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.     

Automorphic forms for elliptic function fields

Let F be the function field of an elliptic curve X over ${\mathbb{F}_q}$ . In this paper, we calculate explicit formulas for unramified Hecke operators acting on automorphic forms over F. We determine these formulas in the language of the graph of an Hecke operator, for which we use its interpretation in terms of ${\mathbb{P}^1}$ -bundles on X. This allows a purely geometric approach, which involves, amongst others, a classification of the ${\mathbb{P}^1}$ -bundles on X. We apply the computed formulas to calculate the dimension of the space of unramified cusp forms and the support of a cusp form. We show that a cuspidal Hecke eigenform does not vanish in the trivial ${\mathbb{P}^1}$ -bundle. Further, we determine the space of unramified F′-toroidal automorphic forms where F′ is the quadratic constant field extension of F. It does not contain non-trivial cusp forms. An investigation of zeros of certain Hecke L-series leads to the conclusion that the space of unramified toroidal automorphic forms is spanned by the Eisenstein series E( · , s) where s?+?1/2 is a zero of the zeta function of X—with one possible exception in the case that q is even and the class number h equals q?+?1.     

Spectral analysis and the Riemann hypothesis

The explicit formulas of Riemann and Guinand-Weil relate the set of prime numbers with the set of nontrivial zeros of the zeta function of Riemann. We recall Alain Connes’ spectral interpretation of the critical zeros of the Riemann zeta function as eigenvalues of the absorption spectrum of an unbounded operator in a suitable Hilbert space. We then give a spectral interpretation of the zeros of the Dedekind zeta function of an algebraic number field K of degree n in an automorphic setting.

If K is a complex quadratic field, the torical forms are the functions defined on the modular surface X, such that the sum of this function over the “Gauss set” of K is zero, and Eisenstein series provide such torical forms.

In the case of a general number field, one can associate to K a maximal torus T of the general linear group G. The torical forms are the functions defined on the modular variety X associated to G, such that the integral over the subvariety induced by T is zero. Alternately, the torical forms are the functions which are orthogonal to orbital series on X.

We show here that the Riemann hypothesis is equivalent to certain conditions bearing on spaces of torical forms, constructed from Eisenstein series, the torical wave packets. Furthermore, we define a Hilbert space and a self-adjoint operator on this space, whose spectrum equals the set of critical zeros of the Dedekind zeta function of K.     

Poincaré series of Drinfeld type

Let K be the rational function field $\mathbb{F}_q (t)$ . We construct Poincaré series on the Bruhat-Tits tree of GL₂ over K _∞ and show that they generate the space of automorphic cusp forms of Drinfeld type.     

A kernel for automorphic L-functions

The full multiple Dirichlet series of an automorphic cusp form is defined, in classical language, as a Dirichlet series of several complex variables over all the Fourier coefficients of the cusp form. It is different from the L-function of Godement and Jacquet, which is defined as a Dirichlet series in one complex variable over a one-dimensional array of the Fourier coefficients. In GL(2) and GL(3), the two notions are simply related. In this paper, we construct a kernel function that gives the full multiple Dirichlet series of automorphic cusp forms on GL(n,R). The kernel function is a new Poincaré series. Specifically, the inner product of a cusp form with this Poincaré series is the product of the full multiple Dirichlet series of the form times a function that is essentially the Mellin transform of Jacquet's Whittaker function. In the proof, the full multiple Dirichlet series is produced by applying the Lipschitz summation formula several times and by an integral which collapses the sum over SL(n−1,Z) in the Fourier expansion of the cusp form.     

Sur les zéros des séries d'Eisenstein dans le demi-plan de Drinfeld

Eisenstein series for GL₂(F_q[T]) of weight q^k − 1 have zeroes in the Drinfeld upper half-plane. Let F be a fundamental domain for the GL₂(A)-action. We determine the number of zeroes in F of these series. Our method is essentially based on an assocíation between Eisenstein series and some functions defined on the edges of the Bruhat-Tits tree.     

