Do the elliptic curves in the same isogeny class after reduction modulo $p$ have the same number of points over the finite field $\mathbb{F}_{p} ?$ 2) Do the ...
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430 views

State of knowledge of $a^n+b^n=c^n+d^n$ vs. $a^n+b^n+c^n=d^n+e^n+f^n$

As far as I understand, both of the Diophantine equations $$a^5 + b^5 = c^5 + d^5$$ and $$a^6 + b^6 = c^6 + d^6$$ have no known nontrivial solutions, but $$24^5 + 28^5 + 67^5 = 3^5+64^5+62^5$$ and ...
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Gross's paper on Heegner points

I try to read Gross's paper on Heegner points and it seems ambiguous for me on some points: Gross (page 87) said that $Y=Y_{0}(N)$ is the open modular curve over $\mathbb{Q}$ which classifies ordered ...
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Automorphisms of the L-function associated to an elliptic $\mathbb{Q}$-curve

Edited after Noam Elkies' comment: From what I understand (very few actually), there exist elliptic curves defined over some number fields $\mathbb{K}$ Galois over $\mathbb{Q}$ which are isogenous to ...
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178 views

The existence of elliptic curves with prescribed supersingular primes

For a given infinite set of primes, not too big, eg, satisfying Lang-Trotter conjecture, can we always find an E.C. with supersingular reduction (at least) at these primes? How about E.C. without CM? ...
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90 views

list of supersingular elliptic curves [closed]

Who can give me a table of supersingualr elliptic curves over F_p? At lsist for small p. If I make such a table using magma on my laptop, how long (as a function of p) will I use.
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The existence of infinitely many supersingular primes for every elliptic curve over Q

Elkies proved The existence of infinitely many supersingular primes for every elliptic curve over Q. I read his paper, but found the supersingular primes he constructed are all 3(mod 4) type. So, how ...
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Fundamental group of the moduli stack of ordinary generalized elliptic curves

Let $M$ be the moduli stack of ordinary but possibly nodal elliptic curves over the field $\overline{\mathbf{F}_p}$. Then $M$ has a $\mathbb{Z}_p^{\times}$-torsor over it, given by the moduli scheme ...
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218 views

Isogeny classes and reduction types of elliptic curves at primes of bad reduction

Fix a conductor. Then 1) Do the elliptic curves in the same isogeny class have the same reduction type at a prime of bad reduction of the curve ? 2) Do the elliptic curves belonging to two ...
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in the preface of “A first course in modular forms”

In the preface of "A first course in modular forms", the author considered the quadratic equation $Q: x^2=d, \ \ d \in \mathbb{Z}, d \not=0$, and for each prime $p$ define an integer $a_p(Q)=\#\tilde ...
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189 views

Elliptic curve is to Tate module as genus one curve is to blank?

Let $E$ be an elliptic curve over $\mathbf{Q}$, and let $\rho:\mathrm{Gal}(\mathbf{Q}) \to \mathrm{GL}_2(\mathbf{Z}_\ell)$ be its Tate module. Let $T$ be the set of isomorphism classes of ...
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442 views

Average rank of elliptic curves, excluding those of low rank

It's conjectured that, asymptotically, half of elliptic curves have rank 0, half have rank 1, and elliptic curves of rank $\geq 2$ have density 0. But what if we disregard elliptic curves of rank 0 or ...
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369 views

transcendence of periods of CM elliptic curves

Let $E$ be an elliptic curve over $\overline{\mathbb{Q}}$ defined by a Weierstrass equation $$ y^2=4x^3+g_2x+g_3. $$ Then $H^1_{dR}(E/\overline{\mathbb{Q}})$ is spanned by the classes of the ...
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$Pic$ of the stack of elliptic curves vs. $Pic$ of the coarse space

There's a natural map $f:\overline{\mathcal{M}}_{1,1}\to \overline{M}_{1,1}\cong \mathbb{P}^1$ from the stack of elliptic curves to the coarse space. Both spaces have $Pic=\mathbb{Z}$ hence ...
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1answer
172 views

Legendre relation for elliptic curves

Let $E$ be an elliptic curve over some subfield $k$ of $\mathbb{C}$, say given by an equation $$ y^2=4x^3+ax+b. $$ Then: $E(\mathbb{C})$ is a complex torus, so $H_1(E(\mathbb{C}), \mathbb{Q})$ is ...
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134 views

Can one find elliptic curve and a point of known order over $\mathbb{Z}/n\mathbb{Z}$?

Let $n$ be composite with unknown factorization. Can one efficiently find elliptic curve $E$ over $\mathbb{Z}/n\mathbb{Z}$ and a point $P$ on $E$ of known order $m > 5$? $P$ should be nontorsion ...
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131 views

picard group of moduli of elliptic r-prym curves

Let $\overline{\mathcal{M}}_{1,1}$ be the DM compactification of the moduli stack of elliptic curves. Its Picard group is $\mathbb{Z}$. Let us now consider stack of $r$-prym curves ...
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104 views

One parameter families of elliptic curves over rings of integers of number fields

Let $A(n), B(n) \in \mathbb{Z}[n]$ be polynomials, not both constant, such that $4A^3(n) + 27B^2(n)$ is not the zero polynomial and the polynomial (in variables $x, y$) $$y^2 - x^3 - A(n)x - B(n) \in ...
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1answer
209 views

Example of non-modular elliptic surface?

In "On elliptic modular surfaces", Shioda proves some interesting theorems on smooth elliptic surfaces (admitting a section); he then focuses on "modular elliptic surfaces" and proves some more ...
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371 views

Rational points or a Weierstrass model for degree 8 elliptic curve

Related to rationally derived polynomials. Neither Maple nor Magma online couldn't solve it. Choose $s,h \in \mathbb{Q}$. I am looking for rational points (possibly of finite order) on this ...
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1answer
119 views

Hodge bundle on F-curves

Let $\mathbb{E}\rightarrow\overline{M}_{g,n}$ be the Hodge bundle. Let us cosider an $F$-curve of type $\overline{M}_{1,1}\subseteq\overline{M}_{g,n}$. Is the degree of the restriction of $\mathbb{E}$ ...
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242 views

what is complexity of finding a non-torsion point on elliptic curve

Let E be an elliptic curve over $Q$ with positive rank $r$. I am looking for algorithms which find a rational point on $E$. I think the algorithms find points with the lowest height. But when I use ...
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123 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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175 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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211 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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174 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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280 views

Determining $\mu$-invariants 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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226 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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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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252 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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611 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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174 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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539 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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376 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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182 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 ...
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1answer
344 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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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 + ...
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181 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 ...
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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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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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Picard-Fuchs equation of elliptic curves with level $N$ structure.

Let $X_N$ be the moduli space of elliptic curves with level $N$ structure. Is the Piard Fuchs equation of the universal family over $X_N$ (take some covering if $X_N$ is not a fine moduli space)? In ...
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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). ...
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129 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 ...
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127 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 ...
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190 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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280 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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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 $k$ ...
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272 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 ...
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1answer
98 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 ...
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1answer
227 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 ...