## Elliptic Curves

Venue: | in [Buhler and Stevenhagen 2007]. Citations in this document: §4 |

Citations: | 1 - 0 self |

### BibTeX

@INPROCEEDINGS{Poonen_ellipticcurves,

author = {Bjorn Poonen},

title = {Elliptic Curves},

booktitle = {in [Buhler and Stevenhagen 2007]. Citations in this document: §4},

year = {}

}

### OpenURL

### Abstract

. This is a introduction to some aspects of the arithmetic of elliptic curves, intended for readers with little or no background in number theory and algebraic geometry. In keeping with the rest of this volume, the presentation has an algorithmic slant. We also touch lightly on curves of higher genus. Readers desiring a more systematic development should consult one of the references for further reading suggested at the end. 1. Plane curves Let k be a field. For instance, k could be the field Q of rational numbers, the field R of real numbers, the field C of complex numbers, the field Q p of p-adic numbers (see [Kob] for an introduction), or the finite field F q of q elements (see Chapter I of [Ser1]). Let k be an algebraic closure of k. A plane curve 1 X over k is defined by an equation f(x, y) = 0 where f(x, y) = # a ij x i y j # k[x, y] is irreducible over k. One defines the degree of X and of f by deg X = deg f = max{i + j : a ij #= 0}. A k-rational point (or simply k-point) on X is a point (a, b) with coordinates in k such that f(a, b) = 0. The set of all k-rational points on X is denoted X(k). Example: The equation x 2 y - 6y 2 - 11 = 0 defines a plane curve X over Q of degree 3, and (5, 1/2) # X(Q). Already at this point we can state an open problem, one which over the centuries has served as motivation for the development of a huge amount of mathematics. Question. Is there an algorithm, that given a plane curve X over Q, determines X(Q), or at least decides whether X(Q) is nonempty? Although X(Q) need not be finite, we will see later that it always admits a finite description, so this problem of determining X(Q) can be formulated precisely using the notion of Turing machine: see [HU] for a definition. For the relationship of this questi...

### Citations

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Citation Context ...man and Tate is an introduction to elliptic curves at the advanced undergraduate level; among other things, it contains a treatment of the elliptic curve factoring method. The graduate level textbook =-=[Sil]-=- by Silverman uses more algebraic number theory and algebraic geometry, but most definitions and theorems are recalled as they are used, so that the book is readable even by those with minimal backgro... |

285 |
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(Show Context)
Citation Context ...Although X(Q) need not be finite, we will see later that it always admits a finite description, so this problem of determining X(Q) can be formulated precisely using the notion of Turing machine: see =-=[HU] f-=-or a definition. For the relationship of this question to Hilbert’s 10th Problem, see the survey [Po2]. The current status is that there exist computational methods that often answer the question fo... |

256 |
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Citation Context ...ed for mathematicians outside algebraic geometry, but the topics covered are so diverse that even specialists are likely to find a few things that are new to them. Finally, Hartshorne’s graduate tex=-=t [Ha]-=-, although more demanding, contains a thorough development of the language of schemes and sheaf cohomology, as well as applications to the theory of curves of surfaces, mainly over algebraically close... |

240 |
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Citation Context ...1, c2, c3 > 0, c4 ≥ 0 are constants depending on X, and the estimates hold as B → ∞. The fact that X(Q) is finite when the genus is ≥ 2 was conjectured by Mordell [Mo1]. Proofs were given by F=-=altings [Fa] a-=-nd Vojta [Vo], and a simplified version of Vojta’s proof was given by Bombieri [Bo]. All known proofs are ineffective: it is not known whether there exists an algorithm to determine X(Q), although t... |

189 |
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Citation Context ...t the end. 1. Plane curves Let k be a field. For instance, k could be the field Q of rational numbers, the field R of real numbers, the field C of complex numbers, the field Qp of p-adic numbers (see =-=[Kob]-=- for an introduction), or the finite field Fq of q elements (see Chapter I of [Ser1]). Let k be an algebraic closure of k. A plane curve 1 X over k is defined by an equation f(x, y) = 0 where f(x, y) ... |

