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**On \(pq\)-hyperelliptic Riemann surfaces.**
*(English)*
Zbl 1080.30037

Let \(X\) be a compact Riemann surface of genus \(g\geq 2\). The surface \(X\) is said to be \(p\)-hyperelliptic if it admits a conformal involution \(\rho \), called a \(p\)-hyperelliptic involution, such that the quotient \(X/\rho \) has genus \(p\). Let us suppose that \(X\) also admits a \(q\)-hyperelliptic involution. In such a case we say that \(X\) is a \(pq\)-hyperelliptic surface. In this paper \(pq\)-hyperelliptic surfaces are studied and several bounds for the genus of the surface in terms of \(p\) and \(q\) are obtained.

It is proved that for arbitrary integers \(0\leq p\leq q \) with \((p,q)\neq (0,0)\) the genus \(g\) of a \(pq\)-hyperelliptic Riemann surface satisfies \(2q-1\leq g\leq 2p+2q+1\). Conversely let us suppose that these numerical conditions are satisfied then there exists a Riemann surface admitting commuting \(p\)- and \(q\)-hyperelliptic involutions whose product is a \(t\) -hyperelliptic involution if and only if \(t\) is a nonnegative integer satisfying \( (g+1)/2-(p+1)\leq t\leq (g+1)/2\) and such that \(p+q+t-g\) is an even and nonnegative integer. Moreover if \(g\geq 2p+2q-2\) then \(t=g-p-q\). As a consequence there exist Riemann surfaces of genus \(2\) or \(3\) which are simultaneously hyperelliptic and elliptic-hyperelliptic (\(p=0\), \(q=1\)) being the product of the \(0\)- and \(1\)-involutions a \(1\)- or a \(2\)-involution, respectively.

The author obtains several other results: if \(q>2p+1\) then a \(p\)-hyperelliptic involution is unique and central in the full group of automorphisms of a \(pq\) -hyperelliptic surface. It is proved that if \((p,q)\neq (0,0)\) and the genus \(g\) satisfies \( 3p+3q+2<2g\leq 4p+4q+2\) then any \(p\)- and \(q\)-hyperelliptic involutions commute. Finally if \(p<q<2p\) and \(3q+1<g\leq2p+2q+1\) then \(p\)- and \(q\)-hyperelliptic involutions in a \(pq\)-hyperelliptic Riemann surface are unique and central in the full group of automorphisms. Hence if \(3q+2<g\leq4q+1\), a Riemann surface of genus \(g\) can admit at most two \(q\)-involutions.

Fuchsian signatures, the Hurwitz-Riemann formula and a result from A. M. Macbeath [Bull. Lond. Math. Soc. 5, 103–108 (1973; Zbl 0259.30016)] about the number of fixed points of an automorphism of a Riemann surface, are the main tools in the proofs, which are short and clear.

It is proved that for arbitrary integers \(0\leq p\leq q \) with \((p,q)\neq (0,0)\) the genus \(g\) of a \(pq\)-hyperelliptic Riemann surface satisfies \(2q-1\leq g\leq 2p+2q+1\). Conversely let us suppose that these numerical conditions are satisfied then there exists a Riemann surface admitting commuting \(p\)- and \(q\)-hyperelliptic involutions whose product is a \(t\) -hyperelliptic involution if and only if \(t\) is a nonnegative integer satisfying \( (g+1)/2-(p+1)\leq t\leq (g+1)/2\) and such that \(p+q+t-g\) is an even and nonnegative integer. Moreover if \(g\geq 2p+2q-2\) then \(t=g-p-q\). As a consequence there exist Riemann surfaces of genus \(2\) or \(3\) which are simultaneously hyperelliptic and elliptic-hyperelliptic (\(p=0\), \(q=1\)) being the product of the \(0\)- and \(1\)-involutions a \(1\)- or a \(2\)-involution, respectively.

The author obtains several other results: if \(q>2p+1\) then a \(p\)-hyperelliptic involution is unique and central in the full group of automorphisms of a \(pq\) -hyperelliptic surface. It is proved that if \((p,q)\neq (0,0)\) and the genus \(g\) satisfies \( 3p+3q+2<2g\leq 4p+4q+2\) then any \(p\)- and \(q\)-hyperelliptic involutions commute. Finally if \(p<q<2p\) and \(3q+1<g\leq2p+2q+1\) then \(p\)- and \(q\)-hyperelliptic involutions in a \(pq\)-hyperelliptic Riemann surface are unique and central in the full group of automorphisms. Hence if \(3q+2<g\leq4q+1\), a Riemann surface of genus \(g\) can admit at most two \(q\)-involutions.

Fuchsian signatures, the Hurwitz-Riemann formula and a result from A. M. Macbeath [Bull. Lond. Math. Soc. 5, 103–108 (1973; Zbl 0259.30016)] about the number of fixed points of an automorphism of a Riemann surface, are the main tools in the proofs, which are short and clear.

Reviewer: Ernesto MartĂnez (Madrid)

### MSC:

30F10 | Compact Riemann surfaces and uniformization |

20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |