Cyclic covers of prime power degree, jacobians and endomorphisms

Publication date: 27 December 2003

Abstract: Suppose that

K

is a field of characteristic 0,

K_{a}

is its algebraic closure,

p

is a prime,

q = p^{r}

is a power prime. Suppose that

f (x) i n K [x]

is a polynomial of degree

n > 4

without multiple roots. Let us consider the superelliptic curve

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

and its jacobian

J (C)

. We study the endomorphism algebra

E n d^{0} (J (C))

of all

K_{a}

-endomorphisms of

J (C)

. We prove that

E n d^{0} (J (C))

is "as small as possible" if the Galois group of

f

over

K

is either the full symmetric group

S_{n}

or the alternating group

A_{n}

.

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