Endomorphisms of ordinary superelliptic jacobians

Abstract: Let

K

be a field of prime characteristic

p

,

n > 4

an integer,

f (x)

an irreducible polynomial over

K

of degree

n

, whose Galois group is either the full symmetric group

S_{n}

or the alternating group

A_{n}

. Let

l

be an odd prime different from

p

,

Z [z e t a_{l}]

the ring of integers in the

l

th cyclotomic field,

C_{f, l} : y^{l} = f (x)

the corresponding superelliptic curve and

J (C_{f, l})

its jacobian. We prove that the ring of all endomorphisms of

J (C_{f, l})

coincides with

Z [z e t a_{l}]

if

J (C_{f, l})

is an ordinary abelian variety and

(l, n) e (5, 5)

.

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