Classifying abelian threefolds of p-rank 0, 1, 2 and 3

Dogger, Floor (2021) Classifying abelian threefolds of p-rank 0, 1, 2 and 3. Master's Thesis / Essay, Mathematics.

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Abstract

The aim of the thesis is to find relations between the endomorphism algebra, the p-rank and the factorization of the rational prime p into prime ideals in the number field generated by the Frobenius endomorphism for abelian threefolds over finite fields of characteristic p. For elliptic curves and abelian surfaces, these relations are discussed in Chapter 1 and Chapter 2. The main theorem follows in Chapter 3 and gives a complete classification of the p-rank in terms of the splitting behaviour of the rational prime p in the maximal order of the number field generated by the Frobenius endomorphism of an absolutely simple abelian threefold over a finite field of characteristic p. In Chapter 4, the reduction of absolutely simple CM abelian threefolds is studied. Only abelian threefolds with CM by a CM field of which the Galois group of the normal closure is cyclic or isomorphic to the dihedral group D6 are considered. For both options, the endomorphism algebra and the p-rank of the reduced CM abelian threefold is determined from the prime factorization of a rational prime p in the ring of integers of the CM field.

Item Type:	Thesis (Master's Thesis / Essay)
Supervisor name:	Kilicer, P.
Degree programme:	Mathematics
Thesis type:	Master's Thesis / Essay
Language:	English
Date Deposited:	23 Jul 2021 08:16
Last Modified:	23 Jul 2021 08:16
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/25373

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