Metadata only
Datum
2022-05Typ
- Journal Article
ETH Bibliographie
yes
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Abstract
A complex algebraic curve is said to have many automorphisms if it cannot be deformed nontrivially together with its automorphism group. Oort asked whether the jacobian of any such curve has complex multiplication. We answer this question completely when the curve is hyperelliptic. For this, we first classify all hyperelliptic curves with many automorphisms, finding 3 infinite sequences and 15 separate isomorphism classes. Of these, all members of the infinite sequences and 5 of the isolated curves have a jacobian with complex multiplication, but the remaining 10 curves do not.
The methods used include a criterion of Wolfart, a representation theoretic criterion of streit and the non-integrality of the j-invariants of certain quotient curves of genus 1. For the hardest cases, we use an ad hoc developed computational criterion depending on the splitting fields of the characteristic polynomials of Frobenius acting on the Tate module of the Jacobian. Mehr anzeigen
Publikationsstatus
publishedExterne Links
Zeitschrift / Serie
International Journal of Number TheoryBand
Seiten / Artikelnummer
Verlag
World ScientificThema
Algebraic curves; hyperelliptic; automorphisms; complex multiplicationETH Bibliographie
yes
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