Toroidal automorphic forms for function fields

The space of toroidal automorphic forms was introduced by Zagier in 1979. Let F be a global field. An automorphic form on GL(2) is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems from the fact (amongst others) that an Eisenstein series of weight s is toroidal if s is a non-trivial zero of the zeta function, and thus a connection with the Riemann hypothesis is established. In this paper, we concentrate on the function field case. We show the following results. The (n ^?1)-th derivative of a non-trivial Eisenstein series of weight s and Hecke character x is toroidal if and only if L(x, s+1/2) vanishes in s to order at least n (for the “only if” part we assume that the characteristic of F is odd). There are no non-trivial toroidal residues of Eisenstein series. The dimension of the space of derivatives of unramified Eisenstein series equals h(g ^?1)+1 if the characteristic is not 2; in characteristic 2, the dimension is bounded from below by this number. Here g is the genus and h is the class number of F. The space of toroidal automorphic forms is an admissible representation and every irreducible subquotient is tempered.     

On p-adic quaternionic Eisenstein series

We show that certain p-adic Eisenstein series for quaternionic modular groups of degree 2 become “real” modular forms of level p in almost all cases. To prove this, we introduce a U(p) type operator. We also show that there exists a p-adic Eisenstein series of the above type that has transcendental coefficients. Former examples of p-adic Eisenstein series for Siegel and Hermitian modular groups are both rational (i.e., algebraic).     

On the spectral expansion of hyperbolic Eisenstein series

In this article, we determine the spectral expansion, meromorphic continuation, and location of poles with identifiable singularities for the scalar-valued hyperbolic Eisenstein series. Similar to the form-valued hyperbolic Eisenstein series studied in Kudla and Millson (Invent Math 54:193–211, 1979), the scalar-valued hyperbolic Eisenstein series is defined for each primitive, hyperbolic conjugacy class within the uniformizing group associated to any finite volume hyperbolic Riemann surface. Going beyond the results in Kudla and Millson (Invent Math 54:193–211, 1979) and Risager (Int Math Res Not 41:2125–2146, 2004), we establish a precise spectral expansion for the hyperbolic Eisenstein series for any finite volume hyperbolic Riemann surface by first proving that the hyperbolic Eisenstein series is in L ². Our other results, such as meromorphic continuation and determination of singularities, are derived from the spectral expansion.     

Eisenstein series on quaternion half-space of degree N

In Kim [7], we studied an Eisenstein series on quaternion half-space of degree 2. By calculating the Siegel series using the method of Karel [5], we obtained the analytic continuation and functional equation of the Eisenstein series. In this note we study an Eisenstein series on quaternion half-space of degreen. By calculating the Siegel series in an analogous way as in Shimura [15] and Kitaoka [8], we obtain singular modular forms of weightk, k<2n and 4/k. Furthermore, we obtain the analytic continuation and functional equation of the Eisenstein series.     

On the construction of normal wildly ramified extensions over Q 2

It is well known that a finite totally ramified extension of a local field can be generated by a uniformising element the minimal polynomial of which is also Eisenstein. The quadratic and the quartic normal totally ramified extensions of Q ₂ are well known and well characterized. In this note we characterize the Eisenstein polynomials of degree 4 with coefficients in Z ₂ that define normal totally ramified extensions of Q ₂. Furthermore we give some necessary conditions for the cyclic case of degree 2 ⁿ . Also examples are given.     

An inner product formula on two-dimensional complex hyperbolic space

We study the Eisenstein series for a convex cocompact discrete subgroup on a two-dimensional complex hyperbolic space ℍ_ℂ². We find an inner product formula which gives the connection between Eisenstein series and automorphic Green functions on a two-dimensional complex hyperbolic space ℍ_ℂ². As an application of our inner product formula, we obtain the functional equations of Eisenstein series.     

Ramanujan's Eisenstein series and new hypergeometric-like series for

Using hypergeometric identities and certain representations for Eisenstein series, we uniformly derive several new series representations for 1/π².     

