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Let k be an algebraically closed non-Archimedean non-trivially real valued field which is complete with respect to its valuation. Let $$\phi :C' \rightarrow C$$ be a finite morphism between smooth projective irreducible k-curves. The morphism $$\phi $$ induces a morphism $$\phi ^{\mathrm {an}} :C'^{\mathrm {an}} \rightarrow C^{\mathrm {an}}$$ between the Berkovich analytifications of the curves. We construct a pair of deformation retractions of $$C'^{\mathrm {an}}$$ and $$C^{\mathrm {an}}$$ which are compatible with the morphism $$\phi ^{\mathrm {an}}$$ and whose images $$\Sigma _{C'^{\mathrm {\mathrm {an}}}},\Sigma _{C^{\mathrm {\mathrm {an}}}}$$ are closed subspaces of that are homeomorphic to finite metric graphs. We refer to such closed subspaces as skeleta. In addition, the subspaces $$\Sigma _{C'^{\mathrm {\mathrm {an}}}}$$ and $$\Sigma _{C^{\mathrm {\mathrm {an}}}}$$ are such that their complements in their respective analytifications decompose into the disjoint union of isomorphic copies of Berkovich open disks. The pair of compatible deformation retractions forces the morphism $$\phi ^{\mathrm {an}}$$ to restrict to a map $$\Sigma _{C'^{\mathrm {\mathrm {an}}}} \rightarrow \Sigma _{C^{\mathrm {\mathrm {an}}}}$$. The first Betti number of the skeleton C is well defined and an invariant of the curve which we call $$g^{\mathrm {an}}(C)$$. We study how the genus of $$\Sigma _{C'^{\mathrm {\mathrm {an}}}}$$ can be calculated using the morphism $$\phi ^{\mathrm {an}}_{|\Sigma _{C'^{\mathrm {\mathrm {an}}}}}$$ and invariants defined on $$\Sigma _{C^{\mathrm {an}}}$$ via the deformation retractions.

Zhang and Zheng (1990) conjectured on the structure of a solution for a two-dimensional Riemann problem for Euler equation. To resolve this illuminating conjecture, many researchers have studied the simplified 2 × 2 systems. In this paper, 3-pieces Riemann problem for two-dimensional 2 × 2 hyperbolic system is considered without the restriction that each jump of the initial data projects one planar elementary wave. We classify twelve topologically distinct solutions and construct analytical and numerical solutions. The computed numerical solutions clearly confirm the constructed analytic solutions.

In this paper we study harmonic maps from super Riemann surfaces in complex projective spaces and projective spaces associated with the super skew-field D . In both cases, we develop the theory of Gaus transforms and study the notion of isotropy, in particular its relation to holomorphic differentials on the super Riemann surface. Moreover, we give a definition of finite type harmonic maps for a special class of maps into C P n | n + 1 and thus obtain a classification for certain harmonic super tori. Furthermore, we investigate the equations satisfied by the underlying objects and give an example of a harmonic super torus in D P 2 whose underlying map is not harmonic.

Given a finite morphism $$\varphi :Y\rightarrow X$$ of quasi-smooth Berkovich curves over a complete, non-archimedean, nontrivially valued algebraically closed field k of characteristic 0, we prove a Riemann–Hurwitz formula relating their Euler–Poincare characteristics (calculated using De Rham cohomology of their overconvergent structure sheaf). The main tools are p-adic Runge’s theorem together with valuation polygons of analytic functions. Using the results obtained, we provide another point of view on Riemann–Hurwitz formula for finite morphisms of curves over algebraically closed fields of positive characteristic.

The criteria for continuous and homeomorphic extensions to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends by Carath´eodory are proved.

We describe the limit surfaces of some equisymmetric $1$-complex dimensional families of Riemann surfaces in the boundary of the Deligne-Mumford compactification of moduli space. We provide a description of such nodal Riemann surfaces in terms of the group of automorphisms defining the family. We apply our method to some known examples.

We suggest an approximate method to find an elliptic function uniformizing a compact Riemann surface of genus 1 which is given as a ramified covering of the Riemann sphere. The method is based on including the surface into a smooth one-parametric family. We deduce a system of ordinary differential equations for critical points of elliptic functions uniformizing surfaces of the family.

Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells having orthogonal diagonals. We discuss discrete coverings, discrete exterior calculus, and discrete Abelian integrals. Our presentation includes several new notions and results such as branched coverings of discrete Riemann surfaces, the discrete Riemann-Hurwitz Formula, double poles of discrete one-forms and double values of discrete meromorphic functions that enter the discrete Riemann-Roch Theorem, and a discrete Abel-Jacobi map., Comment: 35 pages, 9 figures. Minor corrections, new Figures 3, 5, 6, and 7

Text In this paper, we study the distribution of the a-values of the Selberg zeta-function attached to finite volume Riemann surfaces. We obtain a-value free regions, Weyl's law, and investigate the horizontal distribution of a-values. Video For a video summary of this paper, please visit https://youtu.be/cREe7owxYNo .

The original proof-argument of Riemann in 1851 for the Riemann mapping theorem, one of the most central theorems in Complex analysis, was found faulty and essentially buried underneath the proof by $Carath{\acute{e}}odory$ of 1929, now accepted as the "textbook" proof. On the other hand,...

We study the Riemann boundary-value problem on non-rectifiable curves for holomorphic matrices with Fokas–Its–Kitaev asymptotics by means of the Cauchy transforms of certain distributions with supports on that curves. The main results concern the existence of solutions of sufficiently large degree.

We disprove, by means of numerical examples and theoretical arguments, illustrated with p = 3 , the existence of a Riemann–Hurwitz formula for the p-ranks of relative class groups in a p-ramified p-extension K / k of number fields of CM-type containing μ p in contradiction with a result published in 1996. In the cyclic case of degree p, under some assumptions on the p-class group of k and the decomposition of the p-places, we prove some results on the structure of the p-class group of K and justify that some theoretical structures do not exist in this particular situation. In this context, an analogue of Kida's formula is valid for the p-ranks if and only if the p-class group of K is reduced to the group of ambiguous classes, which is not always the case, as shown by a numerical table for p = 3 .

A one-dimensional generalization of the Riemann–Hilbert problem from the Riemann sphere to an elliptic curve is considered. A criterion for its positive solvability is obtained and the explicit form of all possible solutions is found. As in the spherical case, the solutions turn out to be isomonodromic.

Given a function f : E → C ‾ from a closed subset of a Riemann surface R to the Riemann sphere C ‾ , we seek to approximate f in the spherical distance by functions meromorphic on R. As a consequence we generalize a recent extension of Mergelyan's theorem, due to Fragoulopoulou, Nestoridis and Papadoperakis [12] . The problem of approximating by meromorphic functions pole-free on E is equivalent to that of approximating by meromorphic functions zero-free on E, which in turn is related to Voronin's spectacular universality theorem for the Riemann zeta-function.

We present an example, different from Volterra's, of a bounded derivative that is not Riemann integrable. The existence of such functions was one of the motivations for Lebesgue to devise a stronge...

The present paper is devoted to the further development of the discrete theory of Riemann surfaces, which was started in the papers by M. Baker and S. Norine and their followers at the beginning of the century. This theory considers finite graphs as analogs of Riemann surfaces and branched coverings of graphs as holomorphic mappings. The genus of a graph is defined as the rank of its fundamental group. The main object of investigation in the paper is automorphism groups of a graph acting freely on the set of half-edges of the graph. These groups are discrete analogs of groups of conformal automorphisms of a Riemann surface. The celebrated Hurwitz theorem (1893) states that the order of the group of conformal automorphisms of a compact Riemann surface of genus g > 1 does not exceed 84(g — 1). Later, K. Oikawa and T. Arakawa refined this bound in the case of groups that fix several finite sets of prescribed cardinalities. This paper provides proofs of discrete versions of the mentioned theorems. In addition, a discrete analog of the E. Bujalance and G. Gromadzki theorem improving one of Arakawa’s results is obtained.

A general differential-algebraic approach is devised for constructing multi-component Hamiltonian operators as differentiations on suitably constructed loop Lie algebras. The related Novikov-Leibniz algebraic structures are presented and a new non-associative “Riemann” algebra is constructed, which is closely related to the infinite multi-component Riemann integrable hierarchies. A close relationship to the standard symplectic analysis techniques is also discussed.

