Abstract
We want to give an overview on arithmetical aspects of abelian varieties and their torsion structures, isogenies, and resulting Galois representations. This is a wide and deep territory with a huge amount of research activity and exciting results ranging from the highlights of pure mathematics like the proof of Fermat’s last theorem to stunning applications to public-key cryptography. Necessarily we have to be rather superficial, and thus specialists in the different aspects of the topics may be disappointed. But I hope that for many, and in particular for young researchers, the chapter may serve as an appetizer and will raise interest for a fascinating area of mathematics with many open problems (some are very hard and worth a Fields Medal but others are rather accessible).
The first section of the chapter gives basic notions, definitions, and properties of abelian varieties. Disguised as examples one will find their theory over the complex numbers \(\mathbb{C}\) and the special case of elliptic curves. The second section discusses the situation over finite fields, in particular the role of the Frobenius endomorphism, and over number fields where the most interesting results and challenging conjectures occur. Finally we discuss algorithmic aspects of isogenies, mostly of elliptic curves, and relations to cryptography.
This chapter is based on a lecture presented at the conference “Open Problems in Mathematical and Computational Sciences” in Istanbul. I would like to thank the organizers for the opportunity to participate in this very interesting and inspiring conference and for their warm hospitality in the most beautiful environment.
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Notes
- 1.
That is the tangent space of every point of E a has dimension 1; see [ACF], Sect. 4.4.1.
- 2.
That is, irreducible as variety over K s .
- 3.
For example, by homogenous equations.
- 4.
See Definition 2.81 in [ACF] or any textbook on algebraic number theory.
- 5.
Caution for specialists: because of the existence of twists, Y 0 is only a coarse moduli space.
- 6.
\(\mathbb{Z}_{\ell}\) is the ring of l-adic integers and \(\mathbb{Q}_{\ell}\) the field of ℓ-adic numbers (see [ACF]).
- 7.
The tangent space of every point of C has dimension 1, see [ACF], Sect. 4.4.1
- 8.
Poles give rise to negative “order of vanishing”.
- 9.
For p | 6, see [ACF] 13.1.1 and 13.3.
- 10.
That is, the normalized valuation attached to \(\tilde{\mathfrak{p}}\) is a continuation of the one attached to \(\mathfrak{p}.\)
- 11.
In the following we simplify by looking at abelian varieties with principal polarization (e.g., Jacobian varieties) and then neglect some more subtle points concerning these polarizations.
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Frey, G. (2014). Isogenies in Theory and Praxis. In: Koç, Ç. (eds) Open Problems in Mathematics and Computational Science. Springer, Cham. https://doi.org/10.1007/978-3-319-10683-0_3
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