We obtain an explicit expression for the number of ramified coverings of the sphere by the torus with given ramification type for a small number of ramification points, and conjecture this to be true for an arbitrary number of ramification points. In addition, the conjecture is proved for simple coverings of the sphere by the torus. We obtain corresponding expressions for surfaces of higher genera for a small number of ramification points, and conjecture the general form for this number in terms of a symmetric polynomial that appears to be new. The approach involves the analysis of the action of a transposition to derive a system of linear partial differential equations that give the generating series for the desired numbers.
An explicit expression is obtained for the generating series for the number of ramified coverings of the sphere by the torus, with elementary branch points and prescribed ramification type over infinity. This proves a conjecture of Goulden, Jackson and Vainshtein for the explicit number of such coverings.
An explicit expression is obtained for the generating series for the number of ramified coverings of the sphere by the double torus, with elementary branch points and prescribed ramification type over infinity. Thus we are able to prove a conjecture of Graber and Pandharipande, giving a linear recurrence equation for the number of these coverings with no ramification over infinity. The general form of the series is conjectured for the number of these coverings by a surface of arbitrary genus that is at least two.
To any connected reductive group G over a non-archimedean local field F (of characteristic p > 0) and to any maximal torus T of G, we attach a family of extended affine Deligne-Lusztig varieties (and families of torsors over them) over the residue field of F. This construction generalizes affine Deligne-Lusztig varieties of Rapoport, which are attached only to unramified tori of G. Via this construction, we can attach to any maximal torus T of G and any character of T a representation of G. This procedure should conjecturally realize the automorphic induction from T to G. For G = GL_2, we prov…
Let $φ: Y \rightarrow X$ be a finite surjective morphism between smooth complex projective curves, where $X$ is irreducible but $Y$ need not be so. Let $V_*$ be a parabolic vector bundle on $Y$. We construct a parabolic structure on the direct image $φ_* V$ on $X$, where $V$ is the vector bundle underlying $V_*$. The parabolic vector bundle $φ_* V_*$ on $X$ obtained this way has a ramified torus sub-bundle; it is a torus bundle of $\text{Ad}(φ_* V)$ outside the parabolic divisor for $φ_* V_*$ that satisfies certain conditions at the parabolic points. Conversely, given a parabolic vector bundle…
The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to obtain recurrence relations for certain numbers of ramified coverings of a sphere by a sphere with prescribed ramifications. Generating functions for these numbers belong to a very particular subalgebra of the algebra of power series.
A bracket is a function that assigns a number to each monomial in variables τ_0, τ_1, ... We show that any bracket satisfying the string and the dilaton relations gives rise to a power series lying in the algebra A generated by the series \sum n^{n-1} q^n/n! and \sum n^n q^n /n! . As a consequence, various series from A appear in the intersection theory of moduli spaces of curves. A connection between the counting of ramified coverings of the sphere and the intersection theory on moduli spaces allows us to prove that some natural generating functions enumerating the ramified coverings lie, yet…
We consider a resolution of ramified irregular singularities of meromorphic connections on a formal disk via local Fourier transforms. A necessary and sufficient condition for an irreducible connection to have a resolution of the ramified singularity is determined as an analogy of the blowing up of plane curve singularities. We also relate the irregularity of Komatsu and Malgrange of connections to the intersection numbers and the Milnor numbers of plane curve germs. Finally, we shall define an analogue of Puiseux characteristics for connections and find an invariant of the family of connectio…
We are interested in studying the variation of the Hitchin fibration in moduli spaces of parabolic Higgs bundles, under the action of a ramified covering. Given a degree two map $π$ : Y $\rightarrow$ X between compact Riemann surfaces, we may pull back a Higgs bundle from X to Y , the lifted Higgs bundle tends to have many apparent singularities, then we perform a suitable birational transformation in order to eliminate them. This correspondence preserves the Hitchin fibrations and then its restriction to a general fiber gives a map between Abelian varieties. The aim of this paper is to descri…
Let A be the algebra generated by the power series \sum n^{n-1} q^n/n! and \sum n^n q^n /n! . We prove that many natural generating functions lie in this algebra: those appearing in graph enumeration problems, in the intersection theory of moduli spaces M_{g,n} and in the enumeration of ramified coverings of the sphere. We argue that ramified coverings of the sphere with a large number of sheets provide a model of 2-dimensional gravity. Our results allow us to compute the asymptotic of the number of coverings as the number of sheets goes to infinity. The leading terms of such asymptotics are t…
An almost Belyi covering is an algebraic covering of the projective line, such that all ramified points except one simple ramified point lie above a set of 3 points of the projective line. In general, there are 1-dimensional families of these coverings with a fixed ramification pattern. (That is, Hurwitz spaces for these coverings are curves.) In this paper, three almost Belyi coverings of degrees 11, 12, and 20 are explicitly constructed. We demonstrate how these coverings can be used for computation of several algebraic solutions of the sixth Painleve equation.