Spectral Scattering Theory for Automorphic Forms

We construct a scattering process for L²-automorphic forms on the quotient of the upper half plane by a cofinite discrete subgroup Γ of . The construction is algebraic besides being analytic in the sense that we use some relations satisfied by real-analytic Eisenstein series with a complex parameter. Thanks to this feature, the construction of our operators and spaces is explicit. We show some properties of the Lax-Phillips generator on a scattering subspace carved out from this process. We prove that the spectrum of this operator consists only of eigenvalues, which correspond to the nontrivial zeros, counted with multiplicity, of the Dirichlet series appearing in the functional equation of the Eisenstein series. In particular, in the case of the (full) modular group , the Dirichlet series reduces to the Riemann zeta function ζ, thereby we obtain a spectral interpretation of the nontrivial zeros of ζ.      

A note on the non-holomorphic Eisenstein series

We give a sufficient condition of bounded growth for the non-holomorphic Eisenstein series on SL ₂(ℤ). The C ^∞-automorphic forms of bounded growth are introduced by Sturm (Duke Math. J. 48(2), 327–350, 1981) in the study of automorphic L-functions. We also give a Laplace-Mellin transform of the Fourier coefficients of the Eisenstein series. The transformation constructs a projection of the Eisenstein series to the space of holomorphic cusp forms.      

Eisenstein series associated with Γ<Subscript>0</Subscript>(2)

In this paper, we define the normalized Eisenstein series ℘, e, and associated with Γ₀(2), and derive three differential equations satisfied by them from some trigonometric identities. By using these three formulas, we define a differential equation depending on the weights of modular forms on Γ₀(2) and then construct its modular solutions by using orthogonal polynomials and Gaussian hypergeometric series. We also construct a certain class of infinite series connected with the triangular numbers. Finally, we derive a combinatorial identity from a formula involving the triangular numbers.      

Some New Applications of the Subspace Theorem

We present some applications of the Subspace Theorem to the investigation of the arithmetic of the values of Laurent series f(z) at S-unit points. For instance we prove that if f(q ⁿ ) is an algebraic integer for infinitely many n, then h(f(q ⁿ )) must grow faster than n. By similar principles, we also prove diophantine results about power sums and transcendency results for lacunary series; these include as very special cases classical theorems of Mahler. Our arguments often appear to be independent of previous techniques in the context.     

The Chow Motive of a K3 Surface

In this note we present some results on the Chow motive h(X) of an algebraic surface X and relate them to the conjectures of Bloch, Beilinson and Murre. In particular we illustrate the relations between the finite-dimensionality of h(X) and the geometric properties of the surface. Then we focus on the case, where the conjectures are still open, of a complex K3 surface and prove some results which give some evidence to the finite-dimensionality of h(X).     

Rigid extensions of algebraic frames

An extension G ≤ H of lattice-ordered groups is said to be a rigid extension if for each ${h \in H}An extension G ≤ H of lattice-ordered groups is said to be a rigid extension if for each h ? H{h \in H} there exists a g ? G{g \in G} such that h ^⊥⊥ = g ^⊥⊥. In this paper, we will define rigid extensions and some other generalizations in the context of algebraic frames satisfying the FIP. One of the main results is a characterization of rigid extensions using d-elements of the frame. We also show that a rigid extension between two algebraic frames satisfying the FIP will induce a homeomorphism between their corresponding minimal prime spaces with respect to both the hull-kernel topology and the inverse topology. Moreover, basic open sets map to basic open sets.     

Solving the equation −uxx − ϵuyy = f(x,y, u) by an O(h4) finite difference method

The semi‐linear equation −u_xx − ϵu_yy = f(x, y, u) with Dirichlet boundary conditions is solved by an O(h⁴) finite difference method, which has local truncation error O(h²) at the mesh points neighboring the boundary and O(h⁴) at most interior mesh points. It is proved that the finite difference method is O(h⁴) uniformly convergent as h → 0. The method is considered in the form of a system of algebraic equations with a nine diagonal sparse matrix. The system of algebraic equations is solved by an implicit iterative method combined with Gauss elimination. A Mathematica module is designed for the purpose of testing and using the method. To illustrate the method, the equation of twisting a springy rod is solved. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 395–407, 2000     

Algebraic Order Bounded Disjointness Preserving Operators and Strongly Diagonal Operators

Let T be an order bounded disjointness preserving operator on an Archimedean vector lattice. The main result in this paper shows that T is algebraic if and only if there exist natural numbers m and n such that n ≥ m, and T^n!, when restricted to the vector sublattice generated by the range of T^m, is an algebraic orthomorphism. Moreover, n (respectively, m) can be chosen as the degree (respectively, the multiplicity of 0 as a root) of the minimal polynomial of T. In the process of proving this result, we define strongly diagonal operators and study algebraic order bounded disjointness preserving operators and locally algebraic orthomorphisms. In addition, we introduce a type of completeness on Archimedean vector lattices that is necessary and sufficient for locally algebraic orthomorphisms to coincide with algebraic orthomorphisms.     