Denote the set of all holomorphic mappings of a genus 3 Riemann surface S3 onto a genus 2 Riemann surface S2 by Hol(S3, S2). Call two mappings f and g in Hol(S3, S2) equivalent whenever there exist conformal automorphisms α and β of S3 and S2 respectively with f ◦ α = β ◦ g. It is known that Hol(S3, S2) always consists of at most two equivalence classes.

We solve the Riemann boundary value problem for holomorphic functions on a periodic non-rectifiable curve. Probably, this is the first study of the Riemann problem on infinite curve, which is not rectifiable. The solution is obtained in terms of generalized integrals over non-rectifiable path.

We give new proofs of effective versions of the Riemann mapping theorem, its extension to multiply connected domains and the uniformization on Riemann surfaces. Astonishingly, in the presented proofs, we need barely more than computational compactness and the classical results.

A theory of multiplicative functions and Prym differentials for the case of special characters on compact Riemann surfaces has found applications in geometrical function theory of complex variable, analytical number theory and in equations of mathematical physics. Theory of functions on compact Riemann surfaces differs from the theory of functions on finite Riemann surfaces even for the class of single meromorphic functions and Abelian differentials. In this article we continue the construction of the general function theory on finite Riemann surfaces for multiplicative meromorphic functions and differentials. We have proved analogues of the theorem on the full sum of residues for Prym differentials of every integral order and P. Appell's formula on expansion of the multiplicative function with poles of arbitrary multiplicity in the sum of elementary Prym integrals.

In this article, a new canonical function has been established to deal with Riemann boundary-value problem of periodic analytic functions discussed in [16]. In comparison with the corresponding result in [16], the expression of solution obtained here is much simpler. Then, we demonstrate the equivalence of solutions for the homogeneous Riemann problem. What's more, we obtain the precise rank of matrix of coefficients for the system of linear algebraic equations (4.35) in [16]. Those results can simplify the discussion of Riemann problem of single-periodic polyanalytic functions in [16].

Explicit examples of both hyperelliptic and non-hyperelliptic curves which cannot be defined over their field of moduli are known in the literature. In this paper, we construct a tower of explicit examples of such kind of curves. In that tower there are both hyperelliptic curves and non-hyperelliptic curves.

Given a finite Riemann surface of genus h ≥ 1 with boundary composed of m+1 connected components we consider a system of m+h real congruences analogous to the classical Jacobi inversion problem. We provide a solution to this system and its applications to boundary value problems.

We consider single-valued meromorphic functions and differentials on finite Riemann surfaces. We construct elementary q-differentials of all types. We find the dimension of two important quotient spaces of Abelian q-differentials, in particular, the first holomorphic de Rham cohomology group of 1-differentials. For these quotient spaces we construct bases that locally holomorphically depend on the moduli of the Riemann surfaces. Bibliography: 6 titles.

In the moduli space Mg of Riemann surfaces of genus g ≥ 2 there is important, so-called, real locus R_g, consisting of points representing Riemann surfaces having symmetries, by which we understand antiholomorphic involutions. R_g itself is covered by the strata R^k_g, each being formed by the points representing surfaces having a symmetry of given topological type k. These strata are known to be real analytic varieties of dimension 3(g-1). Also, their topological structure is pretty well known. Natanzon-Seppälä have realized that they are connected orbifolds homeomorphic to the factors of R^(3g-3) with respect to actions of some discrete groups and Goulden–Jackson–Harer and Harer–Zagier have found their Euler characteristics, expressing them through the Riemann zeta function. However, topological properties of the whole real locus R_g were less studied. The most known fact is its connectivity, proved independently by Buser-Seppälä-Silhol, Frediani and Costa–Izquierdo. This paper can be seen as a further contribution to the study of topology of R_g, which was possible through the notion of the nerve N_g, associated to R_g and called the real nerve. We find upper bounds for its geometrical and homological dimensions and we show their sharpness for infinitely many values of odd g. Precise values of these dimensions for even g have been found by the authors in an earlier paper.

It is shown, that the mapping class group of a surface of the genus g > 1 admits a faithful representation into the matrix group GL (6g-6, Z). The proof is based on a categorical correspondence between the Riemann surfaces and the so-called toric AF-algebras.