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.     

k-Hypermonogenic automorphic forms

In this paper we deal with monogenic and k-hypermonogenic automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. Monogenic automorphic forms are exactly the 0-hypermonogenic automorphic forms. In the first part we establish an explicit relation between k-hypermonogenic automorphic forms and Maaß wave forms. In particular, we show how one can construct from any arbitrary non-vanishing monogenic automorphic form a Clifford algebra valued Maaß wave form. In the second part of the paper we compute the Fourier expansion of the k-hypermonogenic Eisenstein series which provide us with the simplest non-vanishing examples of k-hypermonogenic automorphic forms.     

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.      

A spectral identity between symmetric spaces

Let π be a cuspidal automorphic representation ofGL _2n . We prove an identity between two spectral distributions onSp _2n andGL _2n respectively. The first is the spherical distribution with respect toSp _n×Sp _nof the residual Eisenstein series induced from π. The second is the weighted spherical distribution of π with respect toGL _n×GL _nand a certain degenerate Eisenstein series. A similar identity relates the pair (U _2n ,Sp _n) and (GL _n/E,GL _n/F) whereE/F is the quadratic extension defining the quasi-split unitary groupU _2n . We also have a Whittaker version of these trace identities. First-named author partially supported by NSF grant DMS 0070611. Second-named author partially supported by NSF grant DMS 9970342.     

The Eisenstein series for <Emphasis Type="Italic">GL</Emphasis>(3,<Emphasis Type="Bold">Z</Emphasis>) induced from cusp forms

We study the Eisenstein series for GL(3,Z) induced from cusp forms. We give the expression of the Fourier-Whittaker coefficients of the Eisenstein series in terms of the Jacquet integrals. Moreover, by evaluating the Jacquet integrals, we give the Mellin-Barnes type integral expressions of those at the minimal K-type.     

Bessel functions for GL 2

In classical analytic number theory there are several trace formulas or summation formulas for modular forms that involve integral transformations of test functions against classical Bessel functions. Two prominent such are the Kuznetsov trace formula and the Voronoi summation formula. With the paradigm shift from classical automorphic forms to automorphic representations, one is led to ask whether the Bessel functions that arise in the classical summation formulas have a representation theoretic interpretation. We introduce Bessel functions for representations of GL ₂ over a finite field first to develop their formal properties and introduce the idea that the γ-factor that appears in local functional equations for L-functions should be the Mellin transform of a Bessel function. We then proceed to Bessel functions for representations of GL ₂(?) and explain their occurrence in the Voronoi summation formula from this point of view. We briefly discuss Bessel functions for GL ₂ over a p-adic field and the relation between γ-factors and Bessel functions in that context. We conclude with a brief discussion of Bessel functions for other groups and their application to the question of stability of γ-factors under highly ramified twists.     

Theta lifting from elliptic cusp forms to automorphic forms on <Emphasis Type="Italic">Sp</Emphasis>(1, <Emphasis Type="Italic">q</Emphasis>)

Tsuneo Arakawa formulated a theta lifting from elliptic cusp forms to automorphic forms on Sp(1,q) in his unpublished note, which was inspired by “Kudla lifting”, i.e. a theta lifting from elliptic modular forms to holomorphic automorphic forms on SU(1,q). We prove that the images of Arakawa’s theta lifting belong to the space of bounded automorphic forms generating quaternionic discrete series, which are non-holomorphic forms. In the appendix we provide the construction of Eisenstein series and Poincaré series generating such discrete series. The author was partially supported by JSPS Research Fellowships for Young Scientist for April 2002 to March 2005. The results of this paper were obtained in this period.     