170 |
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Citation Context ...subset of P 2 (Fq), E(Fq) is a finite abelian group. Hasse proved #E(Fq) = q + 1 − a where |a| ≤ 2 √ q. This is a special case of the “Weil conjectures” (now proven). Moreover, an algorithm =-=of Schoof [Sch] com-=-putes #E(Fq) in time (log q) O(1) as follows: an algorithm we will not explain determines #E(Fq) mod ℓ for each prime ℓ up to about log q, and then the Chinese Remainder Theorem recovers #E(Fq). E... |

74 |
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Citation Context ... and a, we first bound K: K ≤ � ℓ ⌊logℓP � ⌋ ≤ P ≤ P y , ℓ≤y ℓ≤y log K ≤ y log P = L(P ) a+o(1) . Next we need an estimate of the smoothness probability s. A theorem of Can=-=field, Erdös, and Pomerance [CEP] state-=-s that the probability that a random integer in [1, x] is L(x) a - smooth is L(x) −1/(2a)+o(1) as x → ∞. Using a formula of Deuring for the number of elliptic curves of given order over Z/p, one... |

63 |
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Citation Context ... those with minimal background. Cremona’s book [Cr] contains extensive tables of elliptic curves over Q, and discusses many elliptic curve algorithms in detail. 9.2. Algebraic geometry. Fulton’s t=-=ext [Ful] r-=-equires only a knowledge of abstract algebra at the undergraduate level; it develops the commutative algebra as it goes along. Shafarevich’s text, now in two volumes [Sha1],[Sha2], is a extensive su... |

46 |
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Citation Context ...ore algebraic number theory and algebraic geometry, but most definitions and theorems are recalled as they are used, so that the book is readable even by those with minimal background. Cremona’s boo=-=k [Cr] c-=-ontains extensive tables of elliptic curves over Q, and discusses many elliptic curve algorithms in detail. 9.2. Algebraic geometry. Fulton’s text [Ful] requires only a knowledge of abstract algebra... |

45 |
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(Show Context)
Citation Context ...some of these are discussed in [Po1] and [Po2]. Also, there are many other conjectural approaches towards a proof that X(Q) can be determined explicitly for all curves X over Q. (See Section F.4.2 of =-=[HS]-=- for a survey of some of these.) But none have yet been successful. We need some new ideas!s18 BJORN POONEN 9. Further reading For the reader who wants more, we suggest a few other books and survey ar... |

42 |
Basic algebraic geometry. 1. Varieties in projective space. Second edition. Translated from the 1988 Russian edition and with notes by Miles Reid
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Citation Context ...ic geometry. Fulton’s text [Ful] requires only a knowledge of abstract algebra at the undergraduate level; it develops the commutative algebra as it goes along. Shafarevich’s text, now in two volu=-=mes [Sha1]-=-,[Sha2], is a extensive survey of the main ideas of algebraic geometry and its connections to other areas of mathematics. It is intended for mathematicians outside algebraic geometry, but the topics c... |

41 | Detecting perfect powers in essentially linear time
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(Show Context)
Citation Context ...The elliptic curve method. Assume that the integer N to be factored satisfies gcd(N, 6) = 1 and that N �= n r for any integers n, r ≥ 2. (The latter can be tested very quickly, in (log N) 1+o(1) t=-=ime [Ber].) T-=-o search for factors of N of size less than about P : 1. Fix a “smoothness bound” y much smaller than P , and let K be the LCM of all y-smooth integers less than or equal to P . 2. Choose random i... |

24 |
Siegel’s theorem in the compact case
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(Show Context)
Citation Context ...c4 ≥ 0 are constants depending on X, and the estimates hold as B → ∞. The fact that X(Q) is finite when the genus is ≥ 2 was conjectured by Mordell [Mo1]. Proofs were given by Faltings [Fa] an=-=d Vojta [Vo], -=-and a simplified version of Vojta’s proof was given by Bombieri [Bo]. All known proofs are ineffective: it is not known whether there exists an algorithm to determine X(Q), although there probably i... |

