ABSTRACT
In this article, we study the problem of how to determine all elliptic curves defined over an arbitrary number field K with good reduction outside a given finite set of primes S of K by solving S-unit equations. We give examples of elliptic curves over and quadratic fields.
2000 AMS Subject Classification:
Acknowledgments
I would like to thank my supervisor, professor John Cremona for suggesting to me this problem, for the long discussions we had, and his constant encouragement, and also for the suggestion of Proposition 5.2 and his comments on an earlier draft of this article. Moreover, I want to thank ICERM and Brown University because a big part of the article was written during my visit in the institute. I would also like to thank the anonymous referee for the valuable suggestions and comments.
Funding
I am grateful to Bekiari-Vekri foundation and the Academy of Athens for funding a part of my studies and EPSRC for covering a part of my fees.
Notes
1 Suggested to John Cremona in a personal communication with Noam Elkies in June 2010.
2 A cyclic extension L/K is a Galois extension with cyclic Galois group.
3 Class Field theory can also be used for constructing cyclic extensions with the desired properties. See [CitationCohen 99] for a detailed description.
4 The details of computing general Cp extensions can be found in [CitationCohen 99, Chapter 5].
5 Dp does not have a normal subgroup of order 2 while C2p has one.
6 If a1 + b1 = 0 then a1 + b2 ≠ 0 since b1 ≠ b2. So we can change h(x) with h′(x) = (x − a1 − σ(b1))(x − a2 − σ(b2))…(x − ap − σ(bp)).
7 Fj(x) may not be irreducible. The crucial thing is that Fj(x) has coefficients in K.
8 We recall that in this case Gλ = Gμ.
9 The same holds for Sμ in the trivial, cubic, and S3 case but not for the quadratic.
10 In case [L: K] = 6, we can prove that where Lc is the cubic subfield of L/K.
11 Note that .
12 For this case we have that which does not usually have small rank.
13 We have to mention that our computations with Laska’s in [CitationLaska 83].