The aim of these notes is to generalize Laumon's construction [18] of automorphic sheaves corresponding to local systems on a smooth, projective curve $C$ to the case of local systems with indecomposable unipotent ramification at a finite set of points. To this end we need an extension of the notion of parabolic structure on vector bundles to coherent sheaves. Once we have defined this, a lot of arguments from the article "On the geometric Langlands conjecture" by Frenkel, Gaitsgory and Vilonen [10] carry over to our situation. We show that our sheaves descend to the moduli space of parabolic …
We extend the modularity lifting result of the arXiv:1111.2804 to allow Galois representations with some ramification at p. We also prove modularity mod 2 and 5 of certain Galois representations. We use these results to prove many new cases of the strong Artin conjecture over totally real fields in which 5 is unramified. As an ingredient of the proof, we provide a general result on the automatic analytic continuation of overconvergent p-adic Hilbert modular forms of finite slope which substantially generalizes a similar result in arXiv:1111.2804.
We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $π: X \to A$, where $A$ is an abelian variety over $k$ with a dense set of $k$-rational points, we prove that there is a finite-index coset $C \subset A(k)$ such that $π(X(k))$ is disjoint from $C$. Our results do not seem to be in the range of other methods available at present; they confirm predictions coming from Lang's conjectures on rational points, and also go in the direction of an issue raised …
Local models are schemes defined in linear algebra terms that describe the 'etale local structure of integral models for Shimura varieties and other moduli spaces. We point out that the flatness conjecture of Rapoport-Zink on local models fails in the presence of ramification and that in that case one has to modify their definition. We study in detail certain modifications of the local models for G=R_{E/F}GL(n), with E/F a totally ramified extension, and for a maximal parahoric level subgroup. The special fibers of these models are subschemes of the affine Grassmannian. We show that the new lo…
The classical additive Deligne-Simpson problem is the existence problem for Fuchsian connections with residues at the singular points in specified adjoint orbits. Crawley-Boevey found the solution in 2003 by reinterpreting the problem in terms of quiver varieties. A more general version of this problem, solved by Hiroe, allows additional unramified irregular singularities. We apply the theory of fundamental and regular strata due to Bremer and Sage to formulate a version of the Deligne-Simpson problem in which certain ramified singularities are allowed. These allowed singular points are called…
We study ramified covers of the projective plane. Given a smooth projective surface S and a generic enough projection of S to the projective plane, we get a cover of the plane ramified over a plane curve. The branch curve is usually singular, but is classically known to have only cusps and nodes as singularities for a generic projection. Several questions arise. First, what is the_geography_ of branch curves among all nodal-cuspidal curves? Second: what is the_geometry_ of branch curves? In other words, how can one distinguish a branch curve from a non-branch curve with the same numerical inva…
Let $k$ be a complete nonarchimedean field and let $X$ be an affinoid closed disc over $k$. We classify the tamely ramified twisted forms of $X$. Generalizing work of P. Russell on inseparable forms of the affine line we construct explicit families of wildly ramified forms of $X$. We compute the class group and the Grothendieck group of forms of $X$ in certain cases.
In this paper, we give a simple description of the deformations of a map between two smooth curves with partially prescribed branching, in the cases that both curves are fixed, and that the source is allowed to vary. Both descriptions work equally well in the tame or wild case. We then apply this result to obtain a positive-characteristic Brill-Noether-type result for ramified maps from general curves to the projective line, which even holds for wild ramification indices. Lastly, in the special case of rational functions on the projective line, we examine what we can say as a result about fami…
We introduce notions of unramified and totally ramified maps in great generality - for commutative rings, schemes, ring spectra, or derived schemes. We prove that the definition is equivalent to the classical definition in the case of rings of integers in number fields. The new definition leads directly (without computational techniques) to a calculation of topological Hochschild homology for rings of integers. We show that THH(R) is the homotopy cofiber of a map $R[ΩS^3]\otimes_RΩ^1_{R/\mathbb{Z}}\to R[ΩS^3\langle 3\rangle]$, so there is a long exact sequence $\cdots\to H_\ast(ΩS^3;Ω^1_{R/\ma…
We revisit a statement of Birch that the field of moduli for a marked three-point ramified cover is a field of definition. Classical criteria due to Dèbes and Emsalem can be used to prove this statement in the presence of a smooth point, and in fact these results imply more generally that a marked curve descends to its field of moduli. We give a constructive version of their results, based on an algebraic version of the notion of branches of a morphism and allowing us to extend the aforementioned results to the wildly ramified case. Moreover, we give explicit counterexamples for singular curve…
We consider ramified coverings of P^1 with arbitrary ramification type over 0 and infinity and simple ramifications elsewhere and prove that the generating function for the numbers of such coverings is a tau-function for the Toda lattice hierarchy of Ueno and Takasaki.
In Part I, we extend our analysis in [arXiv:0807.1107], and show that a mathematically conjectured geometric Langlands duality for complex surfaces in [1], and its generalizations -- which relate some cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine (sub) G-algebras, where G is any compact Lie group -- can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. In Part II, to the setup in Part I, we introduce Ome…
