A Class of Pseudo-Real Riemann Surfaces with Diagonal Automorphism Group

Abstract: A Riemann surface

m a t h c a l S

having field of moduli

m a t h b b R

, but not a field of definition, is called emph{pseudoreal}. This means that

m a t h c a l S

has anticonformal automorphisms, but non of them is an involution. We call a Riemann surface

m a t h c a l S

emph{plane} if it can be described by a smooth plane model of some degree

d g e q 4

in

m a t h b b P_{m a t h b b C}^{2}

. We characterize pseudoreal-plane Riemann surfaces

m a t h c a l S

, whose conformal automorphism group

o p e r a t o r n a m e {A u t}_{+} (m a t h c a l S)

is

o p e r a t o r n a m e {P G L}_{3} (m a t h b b C)

-conjugate to a finite non-trivial group

G

that leaves invariant infinitely many points of

m a t h b b P_{m a t h b b C}^{2}

. In particular, we show that such pseudoreal-plane Riemann surfaces exist only if

o p e r a t o r n a m e {A u t}_{+} (m a t h c a l S)

is cyclic of even order

n

dividing the degree

d

. Explicit examples are given, for any degree

d = 2 p m

with

m > 1

odd,

p

is prime and

n = d / p

.

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