22 |
Arithmetic on curves
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(Show Context)
Citation Context ...he language of schemes and sheaf cohomology, as well as applications to the theory of curves of surfaces, mainly over algebraically closed fields. 9.3. Surveys on arithmetic geometry. Mazur’s articl=-=e [Maz] is -=-intended for a general mathematical audience: it begins with a discussion of various diophantine problems, and works its way up to a sketch of Faltings’ proof of the Mordell conjecture. Lang’s boo... |

22 |
On the rational solutions of the indeterminate equation of the third and fourth degrees
- Mordell
- 1922
(Show Context)
Citation Context ... 2 finite O(1) In the third column, c1, c2, c3 > 0, c4 ≥ 0 are constants depending on X, and the estimates hold as B → ∞. The fact that X(Q) is finite when the genus is ≥ 2 was conjectured by =-=Mordell [Mo1]. -=-Proofs were given by Faltings [Fa] and Vojta [Vo], and a simplified version of Vojta’s proof was given by Bombieri [Bo]. All known proofs are ineffective: it is not known whether there exists an alg... |

20 |
Rational points on elliptic curves, Undergraduate Texts in Mathematics
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- 1992
(Show Context)
Citation Context ...re listed first. Of course, there are many other books on these topics; those listed here were chosen partly because they are the ones the author is most familiar with. 9.1. Elliptic curves. The book =-=[ST]-=- by Silverman and Tate is an introduction to elliptic curves at the advanced undergraduate level; among other things, it contains a treatment of the elliptic curve factoring method. The graduate level... |

18 | Lectures on the Mordell-Weil Theorem, Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. With a foreword by Brown and Serre. Third edition. Aspects of Mathematics. Friedr. Vieweg and Sohn - Serre - 1997 |

15 |
Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins, Thesis
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(Show Context)
Citation Context ... (ignoring the same subsets as before). In particular, the complete set of rational solutions to x 2 + y 2 = 1 is �� 2 1 − t 2t , 1 + t2 1 + t2 � � : t ∈ Q ∪ {(−1, 0)}. 3.3. d = 3: pla=-=ne cubics. Lind [Lin] a-=-nd Reichardt [Rei] discovered that the Hasse Principle can fail for plane curves of degree 3. Here is a counterexample due to Selmer [Sel]: the curve 3X 3 + 4Y 3 + 5Z 3 = 0 in P 2 has a R-point (((−... |

15 |
A course in arithmetic. Translated from the French. Graduate Texts
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Citation Context ...f rational numbers, the field R of real numbers, the field C of complex numbers, the field Qp of p-adic numbers (see [Kob] for an introduction), or the finite field Fq of q elements (see Chapter I of =-=[Ser1]). L-=-et k be an algebraic closure of k. A plane curve 1 X over k is defined by an equation f(x, y) = 0 where f(x, y) = � aijx i y j ∈ k[x, y] is irreducible over k. One defines the degree of X and of f... |

14 |
The Mordell conjecture revisited, Ann. Scuola Norm
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(Show Context)
Citation Context ...The fact that X(Q) is finite when the genus is ≥ 2 was conjectured by Mordell [Mo1]. Proofs were given by Faltings [Fa] and Vojta [Vo], and a simplified version of Vojta’s proof was given by Bombi=-=eri [Bo]-=-. All known proofs are ineffective: it is not known whether there exists an algorithm to determine X(Q), although there probably is one. 8.5. Jacobians: one tool for studying higher genus curves. Reca... |

14 | Computational aspects of curves of genus at least 2
- Poonen
- 1996
(Show Context)
Citation Context ... the problem of listing the points of X(Q) to a finite computation.) Nevertheless there exist techniques that often succeed in determining X(Q) for particular curves X; some of these are discussed in =-=[Po1]-=- and [Po2]. Also, there are many other conjectural approaches towards a proof that X(Q) can be determined explicitly for all curves X over Q. (See Section F.4.2 of [HS] for a survey of some of these.)... |

