Unitary monodromy of Lamé differential operators

The classical second order Lamé equation contains a so-called accessory parameter B. In this paper we study for which values of B the Lamé equation has a monodromy group which is conjugate to a subgroup of SL(2,ℝ) (unitary monodromy with indefinite hermitian form). We reformulate the problem as a spectral problem and give an asymptotic expansion for the spectrum.      

A family of Schottky groups arising from the hypergeometric equation

We study a complex 3-dimensional family of classical Schottky groups of genus 2 as monodromy groups of the hypergeometric equation. We find non-trivial loops in the deformation space; these correspond to continuous integer-shifts of the parameters of the equation.

     

Monodromy of Rational KZ Equations and Some Related Topics

A special class of integrable Fuchsian systems on C ⁿ related to KZ equations is considered. We survey the construction of such systems and the list of the structural properties their monodromy representations. The relation of the Fuchsian systems obtained by the Veselov construction assosiated with a deformation of the A _n–1-type root system and the Gauss–Manin connection of the natural projection C ⁿ C ^n–1 is described. In this case, we prove that the monodromy representation is equivalent to the Burau representation of the Artin braid group. For a deformations of the other root system, we introduce generalized Burau representations. We conjecture that the integrable Fuchsian systems related to essential new finite sets of the vectors described by Veselov and Chalykh are the result of the Klares–Schlesinger isomonodromic deformations (or transformation) of the integrable Fuchsian system related to the Coxeter root systems.     

Variation of local systems and parabolic cohomology

Given a family of local systems on a punctured Riemann sphere, with moving singularities, its first parabolic cohomology is a local system on the base space. We study this situation from different points of view. For instance, we derive universal formulas for the monodromy of the resulting local system. We use a particular example of our construction to prove that the simple groups PSL₂(p ²) admit regular realizations over the field ℚ(t) for primesp≢ 1, 4, 16 mod 21.     

Construction et classification de certaines solutions algébriques des systèmes de Garnier

IsomonodromicdeformationsofFuchsianequationsoforder2onRiemann sphere are parameterized by the solutions of Garnier system. The purpose of this paper is to construct algebraic solutions exotic, i.e. corresponding to deformations of Fuchsian equation with Zariski dense monodromy. Specifically, we classify all the algebraic solutions (complete) exotic constructed by the method of pull-back of Doran-Kitaev: they are deduced from the data isomonodromic deformations pulling back a Fuchsian equation E given by a family of branched coverings ? _t . We first introduce the structures and associated orbifoldes underlying Fuchsian equation. This allows us to have are fined version of the Riemann Hurwitz formula that allows us quickly to show that E must be hypergeometric. Then we come to limit the degree of ? and exponents, and finally to Painlevé VI. We explicitly construct one of these solutions.     

On Schottky groups arising from the hypergeometric equation with imaginary exponents

In an article by Sasaki and Yoshida (2000), we encountered Schottky groups of genus 2 as monodromy groups of the hypergeometric equation with purely imaginary exponents. In this paper we study automorphic functions for these Schottky groups, and give a conjectural infinite product formula for the elliptic modular function .

     

SL(n+1,R)/S(GL(1,R)×GL(n,R))上线丛的一个超几何方程

本文研究了伪黎曼对称空间SL（n+1,R）/S（GL（1,R）×GL（n,R））线丛上的微分方程.利用李代数方法,即Casimir算子得到这个微分算子.这个微分算子是一个超几何方程,这个结论推广了文献[1,3,5]中的微分方程.     

On the mean square of standard L-functions attached to Ikeda lifts

By using estimates on the frequency of large values of the Riemann zeta-function and modular L-functions attached to the full modular group SL(2, ℤ), we prove sharp upper and lower estimates of the mean square of standard L-functions attached to Siegel cusp forms which are Ikeda lifts, on boundaries and the central line of the critical strip. The mean square of spinor L-functions attached to Saito-Kurokawa lifts is also studied.     

