Automorphic Functions Research Articles

Uniformizing the boundaries of multiply connected quadrature domains using Fuchsian groups

This paper proposes a new method for uniformizing the boundaries of multiply connected quadrature domains by means of conformal maps that are automorphic functions invariant under the action of a Schottky group that is a Fuchsian group of the second kind. The conformal maps are expressed as ratios of Schottky–Klein prime functions. The result is significant because a classical infinite product formula for the Schottky–Klein prime function associated with Fuchsian groups of the second kind is known to be convergent.

Stepanov-like almost automorphic functions and applications to some semilinear equations

This article is concerned with the existence and uniqueness of an almost automorphic solution to the semilinear equation where is the infinitesimal generator of an exponentially stable C 0-semigroup on a Banach space and is Sp -almost automorphic for p > 1 and jointly continuous.

Method of automorphic functions in the study of flow around a stack of porous cylinders

This paper studies the ideal flow around a stack of stationary cylinders with porous walls. The boundary conditions on the surfaces of the cylinders are nonlinear. For small values of the porosity parameters, by applying the asymptotic method, the boundary conditions are linearized. The use of Mobius transformations generating a symmetric Schottky group reduces the problem to a Riemann-Hilbert boundary-value problem for symmetric automorphic functions. Its solution is found in a series form in terms of a quasiautomorphic analogue of the Cauchy integral. The absolute and uniform convergence of the series is guaranteed when the associated flow domain symmetric Schottky group is a first class group. An example of a symmetric Schottky group of divergent type (not of the first class) is given. Formulae for the drag and lift forces acting on the cylinders are derived, and the dependence of the porosity parameters on the forces is studied. In particular, the drag force is zero for a single solid cylinder (d'Alambert's paradox), while for a cylinder with a porous surface this is not true.

Riemann–Hilbert Problem for Automorphic Functions and the Schottky–Klein Prime Function

The construction of analogues of the Cauchy kernel is crucial for the solution of Riemann–Hilbert problems on compact Riemann surfaces. A formula for the Cauchy kernel can be given as an infinite sum over the elements of a Schottky group, and this sum is often used for the explicit evaluation of the kernel. In this paper a new formula for a quasi-automorphic analogue of the Cauchy kernel in terms of the Schottky–Klein prime function of the associated Schottky double is derived. This formula opens the door to finding new ways to evaluate the analogue of the Cauchy kernel in cases where the infinite sum over a Schottky group is not absolutely convergent. Application of this result to the solution of the Riemann–Hilbert problem with a discontinuous coefficient for symmetric automorphic functions is discussed.

Stepanov-like almost automorphic functions and monotone evolution equations

In this paper we are concerned with a (new) class of (Stepanov-like) almost automorphic ( S p -a.a.) functions with values in a Banach space X . This class contains the space AA ( X ) of all (Bochner) almost automorphic functions. We use the results obtained to prove the existence and uniqueness of a weak S p -a.a. solution to the parabolic equation u ′ ( t ) + A ( t ) u = f ( t ) in a reflexive Banach space, assuming some appropriate conditions of monotonicity, coercitivity of the operators A ( t ) and S p ′ -almost automorphy of the forced term f ( t ) . This result extends a known result in the case of almost periodicity. An application is also given.

The multiplicity-one theorem for generic automorphic forms of GSp(4)

We prove the multiplicity-one theorem for irreducible, globally generic, cuspidal, automorphic representations of GSp4(A) occurring in the space of cuspidal automorphic functions on GSp4(A), where A is the ring of adeles of a totally real number field k

Uniformization of triangle modular curves

In the present article, we determine explicit uniformizations of modular curves attached to triangle Fuchsian groups with cusps. Their Hauptmoduln are obtained by integration of non-linear differential equations of the third order. Series expansions involving integral coefficients are calculated around the cusps as wellas around the elliptic points. The method is an updated form of a differential construction of the elliptic modular function j, first performed by Dedekind in 1877. Subtle differences between automorphic functions with respect to conjugate Fuchsian groups become apparent.

Automorphic functions for the Whitehead-link-complement group

We construct automorphic functions on the real $3$-dimensional hyperbolic space $\mathbb{H}^3$ for the Whitehead-link-complement group $W\subset \mathit{GL}_2(\mathbb{Z}[i])$ and for a few groups commensurable with $W$. These automorphic functions give embeddings of the orbit spaces of $\mathbb{H}^3$ under these groups, and arithmetical characterizations of them.

Green currents for modular cycles in arithmetic quotients of complex hyperballs

We obtain a Green current in the sense of Gillet?Soule on an arithmetic quotient of a complex hyperball for the modular cycle stemming from a complex subhyperball of codimension greater than one, generalizing the classical construction of the automorphic Green function for the modular curves.

