2.1. The Cebotarev density theorem. We have seen in the proof of Theorem 1.3 that for any nontrivial ray class character χ : H → C× we could show nonvanishing L(1, χ) = 0 provided that χ factors through J/NL/KJ L · P for some finite extension L/K. By Theorem 1.2 the subgroup P is the norm subgroup of the ray class field K(m)/K. Hence we get nonvanishing for all nontrivial χ and this allows to p...

Overview 1 1. Mymotivation: K-theory of schemes 2 2. First steps in homological algebra 3 3. The long exact sequence 6 4. Derived functors 8 5. Cohomology of sheaves 10 6. Cohomology of a Noetherian Affine Scheme 12 7. Čech cohomology of sheaves 12 8. The Cohomology of Projective Space 14 9. Sheaf cohomology on P̃2 16 10. Pushing around sheaves, especially by the Frobenius 17 11. A first look at...

Let N ≥ 2 and let 1 < a1 < · · · < aN be relatively prime integers. The Frobenius number of this N -tuple is defined to be the largest positive integer that has no representation as PN i=1 aixi where x1, ..., xN are nonnegative integers. More generally, the s-Frobenius number is defined to be the largest positive integer that has precisely s distinct representations like this. We use techniques...

In this paper a Galois representation will be a continuous representation ρ : GQ → GL(n,F) where F is either a topological field of characteristic 0 or a finite field. When the characteristic of F is not two, we say that ρ is odd if the image of complex conjugation is conjugate to a diagonal matrix with alternating 1’s and −1’s on the diagonal. If F has characteristic two, every Galois represen...

Let n ≥ 2 and s ≥ 1 be integers and a = (a1, . . . , an) be a relatively prime integer n-tuple. The s-Frobenius number of this ntuple, Fs(a), is defined to be the largest positive integer that cannot be represented as ∑n i=1 aixi in at least s different ways, where x1, ..., xn are non-negative integers. This natural generalization of the classical Frobenius number, F1(a), has been studied recen...

We present an extension of Perron–Frobenius theory to the spectra and numerical ranges of Perron polynomials, namely, matrix polynomials of the form L(λ) = Iλ − Am−1λm−1 − · · · − A1λ− A0, where the coefficient matrices are entrywise nonnegative. Our approach relies on the companion matrix linearization. First, we recount the generalization of the Perron–Frobenius Theorem to Perron polynomials ...

Our proof is built on Perron-Frobenius theorem, a seminal work in matrix theory (Meyer 2000). By Perron-Frobenius theorem, the power iteration algorithm for predicting top K persuaders converges to a unique C and this convergence is independent of the initialization of C if the persuasion probability matrix P is nonnegative, irreducible, and aperiodic (Heath 2002). We first show that P is nonne...

This paper corrects the statement and the proof of Theorem 1.5 of the paper quoted in the title (Represent. Theory 13 (2009), 427–459). Theorem 1.5 of our paper [1] requires a correction. Below we provide a new statement of this theorem and correct the proof. A mistake in the original proof of Theorem 1.5 is due to missing the multiple 2 at a certain point of the proof (see [1, page 456, line 2...

Recently the first writer [l] gave a characterization of quasiFrobenius rings, introduced formerly by the second writer [3], in terms of a condition proposed by K. Shoda, which reads: A ring A satisfying minimum condition and possessing a unit element is a quasi-Frobenius ring if and only if A satisfies the following condition:1 (a) every (A -left-) homomorphism of a left-ideal of A into A may ...

Let L/K be a Galois extension of number fields. The problem of counting the number of prime ideals p of K with fixed Frobenius class in Gal(L/K) and norm satisfying a congruence condition is considered. We show that the square of the error term arising from the Chebotarëv Density Theorem for this problem is small “on average.” The result may be viewed as a variation on the classical Barban-Dave...