Linearizable Abel equations and the Gurevich–Pitaevskii problem

Applying symmetry reduction to a class of $\mathrm{SL}(2,\mathbb{R})$-invariant third-order ordinary differential equations (ODEs), we obtain Abel equations whose general solution can be parameterized by hypergeometric functions. Particular case of this construction provides a general parametric solution to the Kudashev equation, an ODE arising in the Gurevich–Pitaevskii problem, thus giving the first term of a large-time asymptotic expansion of its solution in the oscillatory (Whitham) zone.     

Transitive Factorizations in the Symmetric Group, and Combinatorial Aspects of Singularity Theory

We consider the determination of the number c_k(α) of ordered factorizations of an arbitrary permutation on n symbols, with cycle distribution α, intok -cycles such that the factorizations have minimal length and the group generated by the factors acts transitively on then symbols. The case k =  2 corresponds to the celebrated result of Hurwitz on the number of topologically distinct holomorphic functions on the 2-sphere that preserve a given number of elementary branch point singularities. In this case the monodromy group is the full symmetric group. For k =  3, the monodromy group is the alternating group, and this is another case that, in principle, is of considerable interest. We conjecture an explicit form, for arbitrary k, for the generating series for c_k(α), and prove that it holds for factorizations of permutations with one, two and three cycles (so α is a partition with at most three parts). Our approach is to determine a differential equation for the generating series from a combinatorial analysis of the creation and annihilation of cycles in products under the minimality condition.     

On the structure of generalized symmetric spaces of SLn𝔽q)

In this paper we extend previous results regarding SL₂(k) over any finite field k by investigating the structure of the symmetric spaces for the family of special linear groups SL_n(k) for any integer n>2. Specifically, we discuss the generalized and extended symmetric spaces of SL_n(k) for all conjugacy classes of involutions over a finite field of odd or even characteristic. We characterize the structure of these spaces and provide an explicit difference set in cases where the two spaces are not equal.     

Curve Fitting, Differential Equations And The Riemann Hypothesis

It is known that the Riemann zeta function ζ (s) in the critical strip 0 < Re(s) < 1, may be represented as the Mellin transform of a certain function φ (x) which is related to one of the theta functions. The function φ (x) satisfies a well known functional equation, and guided by this property we deduce a family of approximating functions involving an arbitrary parameter α. The approximating function corresponding to the value of α = 2 gives rise to a particularly accurate numerical approximation to the function φ (x). Another approximation to φ (x), which is based upon the first one, is obtained by solving a certain differential equation. Yet another approximating function may be determined as a simple extension of the first. All three approximations, when used in conjunction with the Mellin transform expression for ζ (s) in the critical strip, give rise to an explicit expression from which it is clear that Re(s) = 1/2 is a necessary and sufficient condition for the vanishing of the imaginary part of the integral, the real part of which is non-zero. Accordingly, the analogy with the Riemann hypothesis is only partial, but nevertheless Re(s) = 1/2 emerges from the analysis in a fairly explicit manner. While it is generally known that the imaginary part of the Mellin transform must vanish along Re(s) = 1/2, the major contribution of this paper is the presentation of the actual calculation for three functions which approximate φ (x). The explicit nature of these calculation details may facilitate progress towards the corresponding calculation for the actual φ (x), which may be necessary in a resolution of the Riemann hypothesis.2000 Mathematics Subject Classification: Primary—11M06, 11M26     

On reducible monodromy representations of some generalized Lamé equation

In this note, we compute the explicit formula of the monodromy data for a generalized Lamé equation when its monodromy is reducible but not completely reducible. We also solve the corresponding Riemann–Hilbert problem.

     

On the period matrix of a Riemann surface of large genus (with an Appendix by J.H. Conway and N.J.A. Sloane)

Summary Riemann showed that a period matrix of a compact Riemann surface of genusg1 satisfies certain relations. We give a further simple combinatorial property, related to the length of the shortest non-zero lattice vector, satisfied by such a period matrix, see (1.13). In particular, it is shown that for large genus the entire locus of Jacobians lies in a very small neighborhood of the boundary of the space of principally polarized abelian varieties.We apply this to the problem of congruence subgroups of arithmetic lattices in SL₂(). We show that, with the exception of a finite number of arithmetic lattices in SL₂(), every such lattice has a subgroup of index at most 2 which is noncongruence. A notable exception is the modular groupSL ₂().     