Almost automorphic solutions for semilinear boundary differential equations

In this work, we use the extrapolation methods to study the existence and uniqueness of almost automorphic solutions to the semilinear boundary differential equation { x ′ ( t ) a m p ; = A m x ( t ) + h ( t , x ( t ) ) , a m p ; a m p ; t ∈ R , L x ( t ) a m p ; = g ( t , x ( t ) ) , a m p ; a m p ; t ∈ R , \begin{equation} \tag {SBDE} \left \{ \begin {aligned} x’(t) &= A_mx(t)+h(t,x(t)), && t\in \mathbb {R}, \\ Lx(t) &= g(t,x(t)), && t\in \mathbb {R}, \end{aligned} \right . \end{equation} where A := A m | ker ⁡ L A:=A_m|\ker L generates a hyperbolic C 0 C_0 -semigroup on a Banach space X X and h , g h,g are almost automorphic functions which take values in X X and a “boundary space” ∂ X \partial X , respectively. These equations are an abstract formulation of partial differential equations with semilinear terms at the boundary, such as population equations, retarded differential equations and boundary control systems. An application to retarded differential equations is given.

On the almost automorphy of bounded solutions of differential equations with piecewise constant argument

In this paper we give sufficient spectral conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x ′ ( t ) = A x ( [ t ] ) + f ( t ) , t ∈ R , where A is a bounded linear operator in X and f is an X -valued almost automorphic function.

ELLIPTIC AND AUTOMORPHIC DYNAMICAL SYSTEMS ON SURFACES

We derive explicit differential equations for dynamical systems defined on generic surfaces applying elliptic and automorphic function theory to make uniform the surfaces in the upper half of the complex plane with the hyperbolic metric. By modifying the definition of the standard theta series we will determine general meromorphic systems on a fundamental domain in the upper half plane the solution trajectories of which "roll up" onto an appropriate surface of any given genus.

On Kubota's Dirichlet series

Kubota [T. Kubota, Some results concerning reciprocity law and real analytic automorphic functions, in: 1969 Number Theory Institute (Proc. Sympos. Pure Math. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math. Soc., Providence, R.I. (1971), 382–395.] showed how the theory of Eisenstein series on the higher metaplectic covers of SL2 (which he discovered) can be used to study the analytic properties of Dirichlet series formed with n-th order Gauss sums. In this paper we will prove a functional equation for such Dirichlet series in the precise form required by the companion paper [B. Brubaker, D. Bump, G. Chinta, S. Friedberg, and J. Hostein, Weyl group multiple Dirichlet series I, preprint, http://sporadic.stanford.edu/bump/wmd.pdf.]. Closely related results are in Eckhardt and Patterson [C. Eckhardt and S. J. Patterson, On the Fourier coefficients of biquadratic theta series, Proc. London Math. Soc. (3) 64(2) (1992), 225–264.].

The rankin-seiberg convolution for cohen’s eisenstein series of half integral weight

In this paper, we give a meromorphic continuation and a functional equation of the convolution product of certain Eisenstein series of half integral weight of one variable which were introduced by COHEN for a special case. We give also a precise description of the poles of the convolution product and their residues. These results have some applications to the Koecher-Maass series of Siegel-Eisenstein series. To prove our result, we use ZAGIER’S Rankin-Selberg method for automorphic functions which are not of rapid decay. In the final section, we present KUDLA’S version of ZAGIER’S Rankin-Selberg method. This is simpler than ZAGIER’S original one.

Almost automorphic solutions of second order evolution equations

This article is concerned with the existence of almost automorphic mild solutions to second order evolution equations of the form where A generates a strongly continuous semigroup and f is an almost automorphic function. Using the notion of uniform spectrum of a function and the method of sums of commuting operators in previous works for the case of bounded uniformly continuous solutions, we obtain sufficient conditions for the existence of almost automorphic mild solutions to (*) in terms of spectrum of A and uniform spectrum of f. Moreover, we study the nonlinear perturbation of this equation and obtain an extension of results by Diagana and N’Guérékata.

Almost Automorphic Functions in Frechet Spaces and Applications to Differential Equations

In this paper we first develop a theory of almost automorphic functions with values in Frechet spaces. Then, we consider the semilinear differential equation x'(t) = A x(t) + f(t, x(t)), t ∈ ℝ in a Frechet space X, where A is the infinitesimal generator of a C0-semigroup satisfying some conditions of exponential stability. Under suitable conditions on f, we prove the existence and uniqueness of an almost automorphic mild solution to the equation.

Reduction Groups and Automorphic Lie Algebras

We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic Lie algebras. For automorphic Lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly. These algebras have a useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts.

Reflection Operator and Canonical Basis

We construct a canonical basis of the Hilbert space of automorphic functions such that the invariance group of this basis has a nontrivial multiplier. Bibliography: 7 titles.

The twelfth moment of central values of Hecke series

Let H j ( s) be the Hecke L-function attached to the Maass wave form for the jth eigenvalue 1 4 +κ j 2 of the hyperbolic Laplacian acting in the Hilbert space of automorphic functions for the full modular group. The following mean value estimate for the central values H j( 1 2 ) is proved: ∑ κ j⩽K H j 12 1 2 ⪡K 4+ε.

Almost automorphic solutions of evolution equations

This paper is concerned with the existence of almost automorphic mild solutions to equations of the form (*) u(t) = Au(t) + f(t), where A generates a holomorphic semigroup and f is an almost automorphic function. Since almost automorphic functions may not be uniformly continuous, we introduce the notion of the uniform spectrum of a function. By modifying the method of sums of commuting operators used in previous works for the case of bounded uniformly continuous solutions, we obtain sufficient conditions for the existence of almost automorphic mild solutions to (*) in terms of the imaginary spectrum of A and the uniform spectrum of f.