13 |
Einige im Kleinen überall lösbare, im Grossen unlösbare diophantische Gleichungen
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- 1942
(Show Context)
Citation Context ...subsets as before). In particular, the complete set of rational solutions to x 2 + y 2 = 1 is �� 2 1 − t 2t , 1 + t2 1 + t2 � � : t ∈ Q ∪ {(−1, 0)}. 3.3. d = 3: plane cubics. Lind [Lin=-=] and Reichardt [Rei] d-=-iscovered that the Hasse Principle can fail for plane curves of degree 3. Here is a counterexample due to Selmer [Sel]: the curve 3X 3 + 4Y 3 + 5Z 3 = 0 in P 2 has a R-point (((−4/3) 1/3 : 1 : 0) is... |

12 |
The diophantine equation ax 3 +by 3 +cz 3
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- 1951
(Show Context)
Citation Context ... t2 � � : t ∈ Q ∪ {(−1, 0)}. 3.3. d = 3: plane cubics. Lind [Lin] and Reichardt [Rei] discovered that the Hasse Principle can fail for plane curves of degree 3. Here is a counterexample due =-=to Selmer [Sel]: -=-the curve 3X 3 + 4Y 3 + 5Z 3 = 0 in P 2 has a R-point (((−4/3) 1/3 : 1 : 0) is one) and a Qp-point for each prime p, but it has no Q-point. (For p > 5, the existence of Qp-points can be proved by co... |

9 |
I.R.: Basic Algebraic Geometry 2, Schemes and Complex Manifolds
- Shafarevich
- 1994
(Show Context)
Citation Context ...etry. Fulton’s text [Ful] requires only a knowledge of abstract algebra at the undergraduate level; it develops the commutative algebra as it goes along. Shafarevich’s text, now in two volumes [Sh=-=a1],[Sha2]-=-, is a extensive survey of the main ideas of algebraic geometry and its connections to other areas of mathematics. It is intended for mathematicians outside algebraic geometry, but the topics covered ... |

4 |
On the magnitude of the integer solutions of the equation ax 2 + by 2 + cz 2 = 0
- Mordell
(Show Context)
Citation Context ...≡ 0 (mod b) are solvable in integers. Moreover, in this case, aX 2 +bY 2 +cZ 2 = 0 has a nontrivial solution in integers X, Y, Z satisfying |X| ≤ |bc| 1/2 , |Y | ≤ |ac| 1/2 , and |Z| ≤ |ab| 1/=-=2 . See [Mo2]. -=-In the case where the conic X has a Q-point P0, there remains the problem of describing the set of all Q-points. For this there is a famous trick: for each P ∈ X(Q) draw the line through P0 and P , ... |

1 |
Elkies, Heegner point computations, Algorithmic number theory
- D
- 1994
(Show Context)
Citation Context ..., O} � Z/5. Example 3. Let E be the elliptic curve 1063y 2 = x 3 − x. (This is not in Weierstrass form, but it is isomorphic to y 2 = x 3 − 1063 2 x.) Using “Heegner points on modular curves,�=-=�� Elkies [Elk] com-=-puted that E(Q) � Z × Z/2 × Z/2, where E(Q)/E(Q)tors is generated by a point with x-coordinate q 2 /1063, where q = 11091863741829769675047021635712281767382339667434645 31734265754477218073520797... |

1 |
An elliptic curve over Q with rank at least 24, January 2000, electronic announcement on the NMBRTHRY list server (posted May 2
- Martin, McMillen
- 2000
(Show Context)
Citation Context ...e 4. Let E be the elliptic curve y 2 + xy + y = x 3 + ax + b where a = −120039822036992245303534619191166796374, and b = 504224992484910670010801799168082726759443756222911415116. Martin and McMille=-=n [MM] sho-=-wed that E(Q) � Z r , where r ≥ 24. It is a folklore conjecture that as E varies over all elliptic curves over Q, the rank r can be arbitrarily large. 6.2. One-dimensional affine group varieties o... |

1 |
Computing rational points on curves, to appear
- Poonen
(Show Context)
Citation Context ... problem of determining X(Q) can be formulated precisely using the notion of Turing machine: see [HU] for a definition. For the relationship of this question to Hilbert’s 10th Problem, see the surve=-=y [Po2]-=-. The current status is that there exist computational methods that often answer the question for a particular X, although it has never been proved that these methods work in general. Even the followi... |