On the weights of the fixed points of an automorphism of a compact Riemann surface

In this work we study automorphisms of compact Riemann surfaces with more than four fixed points. We obtain a lower bound for the weight of each of these fixed points. The discussion depends on the parities of the order of the automorphism and the number of fixed points. Moreover, we discuss the sharpness of our bounds. Received: 15 February 2005     

p-Groups acting on Riemann surfaces

Let p be a prime integer and let $r\ge 3$ be an integer so that $p\ge 5r?7$. We show that a closed Riemann surface S of genus $g\ge 2$ has at most one p-group H of conformal automorphisms so that $S/H$ has genus zero and exactly r cone points. This, in particular, asserts that, for $r=3$ and $p\ge 11$, the minimal field of definition of S coincides with that of $(S,H)$. Another application of this fact, for the case that S is pseudo-real, is that $\mathrm{Aut}(S)/H$ must be either trivial or a cyclic group and that r is necessarily even. This generalizes a result due to Bujalance–Costa for the case of pseudo-real cyclic p-gonal Riemann surfaces.     

On Singerman symmetries of a class of Belyi Riemann surfaces

By virtue of the Belyi Theorem an algebraic curve can be defined over the algebraic numbers if and only if the corresponding Riemann surface can be uniformized by a subgroup of a Fuchsian triangle group. Such surfaces are known as Belyi surfaces and an important class of them consists of Riemann surfaces having the so-called large group of automorphisms. Necessary and sufficient algebraic conditions for these surfaces to be symmetric were found by Singerman in the middle of the seventies and, by a recent result of Köck and Singerman, the algebraic numbers above can be chosen to be real if and only if the respective surface is symmetric. The aim of this paper is to give, in similar terms, the formulas for the number of ovals of the corresponding symmetries, which we refer to as the Singerman symmetries.     

Sets of periods for automorphisms of compact Riemann surfaces

Let G=〈f〉 be a finite cyclic group of order N that acts by conformal automorphisms on a compact Riemann surface S of genus g≥2. Associated to this is a set A of periods defined to be the subset of proper divisors d of N such that, for some x∈S, x is fixed by f^d but not by any smaller power of f. For an arbitrary subset A of proper divisors of N, there is always an associated action and, if g_A denotes the minimal genus for such an action, an algorithm is obtained here to determine g_A. Furthermore, a set A_max is determined for which g_A is maximal.     

Shared values of meromorphic functions on compact Riemann surfaces

Let S be a compact Riemann surface of genus g and gonality d. We derive upper bounds (in terms of g and/or d) for the number of values that two non-constant meromorphic functions on S can share. The case d = 2 (i.e., the surface is hyperelliptic or elliptic) is studied in more detail.Received: 14 April 2004     

Weierstrass weight of the hyperosculating points of generalized Fermat curves

Let $(S,H)$ be a generalized Fermat pair of the type $(k,n)$. If $F?S$ is the set of fixed points of the non-trivial elements of the group H, then F is exactly the set of hyperosculating points of the standard embedding $S?{\mathbb{P}}^n$. We provide an optimal lower bound (this being sharp in a dense open set of the moduli space of the generalized Fermat curves) for the Weierstrass weight of these points.     

The gonality of Riemann surfaces under projections by normal coverings

A compact Riemann surface X of genus g≥2 which can be realized as a q-fold, normal covering of a compact Riemann surface of genus p is said to be (q,p)-gonal. In particular the notion of (2,p)-gonality coincides with p-hyperellipticity and (q,0)-gonality coincides with ordinary q-gonality. Here we completely determine the relationship between the gonalities of X and Y for an N-fold normal covering X→Y between compact Riemann surfaces X and Y. As a consequence we obtain classical results due to Maclachlan (1971) [5] and Martens (1977) [6].     

Abel maps for curves of compact type

Recently, the first Abel map for a stable curve of genus g≥2 has been constructed. Fix an integer d≥1 and let C be a stable curve of compact type of genus g≥2. We construct two d-th Abel maps for C, having different targets, and we compare the fibers of the two maps. As an application, we get a characterization of hyperelliptic stable curves of compact type with two components via the second Abel map.     

On unramified normal coverings of hyperelliptic curves

It is well known that the number of unramified normal coverings of an irreducible complex algebraic curve C with a group of covering transformations isomorphic to Z₂⊕Z₂ is (2^4g−3⋅2^2g+2)/6. Assume that C is hyperelliptic, say . Horiouchi has given the explicit algebraic equations of the subset of those covers which turn out to be hyperelliptic themselves. There are of this particular type. In this article, we provide algebraic equations for the remaining ones.     

The full automorphism groups of hyperelliptic Riemann surfaces

For every integer g≥2 we obtain the complete list of groups acting as the full automorphisms groups on hyperelliptic Riemann surfaces of genus g. Partially supported by DGICYT PB 89-201 and Science Plan 910021 Partially supported by DGICYT PB 89/379/C02/01 and Science Plan 910021 Partially supported by DGICYT After the typing of this paper we have heard about a Ph.D. Thesis by Britta Krapp on questions related to the problem studied here.     

