Computing All Elliptic Curves Over an Arbitrary Number Field with Prescribed Primes of Bad Reduction

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 $\mathbb{Q}$ and quadratic fields.

KEYWORDS:

• elliptic curves
• bad reduction
• S-unit equations

2000 AMS Subject Classification:

• Number Theory

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 [Cohen 99] for a detailed description.

4 The details of computing general C_p extensions can be found in [Cohen 99, Chapter 5].

5 D_p does not have a normal subgroup of order 2 while C_2p 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 − a_p − σ(b_p)).

7 F_j(x) may not be irreducible. The crucial thing is that F_j(x) has coefficients in K.

8 We recall that in this case G_λ = G_μ.

9 The same holds for S_μ in the trivial, cubic, and S₃ case but not for the quadratic.

10 In case [L: K] = 6, we can prove that ${ord}_{\mathfrak{p}}\left(\Delta (L/K)\right)\equiv 0\mathrm{mod}2\iff {ord}_{\mathfrak{p}}\left(\Delta (L_c/K)\right)\equiv 0\mathrm{mod}2$ where L_c is the cubic subfield of L/K.

11 Note that $\mathfrak{B}\in S_{\lambda }^{\infty }\iff \tau \left(\mathfrak{B}\right)\in S_{\lambda }^{\infty }$.

12 For this case we have that $G_{\mu }={\mathcal{O}}_{L,S_L}^{*}$ which does not usually have small rank.

13 We have to mention that our computations with Laska’s in [Laska 83].