Hyperelliptic jacobians without complex multiplication and Steinberg representations in positive characteristic

Publication date: 16 January 2003

Abstract: In his previous papers (Math. Res. Letters 7 (2000), 123--13; Math. Res. Letters 8 (2001), 429--435; Moscow Math. J. 2 (2002), issue 2, 403-431) the author proved that in characteristic

e 2

the jacobian

J (C)

of a hyperelliptic curve

C : y^{2} = f (x)

has only trivial endomorphisms over an algebraic closure

K_{a}

of the ground field

K

if the Galois group

G a l (f)

of the irreducible polynomial

f (x) i n K [x]

is either the symmetric group

S n

or the alternating group

A_{n}

. Here

n g e 9

is the degree of

f

. The goal of this paper is to extend this result to the case of certain ``smaller doubly transitive simple Galois groups. Namely, we treat the infinite series $n = 2^{m} + 1, G a l (f) = L_{2} (2^{m}) : = P S L_{2} (F_{2^{m}})$ , $n = 2^{4 m + 2} + 1, G a l (f) = S z (2^{2 m + 1}) =^{2} B_{2} (2^{2 m + 1})$ and $n = 2^{3 m} + 1, G a l (f) = U_{3} (2^{m}) : = P S U_{3} (F_{2^{m}})$ .

Mathematics Subject Classification ID

Jacobians, Prym varieties (14H40) Algebraic theory of abelian varieties (14K05)

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