Eisenstein series for SL(2)

This paper gives explicit formulas for the Fourier expansion of general Eisenstein series and local Whittaker functions over SL₂. They are used to compute both the value and derivatives of these functions at critical points.     

Some observations on the arithmetic of Eisenstein series for the modular group SL(2, \Bbb Z)

For even integers k\geqq4k\geqq4, let j_k(X)\varphi_k(X) be the separable rational polynomial that encodes the j-invariants of non-elliptic zeroes of the Eisenstein series E_k for the modular group SL(2,Bbb Z)(2,{Bbb Z}). We prove Kummer-type congruence properties for the j_k\varphi_k and, based on extensive calculations, make observations about the Galois group, the discriminant, and the distribution of zeroes of j_k(X)\varphi_k(X).     

Poincaré series in SL(3, R)

It is shown that in the spectral decomposition of the one-parameter Poincaré series from SL(3, R) involves only the discrete spectrum, induced from SL₂ and not the discrete spectrum of SL₃ itself. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 37–40, 1987.     

A separation property of the zeros of Eisenstein series for SL(2, Z)

Let E_k(z) be the Eisenstein series with weight k for the modulargroup SL(2, ). We prove that the zeros of E_k(eⁱ) interlace withthe zeros of E_k+12(eⁱ) on /2 < < 2/3. That is, any zeroof E_k(eⁱ) lies between two consecutive zeros of E_k+12(eⁱ) on/2 < < 2/3.     

Reducibility of the elementary representations for SL (n,R) and GL(n,R)

Functional Analysis and Its Applications -     

Is the group SL (6, ℤ) (2, 3)-generated?

The problem concerning the generation of the group SL(6, ℤ) by an involution and an element of order three is considered. The problem is reduced to the question of whether SL(6, ℤ) is generated by one of the eight explicitly written pairs of matrices. Bibliography: 14 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 101–130.     

On the analytic continuation of Eisenstein series for Siegel's modular group of degreen

For \(M = \left( {\begin{array}{*{20}c} {A B} \\ {C D} \\ \end{array} } \right)\) ∈ Γ(n)=Sp(n?) andZ=Z+iY,Y > 0, set $$M\left\langle Z \right\rangle = (AZ + B)(CZ + D)^{ - 1} = X_M + iY_M ;M\{ Z\} = CZ + D.$$ Denote with Γ _n (n) the subgroup defined byC=0. Forr∈? and a complex variable ω form the Eisenstein series $$E(n,r,Z,\omega ) = \sum\limits_{M\varepsilon I'_n (n)\backslash \Gamma (n)} {(DetM\{ Z\} )^{ - 2r} (DetY_M )^{\omega - r} } .$$ It is proved thatE(n, r, Z, ω) can be meromorphically continued to the ω-plane and satisfies a functional equation. Forr=1, 2, [(n?1)/2], [(n+1)/2] the functionE(n, r, Z, ω) is holomorphic at ω-r. For 3≤r≤[(n?3)/2] the functionE(n, r, Z, ω) may have poles at ω=r. But the pole-order is for two smaller than known until now. This result says especially that the Eisenstein series has Hecke summation forr=1, 2, [(n?1)/2], [(n+1)/2].     

A series of a discrete group and its application to the Dirichlet series associated with automorphic forms

Starting from the Poincaré-Selberg series, another series is defined for a discrete group; the relationship is established between this series and the Dirichlet series for which the coefficients are products of the Fourier coefficients of the automorphic forms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 63, pp. 3–7, 1976.     

On the space Ext for the group SL(2, q)

We consider the space Ext ^r (A,B) = Ext _KG ^r (A, B), where G = SL(2, q), q = p ⁿ , K is an algebraically closed field of characteristic p, A and B are irreducible KG-modules, and r ? 1. Carlson [6] described a basis of Ext ^r (A, B) in arithmetical terms. However, there are certain difficulties concerning the dimension of such a space. In the present article, we find the dimension of Ext ^r (A, B) for r = 1, 2 (in the above-mentioned article, Carlson presents the corresponding assertions without proofs; moreover, there are errors in their formulations). As a corollary, we find the dimension of the space H ^r (G, A), where A is an irreducible KG-module. This result can be used in studying nonsplit extensions of L ₂(q).     

Raghunathan’s conjectures for SL(2,R)

In this paper I give simple proofs of Raghunathan’s conjectures for SL(2,R). These proofs incorporate in a simplified form some of the ideas and methods I used to prove the Raghunathan’s conjectures for general connected Lie groups. Partially supported by the NSF Grant DMS-8701840.