The Stokes phenomenon in the confluence of the hypergeometric equation using Riccati equation

In this paper we study the confluence of two regular singular points of the hypergeometric equation into an irregular one. We study the consequence of the divergence of solutions at the irregular singular point for the unfolded system. Our study covers a full neighborhood of the origin in the confluence parameter space. In particular, we show how the divergence of solutions at the irregular singular point explains the presence of logarithmic terms in the solutions at a regular singular point of the unfolded system. For this study, we consider values of the confluence parameter taken in two sectors covering the complex plane. In each sector, we study the monodromy of a first integral of a Riccati system related to the hypergeometric equation. Then, on each sector, we include the presence of logarithmic terms into a continuous phenomenon and view a Stokes multiplier related to a 1-summable solution as the limit of an obstruction that prevents a pair of eigenvectors of the monodromy operators, one at each singular point, to coincide.     

The Combinatorial Quantum Cohomology Ring of <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">B</Emphasis>

A purely combinatorial construction of the quantum cohomology ring of the generalized flag manifold is presented. We show that the ring we construct is commutative, associative and satisfies the usual grading condition. By using results of our previous papers [12, 13], we obtain a presentation of this ring in terms of generators and relations, and formulas for quantum Giambelli polynomials. We show that these polynomials satisfy a certain orthogonality property, which—for G = SL_n( )—was proved previously in the paper [5].     

The Subelliptic Heat Kernels on SL(2, ℝ) and on its Universal Covering \widetilde{\mathbf{SL}(2,\mathbb{R})}: Integral Representations and Some Functional Inequalities

In this paper, we study a subelliptic heat kernel on the Lie group SL(2, ℝ) and on its universal covering [(SL(2,\mathbbR))\tilde]\widetilde{\mathbf{SL}(2,\mathbb{R})}. The subelliptic structure on SL(2,ℝ) comes from the fibration SO(2)→SL(2,ℝ) →H ² and it can be lifted to [(SL(2,\mathbbR))\tilde]\widetilde{\mathbf{SL}(2,\mathbb{R})}. First, we derive an integral representation for these heat kernels. These expressions allow us to obtain some asymptotics in small times of the heat kernels and give us a way to compute the subriemannian distance. Then, we establish some gradient estimates and some functional inequalities like a Li-Yau type estimate and a reverse Poincaré inequality that are valid for both heat kernels.     

Numerical Schottky Uniformizations

A real algebraic curve of algebraic genus g ≥ 2 is a pair (S, τ), where S is a closed Riemann surface of genus g and τ is a reflection on S (anticonformal involution with fixed points). In this note, we discuss a numerical (Burnside) program which permits to obtain a Riemann period matrix of the surface S in terms of an uniformizing real Schottky group. If we denote by Aut₊(S, τ) the group of conformal automorphisms of S commuting with the real structure τ, then it is a well known fact that |Aut₊(S,τ)| ≥ 12(g−1). We say that (S,τ) is maximally symmetric if |Aut₊(S,τ)|=12(g−1). We work explicitly such a numerical program in the case of maximally symmetric real curves of genus two. We construct a real Schottky uniformization for each such real curve and we use the numerical program to obtain a real algebraic curve, a Riemann period matrix and the accessory parameters in terms of the corresponding Schottky uniformization. In particular, we are able to give for Bolza’s curve a Schottky uniformization (at least numerically), providing an example for which the inverse uniformization theorem is numerically solved.Partially supported by Projects Fondecyt 1030252 1030373 and UTFSM 12.03.21     

Isotrivially fibred isles in the moduli space of surfaces of general type

We complement Catanese's results on isotrivially fibred surfaces by completely describing the components containing an isotrivial surface with monodromy group . We also give an example for deformation equivalent isotrivial surfaces with different monodromy group.