Distribution of values of a real function means,moments, and symmetry

For a distribution functionD we define itsabsolute andsigned moments of orderk∈R, which generalise in a natural way the Hamburger moments of orders an even and an odd natural number. Similarly, for a real functionh we define itsabsolute andsigned asymptotic means of orderk∈R. We show that if the means exist on an infinite and bounded set of values ofk, then they exist on an intervalI and coincide onI ^o with the moments ofD=D _h, the distribution function of the values ofh, which is shown to exist (in the sense of Wintner). We also give a sufficient condition forD _h to be symmetric. These results apply to a class of functionsh that contain in particular error terms related to the Euler phi function and to the sigma divisor function. A further application on a certain class of converging trigonometrical series implies in particular classical results of A. Wintner establishing the existence for such functions of a distribution function as well as Hamburger moments of arbitrarily large orders. The remainder term of the prime number theorem belongs to this class provided the Riemann hypothesis holds, and the distribution function of its values is shown to be “almost” symmetric.     

On a conjecture of Hecke concerning elementary class number formulas

Let K be a totally real algebraic number field of class number h_K and L a totally imaginary quadratic extension of K of class number h_L. Hecke conjectured that there exists an elementary formula for the first factor h_L/h_K of the class number of L. The paper develops a theory which allows computation of h_L/h_K in terms of the periods of certain complex differential forms associated to a manifold defined in a natural way from K. Thus, Hecke's conjecture is reduced to the problem of finding elementary formulas for these periods. The essential idea of the proof consists of establishing a Kronecker limit formula for the non-analytic Eisenstein series for the Hilbert modular group for K.Research supported by NSF Grant GP 20538     

On the topological types of symmetries of elliptic-hyperelliptic Riemann surfaces

LetX be a Riemann surface of genusg. The surfaceX is called elliptic-hyperelliptic if it admits a conformal involutionh such that the orbit spaceX/〈h〉 has genus one. The involutionh is then called an elliptic-hyperelliptic involution. Ifg>5 then the involutionh is unique, see [A]. We call symmetry to any anticonformal involution ofX. LetAut ^±(X) be the group of conformal and anticonformal automorphisms ofX and letσ, τ be two symmetries ofX with fixed points and such that {σ, hσ} and {τ, hτ} are not conjugate inAut ^±(X). We describe all the possible topological conjugacy classes of {σ, σh, τ, τh}. As consequence of our study we obtain that, in the moduli space of complex algebraic curves of genusg (g even >5), the subspace whose elements are the elliptic-hyperelliptic real algebraic curves is not connected. This fact contrasts with the result in [Se]: the subspace whose elements are the hyperelliptic real algebraic curves is connected. The authors are supported by BFM2002-04801.     

p-adic Siegel–Eisenstein series of degree two

We prove an explicit formula for Fourier coefficients of Siegel–Eisenstein series of degree two with a primitive character of any conductor. Moreover, we prove that there exists the p-adic analytic family which consists of Siegel–Eisenstein series of degree two and a certain p-adic limit of Siegel–Eisenstein series of degree two is actually a Siegel–Eisenstein series of degree two.     

Algebraic Versus Homological Equivalence for Singular Varieties

Let Y ? ?^N be a possibly singular projective variety, defined over the field of complex numbers. Let X be the intersection of Y with h general hypersurfaces of sufficiently large degrees. Let d > 0 be an integer, and assume that dimY = n + h and dimY_sing ≤ min {d + h ? 1, n ? 1}. Let Z be an algebraic cycle on Y of dimension d + h, whose homology class in H_2(d+h)(Y; ?) is nonzero. In the present article, we prove that the restriction of Z to X is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case Y is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.