We continue here with previous investigations on the global behavior of general type nonlinear wave equations for a class of small, scale-invariant initial data. In particular, we show that the (4 + 1) dimensional Yang-Mills equations are globally well posed with asymptotically free behavior for a wide class of initial data sets which include general charges. The method here is based on the use of a new set of Strichartz estimates for the linear wave equation which incorporates extra weighted smoothness assumptions with respect to the angular variable, along with the construction of appropriate micro-local function spaces which take into account this type of additional regularity.
(pages 611-664.) Abstract in TexWe give axiomatizations and prove quantifier elimination theorems for first-order theories of unramified valued fields with an automorphism having a close interaction with the valuation. We achieve an analogue of the classical Ostrowski theory of pseudoconvergence. In the outstanding case of Witt vectors with their Frobenius map, we use the ∂-ring formalism from Joyal.
(pages 665-721.) Abstract in TexWe prove that truncated, pseudo-Eisenstein series have compact support on the arithmetic quotient.
(pages 723-784.) Abstract in TexConsider a continuous one parameter family of circles in a complex plane that contains two circles lying in the exterior of one another. We prove that if a continuous function on the union of the circles extends holomorphically into each circle, then the function is holomorphic in the interior of the union of the circles.
(pages 785-790.) Abstract in TexWe study the relationship between the arithmetic and the spectrum of the Laplacian for manifolds arising from congruence arithmetic subgroups of SL(1,D), where D is an indefinite quaternion division algebra defined over a number field F.
We give new examples of isospectral but nonisometric compact, arithmetically defined varieties, generalizing the class of examples constructed by Vigneras. These examples are based on an interplay between the simply connected and adjoint group and depend explicitly on the failure of strong approximation for the adjoint group. The examples can be considered as a geometric analogue and also as an application of the concept and results on L-indistinguishability for SL(1,D) due to Labesse and Langlands.
We verify that the Hasse-Weil zeta functions are equal for the examples of isospectral pair of arithmetic varieties we construct giving further evidence for an archimedean analogue of Tate’s conjecture, which expects that the spectrum of the Laplacian determines the arithmetic of such spaces.
(pages 791-806.) Abstract in TexWe construct a family of rational functions on a Hilbert modular surface from the classical j-invariant and its Hecke translates. These functions are obtained by means of a multiplicative analogue of the Doi-Naganuma lifting and can be viewed as twisted Borcherds products. We then study when the value of at a CM point associated to a nonbiquadratic quartic CM field generates the “CM class field” of the reflex field. For the real quadratic field , we factorize the norm of some of these CM values to numerically.
(pages 807-841.) Abstract in TexThe wave equation ∂_{tt}ψ − Δψ − ψ^{5} = 0 in ℝ^{3} is known to exhibit finite time blowup for data of negative energy. Furthermore, it admits the special static solutions Φ(x, a) = (3a) ¼ (1 + a|x|^{2})−½ for all a > 0 which are linearly unstable. We view these functions as a curve in the energy space Ḣ^{1} ×L^{2}. We prove the existence of a family of perturbations of this curve that lead to global solutions possessing a well-defined long time asymptotic behavior as the sum of a bulk term plus a scattering term. Moreover, this family forms a co-dimension one manifold of small diameter in a suitable topology. Loosely speaking, acts as a center-stable manifold with the curve Φ(·, a) as an attractor in .
(pages 843-913.) Abstract in Tex