Annales Mathematiques du Quebec最新文献

Building on previous results [17, 35], we complete the classification of compact oriented Einstein 4-manifolds with (det (W^+) > 0). There are, up to diffeomorphism, exactly 15 manifolds that carry such metrics, and, on each of these manifolds, such metrics sweep out exactly one connected component of the corresponding Einstein moduli space.

In the early 90’s, Perrin-Riou (Ann Inst Fourier 43(4):945–995, 1993) introduced an important refinement of the Mazur–Swinnerton-Dyer p-adic L-function of an elliptic curve E over (mathbb {Q}), taking values in its p-adic de Rham cohomology. She then formulated a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for this p-adic L-function, in which the formal group logarithms of global points on E make an intriguing appearance. The present work extends Perrin-Riou’s construction to the setting of a Garret–Rankin triple product (f, g, h), where f is a cusp form of weight two attached to E and g and h are classical weight one cusp forms with inverse nebentype characters, corresponding to odd two-dimensional Artin representations (varrho _g) and (varrho _h) respectively. The resulting p-adic Birch and Swinnerton-Dyer conjecture involves the p-adic logarithms of global points on E defined over the field cut out by (varrho _gotimes varrho _h), in the style of the regulators that arise in Darmon et al. (Forum Math 3(e8):95, 2015), and recovers Perrin-Riou’s original conjecture when g and h are Eisenstein series.

We study the spectrum of the Laplacian on the hemisphere with Robin boundary conditions. It is found that the eigenvalues fall into small clusters close to the Neumann spectrum, and satisfy a Szegő type limit theorem. Sharp upper and lower bounds for the gaps between the Robin and Neumann eigenvalues are derived, showing in particular that these are unbounded. Further, it is shown that except for a systematic double multiplicity, there are no multiplicities in the spectrum as soon as the Robin parameter is positive, unlike the Neumann case which is highly degenerate. Finally, the limiting spacing distribution of the desymmetrized spectrum is proved to be the delta function at the origin.

We study how the (ell )-adic valuation of the number of spanning trees varies in regular abelian (ell )-towers of multigraphs. We show that for an infinite family of regular abelian (ell )-towers of bouquets, the (ell )-adic valuation of the number of spanning trees behaves similarly to the (ell )-adic valuation of the class numbers in ({mathbb {Z}}_{ell })-extensions of number fields.

Sommaire

The hyperbolic center of mass of a finite measure on the unit ball with respect to a radially increasing weight is shown to exist, be unique, and depend continuously on the measure. Prior results of this type are extended by characterizing the center of mass as the minimum point of an energy functional that is strictly convex along hyperbolic geodesics. A special case is Hersch’s center of mass lemma on the sphere, which follows from convexity of a logarithmic kernel introduced by Douady and Earle.

For a CM-field K and an odd prime number p, let (widetilde{K}') be a certain multiple (mathbb {Z}_p)-extension of K. In this paper, we study several basic properties of the unramified Iwasawa module (X_{widetilde{K}'}) of (widetilde{K}') as a (mathbb {Z}_pllbracket mathrm{Gal}(widetilde{K}'/K)rrbracket )-module. Our first main result is a description of the order of a Galois coinvariant of (X_{widetilde{K}'}) in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic (mathbb {Z}_p)-extension of K under a certain assumption on the splitting of primes above p. The second result is that if K is an imaginary quadratic field and if p does not split in K, then, under several assumptions on the Iwasawa (lambda )-invariant and the ideal class group of K, we determine a necessary and sufficient condition such that (X_{widetilde{K}}) is (mathbb {Z}_pllbracket mathrm{Gal}(widetilde{K}/K)rrbracket )-cyclic. Here, (widetilde{K}) is the (mathbb {Z}_p^2)-extension of K.