Tits has defined Steinberg groups and Kac-Moody groups for any root system and any commutative ring R. We establish a Curtis-Tits-style presentation for the Steinberg group St of any rank ≥ 3 irreducible affine root system, for any R. Namely, St is the direct limit of the Steinberg groups coming from the 1and 2-node subdiagrams of the Dynkin diagram. This leads to a completely explicit presenta...

In a 1983 paper [M1], I. G. Macdonald introduced his well-known “constant term conjectures.” These conjectures concern a certain polynomial ∆ = ∆(G, k) that is indexed by a semisimple Lie algebra G and a positive integer k. The polynomial ∆ lives in Z[Φ, q], the group ring of the root lattice Φ of G over Z[q]. A basis for this ring, over Z[q], is the set of formal exponentials, ev, for v ∈ Φ th...

We classify complactly generated t-structures on the derived category of modules over a commutative Noetherian ring R in terms of decreasing filtrations by supports on Spec(R). A decreasing filtration by supports φ : Z → Spec(R) satisfies the weak Cousin condition if for any integer i the set φ(i) contains all the inmediate generalizations of each point in φ(i+ 1). Every t-structure on Dbfg(R) ...

Abstract Dedekind type DC sums and their generalizations are defined in terms of Euler functions generalization. Recently, Ma et al. (Adv. Differ. Equ. 2021:30 2021) introduced the poly-Dedekind by replacing function appearing sums, they were shown to satisfy a reciprocity relation. In this paper, we consider two kinds new sums. One is unipoly-Dedekind sum associated with 2 unipoly-Euler expres...

If R is a Dedekind domain, P a prime ideal of R and S ⊆R a finite subset then a P -ordering of S, as introduced by M. Bhargava in (J. Reine Angew. Math. 490:101–127, 1997), is an ordering {ai}i=1 of the elements of S with the property that, for each 1 < i ≤m, the choice of ai minimizes the P -adic valuation of ∏j<i(s− aj ) over elements s ∈ S. If S, S′ are two finite subsets of R of the same ca...

Dedekind symbols generalize the classical Dedekind sums (symbols). The symbols are determined uniquely by their reciprocity laws up to an additive constant. There is a natural isomorphism between the space of Dedekind symbols with polynomial (Laurent polynomial) reciprocity laws and the space of cusp (modular) forms. In this article we introduce Hecke operators on the space of weighted Dedekind...

The article presents an algorithm to compute a C[t]-module basis G for a given subalgebra A over a polynomial ring R = C[x] with a Euclidean domain C as the domain of coefficients and t a given element of A. The reduction modulo G allows a subalgebra membership test. The algorithm also works for more general rings R, in particular for a ring R ⊂ C((q)) with the property that f ∈ R is zero if an...

A Dedekind cut of an ordered abelian group G is a pair (X, Y) of nonempty subsets of G where Y=G−X and every member of X precedes every member of Y. A Dedekind cut (X, Y) is said to be continuous if X has a greatest member or Y has a least member, but not both; if every Dedekind cut of G is a continuous cut, G is said to be (Dedekind) continuous. The ordered abelian group R of real numbers is, ...

The subject of this thesis is the theory of nonholomorphic modular forms of non-integral weight, and its applications to arithmetical functions involving Dedekind sums and Kloosterman sums. As was discovered by Andre Weil, automorphic forms of non-integral weight correspond to invariant funtions on Metaplectic groups. We thus give an explicit description of Meptaplectic groups corresponding to ...