Archimedean zeta integrals on GL n × GL m and SO2n+1 × GL m

In this paper, we evaluate archimedean zeta integrals for automorphic L-functions on GL _n × GL _n-1+? and on SO_2n+1 × GL _n+? , for ? = ?1, 0, and 1. In each of these cases, the zeta integrals in question may be expressed as Mellin transforms of products of class one Whittaker functions. Here, we obtain explicit expressions for these Mellin transforms in terms of Gamma functions and Barnes integrals. When ? = 0 or ? = 1, the archimedean zeta integrals amount to integrals over the full torus. We show that, as has been predicted by Bump for such domains of integration, these zeta integrals are equal to the corresponding local L-factors—which are simple rational combinations of Gamma functions. (In the cases of GL _n × GL _n-1 and GL _n × GL _n this has, in large part, been shown previously by the second author of the present work, though the results here are more general in that they do not require the assumption of trivial central characters. Our techniques here are also quite different. New formulas for GL(n, R) Whittaker functions, obtained recently by the authors of this work, allow for substantially simplified computations). In the case ? = ?1, we express our archimedean zeta integrals explicitly in terms of Gamma functions and certain Barnes-type integrals. These evaluations rely on new recursive formulas, derived herein, for GL(n, R) Whittaker functions. Finally, we indicate an approach to certain unramified calculations, on SO_2n+1 × GL _n and SO_2n+1 × GL _n+1, that parallels our method herein for the corresponding archimedean situation. While the unramified theory has already been treated using more direct methods, we hope that the connections evoked herein might facilitate future archimedean computations.     

Power-moments of SL3(ℤ) Kloosterman sums

Classical Kloosterman sums have a prominent role in the study of automorphic forms on GL₂ and further they have numerous applications in analytic number theory. In recent years, various problems in analytic theory of automorphic forms on GL₃ have been considered, in which analogous GL₃-Kloosterman sums (related to the corresponding Bruhat decomposition) appear. In this note we investigate the first four power-moments of the Kloosterman sums associated with the group SL₃(?). We give formulas for the first three moments and a nontrivial bound for the fourth.     

Correspondance de Jacquet-Langlands pour les corps locaux de caractéristique non nulle

In this article we prove the Jacquet-Langlands local correspondence in non-zero characteristic. Let F be a local field of non-zero charactersitic and G′ an inner form of GL_n(F); then, following [17], we prove relations between the representation theory of G′ and the representation theory of an inner form of GL_n(L), where L is a local field of zero characteristic close to F. The proof of the Jacquet-Langlands correspondence between G′ and GL_n(F) is done using the above results and ideas from the proof by Deligne, Kazhdan and Vignéras [10] of the zero characteristic case. We also get the following, already known in zero characteristic: orthogonality relations for G′, inequality involving conductor and level for representations of G′ and finiteness for automorphic cuspidal representations with fixed component at almost every place for an inner form of GL_n over a global field of non-zero characteristic.     

Eisenstein series attached to lattices¶and modular forms on orthogonal groups

We study certain vector valued Eisenstein series on the metaplectic cover of SL₂(ℝ), which transform with the Weil representation associated with the discriminant group of an even lattice L. We find a closed formula for the Fourier coefficients in terms of Dirichlet L-series and representation numbers of L modulo “bad” primes. Such Eisenstein series naturally occur in the context of Borcherds' theory of automorphic products. We indicate some applications to modular forms on the orthogonal group of L with zeros on Heegner divisors. Received: 27 September 2001     

Kloosterman identities over a quadratic extension II

We prove certain identities between Kloosterman integrals. They constitute the fundamental lemma of a relative trace formula for Hecke functions. The main application of the trace formula in question is the following result. Let E/F be a quadratic extension of number fields. A cuspidal automorphic representation of GL(n,EA) is distinguished by some unitary group if and only if it is the base change of an automorphic cuspidal representation of GL(n,FA).     

Computation of Weng's rank 2 zeta function over an algebraic number field

In this paper, we study the zeta function, named non-abelian zeta function, defined by Lin Weng. We can represent Weng's rank r zeta function of an algebraic number field F as the integration of the Eisenstein series over the moduli space of the semi-stable OF-lattices with rank r. For r=2, in the case of F=Q, Weng proved that it can be written by the Riemann zeta function, and Lagarias and Suzuki proved that it satisfies the Riemann hypothesis. These results were generalized by the author to imaginary quadratic fields and by Lin Weng to general number fields. This paper presents proofs of both these results. It derives a formula (first found by Weng) for Weng's rank 2 zeta functions for general number fields, and then proves the Riemann hypothesis holds for such zeta functions.