Orthogonality and Hecke operators

In this article we analyze orthogonality relations between old forms and the connection to the theory of Hecke operators.     

Hecke operators and an identity for the dedekind sums

If h, k ∈ Z, k > 0, the Dedekind sum is given by

$\text{s(h,k) =}\text{∑}\text{μ=1}\text{k}\text{μ}\text{k}\text{hμ}\text{k}$

, with

$\text{((x)) = x ? [x] ?}\text{1}\text{2}\text{, x?Z}$

,

$\text{=0 , x∈Z}$

. The Hecke operators T_n for the full modular group SL(2, Z) are applied to log η(τ) to derive the identities (n ∈ Z⁺)

$\text{∑ ∑ s(ah+bk,dk) = σ(n)s(h,k)}$

,

$\text{ad=n b(}\text{mod}\text{d)}$

$\text{d>0}$

where (h, k) = 1, k > 0 and σ(n) is the sum of the positive divisors of n. Petersson had earlier proved (1) under the additional assumption k ≡ 0, h ≡ 1 (mod n). Dedekind himself proved (1) when n is prime.     

On the traces of Hecke operators

From Eichler's results on traces of Hecke operators and making essential use of the theory of Atkin-Lehner on newforms, we derive in a simple manner an equality between the traces of Hecke operators acting on different spaces of cusp forms.     

Groups with an element permuting with finitely many of its conjugates

We study the problem as to which is the cardinality of connected components of the graph Γ_α, defined as follows. Let G be a group and a an element of G. The vertex set V(Γ_α) of the graph is the conjugacy class of elements,Cl _G(a), and two vertices x and y of the graph Γ_α are bridged by an edge iff x=y. If the intersectionC _G(a)∩Cl _G(a) is finite, Γ_α is locally finite. We prove that connected components of the locally finite graph Γ_α are finite in some classes of groups. Supported by RFFR grant No. 94-01-01084. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 543–551, September–October, 1996.     

Equivalence systems with finitely many relations

We consider a setX with a finite totally ordered setE of equivalence relations onX. We describe the automorphism group of this system, that is, the group of all those permutations ofX that leave each relation inE invariant.     

Decomposing Rd into finitely many semigroups

We determine the precise structure of those (additive) semigroups in R^d which belong to at least one partition of R^d into finitely many disjoint Borel measurable semigroups.We also find the structure of the convex sets in R^d which belong to some partition of R^d into finitely many disjoint convex sets.     

Groups with finitely many linear orders

It is proved that there exist orderable groups having exactly 6 and 14 distinct linear orders. For any natural number k, we construct examples of orderable groups on which 2(4k+3) linear orders are defined. Supported by RFFR grant No. 96-01-00088. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 176–200, March–April, 1999.