Let X be a connected based space and p be a two-regular prime number. If the fundamental group of X has order p, we compute the two-primary homotopy groups of the homotopy fiber of the trace map A(X) → TC(X) relating algebraic K-theory of spaces to topological cyclic homology. The proof uses a theorem of Dundas and an explicit calculation of the cyclotomic
trace map K(ℤ[Cp])→ TC(ℤ[Cp]). 相似文献
We prove that for every homogeneous and strongly locally homogeneous separable metrizable space there is a metrizable compactification of such that, among other things, for all there is a homeomorphism such that . This implies that is a coset space of some separable metrizable topological group .
We show that the resultants with respect to of certain linear forms in Chebyshev polynomials with argument are again linear forms in Chebyshev polynomials. Their coefficients and arguments are certain rational functions of the coefficients of the original forms. We apply this to establish several related results involving resultants and discriminants of polynomials, including certain self-reciprocal quadrinomials.
We give a new proof of a theorem of P. Mihailescu which states that the equation xp – yq = 1 is unsolvable with x, y integral and p, q odd primes, unless the congruences pqp (mod q2) and qpq (mod p2) hold. 相似文献
The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta function at odd negative integers, are uniformly distributed modulo for every . This is the basis of a well-known heuristic, given by Siegel, estimating the frequency of irregular primes. So far, analyses have shown that if is a real quadratic field, then the values of the zeta function at negative odd integers are also distributed as expected modulo for any . We use this heuristic to predict the computational time required to find quadratic analogues of irregular primes with a given order of magnitude. We also discuss alternative ways of collecting large amounts of data to test the heuristic.
In previous work, the author has extended the concept of regular and irregular primes to the setting of arbitrary totally real number fields , using the values of the zeta function at negative integers as our ``higher Bernoulli numbers'. In the case where is a real quadratic field, Siegel presented two formulas for calculating these zeta-values: one using entirely elementary methods and one which is derived from the theory of modular forms. (The author would like to thank Henri Cohen for suggesting an analysis of the second formula.) We briefly discuss several algorithms based on these formulas and compare the running time involved in using them to determine the index of -irregularity (more generally, ``quadratic irregularity') of a prime number.