Topologically unique maximal elementary abelian group actions on compact oriented surfaces

We determine all finite maximal elementary abelian group actions on compact oriented surfaces of genus σ≥2 which are unique up to topological equivalence. For certain special classes of such actions, we determine group extensions which also define unique actions. In addition, we explore in detail one of the families of such surfaces considered as compact Riemann surfaces and tackle the classical problem of constructing defining equations.     

Scaling group flow and Lefschetz trace formula for laminated spaces with p-adic transversal

In his approach to analytic number theory C. Deninger has suggested that to the Riemann zeta function (resp. the zeta function ζY(s) of a smooth projective curve Y over a finite field Fq, q=pf)) one could possibly associate a foliated Riemannian laminated space (SQ,F,g,?t) (resp. (SY,F,g,?t)) endowed with an action of a flow ?t whose primitive compact orbits should correspond to the primes of Q (resp. Y). Precise conjectures were stated in our report [E. Leichtnam, An invitation to Deninger's work on arithmetic zeta functions, in: Geometry, Spectral Theory, Groups, and Dynamics, in: Contemp. Math. vol. 387, Amer. Math. Soc., Providence, RI, 2005, pp. 201-236] on Deninger's work. The existence of such a foliated space and flow ?t is still unknown except when Y is an elliptic curve (see Deninger [C. Deninger, On the nature of explicit formulas in analytic number theory, a simple example, in: Number Theoretic Methods, Iizuka, 2001, in: Dev. Math., vol. 8, Kluwer Acad. Publ., Dordrecht, 2002, pp. 97-118]). Being motivated by this latter case, we introduce a class of foliated laminated spaces () where L is locally , D being an open disk of C. Assuming that the leafwise harmonic forms on L are locally constant transversally, we prove a Lefschetz trace formula for the flow ?t acting on the leafwise Hodge cohomology (0?j?2) of (S,F) that is very similar to the explicit formula for the zeta function of a (general) smooth curve over Fq. We also prove that the eigenvalues of the infinitesimal generator of the action of ?t on have real part equal to .Moreover, we suggest in a precise way that the flow ?t should be induced by a renormalization group flow “à la K. Wilson”. We show that when Y is an elliptic curve over Fq this is indeed the case. It would be very interesting to establish a precise connection between our results and those of Connes (p. 553 in [A. Connes, Noncommutative Geometry Year 2000, in: Special Volume GAFA 2000 Part II, pp. 481-559], p. 90 in [A. Connes, Symétries Galoisiennes et Renormalisation, in: Séminaire Bourbaphy, Octobre 2002, pp. 75-91]) and Connes-Marcolli [A. Connes, M. Marcolli, Q-lattices: quantum statistical mechanics and Galois theory, in: Frontiers in Number Theory, Physics and Geometry, vol. I, Springer-Verlag, 2006, pp. 269-350; A. Connes, M. Marcolli, From physics to number theory via noncommutative geometry. Part II: renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory, in: Frontiers in Number Theory, Physics and Geometry, vol. II, Springer-Verlag, 2006, pp. 617-713] on the Galois interpretation of the renormalization group.     

Signatures of foliated surface bundles and the symplectomorphism groups of surfaces

For any closed oriented surface Σ_g of genus g?3, we prove the existence of foliatedΣ_g-bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form ω on the fiber. We relate the cohomology class represented by the transverse symplectic form to a crossed homomorphism which is an extension of the flux homomorphism from the identity component to the whole group of symplectomorphisms of Σ_g with respect to the symplectic form ω.     

Serre-Taubes duality for pseudoholomorphic curves

According to Taubes, the Gromov invariants of a symplectic four-manifold X with b₊>1 satisfy the duality Gr(α)=±Gr(κ−α), where κ is Poincaré dual to the canonical class. Extending joint work with Simon Donaldson, we interpret this result in terms of Serre duality on the fibres of a Lefschetz pencil on X, by proving an analogous symmetry for invariants counting sections of associated bundles of symmetric products. Using similar methods, we give a new proof of an existence theorem for symplectic surfaces in four-manifolds with b₊=1 and b₁=0. This reproves another theorem due to Taubes: two symplectic homology projective planes with negative canonical class and equal volume are symplectomorphic.     

Modifying a branched surface to carry a foliation

We consider C¹ nonsingular flows on a closed 3-manifold under which there is no transverse disk that flows continuously back into its own interior. We provide an algorithm for modifying any branched surface transverse to such a flow ? that terminates in a branched surface carrying a foliation F precisely when F is transverse to ?. As a corollary, we find branched surfaces that do not carry foliations but that lift to ones that do.     

On holomorphic principal bundles over a compact Riemann surface admitting a flat connection

Let G be a connected reductive linear algebraic group over , and X a compact connected Riemann surface. Let be a Levi factor of some parabolic subgroup of G, with its maximal abelian quotient. We prove that a holomorphic G-bundle over X admits a flat connection if and only if for every such L and every reduction of the structure group of to L, the -bundle obtained by extending the structure group of is topologically trivial. For , this is a well-known result of A. Weil. Received: 1 December 2000 / Revised version: 2 April 2001 / Published online: 24 September 2001