Stephen S. Kudla

A Regularized Siegel-Weil Formula: The First Term Identity

A process for the conversion of H2S to SO2 in a feed gas containing H2S is effected by oxidation with air or oxygen at temperatures between 300 DEG and 900 DEG F. The oxidation is conducted in the presence of an extremely stable oxidation catalyst comprising an oxide and/or sulfide of vanadium supported on a non-alkaline porous refractory oxide. The preferred catalyst comprises between 5 and 15 wt. % V2O5 on hydrogen mordenite or alumina. Hydrogen, carbon monoxide, light hydrocarbons, and ammonia present in the feed gas are not oxidized. The invention is especially contemplated for use in treating waste gases from geothermal steam power plants.

Splitting metaplectic covers of dual reductive pairs

AbstractThe symplectic group Sp(N, F) over a local fieldF (other than ℂ) has a unique non-trivial twofold central extension. The inclusion of {±1} into the circle ℂ1 induces an extension

Intersection numbers of cycles on locally symmetric spaces and fourier coefficients of holomorphic modular forms in several complex variables

1 \to C^1 \to Mp(n,F) \to Sp(n,F) \to

Degenerate principal series and invariant distributions

. In this paper, an explicit splitting of the restriction of this extension to a dual reductive pair (G, H) in Sp(n, F) is given in all cases in which it exists. Such an explicit splitting is often an essential technical ingredient in the study of the local theta correspondence for the dual pair [4].

An introduction to the Langlands program

Some recent work of Gross and Prasad [14] suggests that the root numbers attached to certain symplectic representations of the Weil-Deligne group of a local field F control certain branching rules for representations of orthogonal groups over F . On a global level, this local phenomenon should have implications for the structure of the value at the center of symmetry for certain L-functions of arithmetic interest [12]. This conjectural picture is based, to some extent, on the now classic work of Tunnell [47] and Waldspurger [49] as well as on the case of the triple product L-function [13, 34, 17]. In all of these examples, the local root number detects the existence of a certain type of invariant linear functional. It is possible to set up analogous conjectures for the isometry groups of Hermitian or quaternion-hermitian forms. It turns out that root numbers also play a role in the local and global theta correspondence. Roughly speaking, certain local root numbers should control the occurrence of representations in the local theta correspondence between groups of the ‘same size’, e.g., for dual pairs of the form (Sp(n), O(2n+1)) and (U(n), U(n)). As will be seen, the theta correspondence for such pairs is connected with a certain induced representation In(s, χ) at the point s = 0 on the unitary axis. By contrast, the correspondence for the pairs (Sp(n), O(2n)) and (Sp(n), O(2n+ 2)), discussed by Prasad [35], is connected to the behavior of a similar induced representation at the points ± 12 , in which case no epsilon factor arises. In this paper we consider the local theta correspondence for unitary groups in the non-archimedean case. Let F be a non-archimedean local field of characteristic not equal to 2, and let E be a quadratic extension of F . For further notation, see the notation section at the end of this introduction. Let V and W be E vector spaces of dimensions m and n equipped, respectively, with a Hermitian form ( , ) : V × V −→ E and a skew-Hermitian form 〈 , 〉 : W ×W −→ E. Then the isometry groups G(V ) and G(W ) of the spaces V and W form a dual reductive pair in the symplectic group Sp(W), where W = V ⊗E W , viewed as an F vector space of dimension 2mn and equipped with the symplectic form

Derivatives of Eisenstein series and Faltings heights

Using the theta correspondence we construct liftings from the cohomology with compact supports of locally symmetric spaces associated to O(p, q) (resp. U(p, q)) of degreenq (resp. Hodge typenq, nq) to the space of classical holomorphic Siegel modular forms of weight (p +q)/2 and genusn (resp. holomorphic hermitian modular forms of weightp +q and genusn). It is important to note that the cohomology with compact supports contains the cuspidal harmonic forms by Borel [3]. We can express the Fourier coefficients of the lift of η in terms of periods of η over certain totally geodesic cycles—generalizing Shintani’s solution [21] of a conjecture of Shimura. We then choose η to be the Poincaré dual of a (finite) cycle and obtain a collection of formulas analogous to those of Hirzebruch-Zagier [8]. In our previous work we constructed the above lifting but we were unable to prove that it took values in theholomorphic forms. Moreover, we were unable to compute the indefinite Fourier coefficients of a lifted class. By Koecher’s Theorem we may now conclude that all such coefficients are zero.

Ramified degenerate principal series representations forSp(n)

In his Inventiones papers in 1995 and 1998, Borcherds constructed holomorphic automorphic forms Ψ(f) with product expansions on bounded domains D associated to rational quadratic spaces V of signature (n2), starting from vector valued modular forms f of weight 1 − n2 for SL2(Z) which are allowed to have poles at the cusp and whose nonpositive Fourier coefficients are integers cμ(−m), m ≥ 0. In this paper, we use the Siegel–Weil formula to give an explicit formula for the integral κ(Ψ(f)) of − log||Ψ(f)||2 over X = Γ\D, where || ||2 is the Petersson norm. This integral is given by a sum for m0 of quantities cμ(−m)κμ(m), where κμ(m) is the limit as Im(τ) → ∞ of the mth Fourier coefficient of the second term in the Laurent expansion at s = n2 of a certain Eisenstein series E(τs) of weight (n2) + 1 attached to V. The possible role played by the quantity κ(Ψ(f)) in the Arakelov theory of the divisors Zμ(m) on X is explained in the last section.

Special Cycles and Derivatives of Eisenstein Series

In this article we give a description of the tempered distributions on a matrix spaceMm,n(R) which are invariant under the linear action of an orthogonal groupO(p, q),p+q=m. We also determine the points of reducibility of the degenerate principal series of the metaplectic group Mp(n,R) induced from a character of the maximal parabolic with GL(n,R) as Levi factor. Finally, we identify the representation of MP(n,R) which is associated to the trivial representation ofO(p, q) under the archimedean theta correspondence.