Volume 28, Issue 3

The aim of this paper is to derive explicitly a connection between the Zagier elliptic trilogarithm and Mahler measures of certain families of three-variable polynomials defining K3 surfaces. In addition, we prove some linear relations satisfied by the elliptic trilogarithm evaluated at torsion points on elliptic curves.

We describe a procedure for constructing morphisms in additive categories, combining Auslander's concept of a morphism determined by an object with the existence of flat covers. Also, we show how flat covers are turned into projective covers and we interpret these constructions in terms of adjoint functors.

We construct Graev ultrametrics on free products of groups with two-sided invariant ultrametrics and HNN extensions of such groups. We also introduce a notion of a free product of general Polish groups and prove, in particular, that two Polish groups G and H can be embedded into a Polish group T in such a way that the subgroup of T generated by G and H is isomorphic to the free product G * H .

In a previous article we had proved an algebraicity result for the central critical value for L -functions for GL n × GL n -1 over ℚ assuming the validity of a nonvanishing hypothesis involving archimedean integrals. The purpose of this article is to generalize that result for all critical values for L -functions for GL n × GL n -1 over any number field F while using certain period relations proved by Freydoon Shahidi and the author, and some additional inputs as will be explained below. Thanks to some recent work of Binyong Sun, the nonvanishing hypothesis has now been proved. The results of this article are unconditional. Applying this to GL 3 × GL 2 , new unconditional algebraicity results for the special values of symmetric cube L -functions for GL 2 over F have been proved. Previously, algebraicity results for the critical values of symmetric cube L -functions for GL 2 have been known only in special cases by the works of Garrett–Harris, Kim–Shahidi, Grobner–Raghuram, and Januszewski.

We settle the simple connectivity of the geometry opposite a chamber in a polar space of rank 3 by completing the job for the non-embeddable polar spaces and some polar spaces with small parameters.

We extend to p = 2 and p = 3 the result of Gillard and Schneps which states that the μ-invariant of Katz p -adic L -functions attached to imaginary quadratic fields along the Coates–Wiles ℤ p -extension is zero for p > 3. The arithmetic interpretation of this fact is given in the introduction.

We present a necessary and sufficient condition for an Einstein metric on a symplectic manifold ( M ,ω) to be ω-compatible. An immediate consequence of the solution of the Goldberg conjecture by Sekigawa is that there exists no Einstein metric with non-negative Ricci curvature satisfying our condition on compact symplectic manifolds that admit no Kähler structure.

The efficacy of using complexifications to understand the structure of real algebraic groups is demonstrated. In particular, the following two results are proved: (i) Let G be a connected solvable linear group whose eigenvalues are all real. If the complexification G ℂ of G is algebraic and operates algebraically on a complex variety V , and some G orbit in V is compact, then this orbit is a point. (ii) If L is a connected subgroup of a connected real linear semisimple group G such that the complexification L ℂ of L is algebraic and L ℂ contains a maximal torus of G ℂ , then L contains a maximal torus of G which complexifies to a maximal torus of G ℂ .

We consider nonlinear Neumann problems driven by a nonhomogeneous differential operator and an indefinite potential. In this paper we are concerned with two distinct cases. We first consider the case where the reaction is ( p -1)-sublinear near ±∞ and ( p -1)-superlinear near zero. In this setting the energy functional of the problem is coercive. In the second case, the reaction is ( p -1)-superlinear near ±∞ (without satisfying the Ambrosetti–Rabinowitz condition) and has a ( p -1)-sublinear growth near zero. Now, the energy functional is indefinite. For both cases we prove “three solutions theorems” and in the coercive setting we provide sign information for all of them. Our approach combines variational methods, truncation and perturbation techniques, and Morse theory (critical groups).

For many classes of models, there are universal members in any cardinal λ which “essentially satisfies GCH, i.e., λ = 2 <λ ,” in particular for the class of models of a complete first-order T (well, if at least λ > | T |). But if the class is “complicated enough”, e.g., the class of linear orders, we know that if λ is “regular and not so close to satisfying GCH”, then there is no universal member. Here, we find new sufficient conditions (which we call the olive property), not covered by earlier cases (i.e., fail the so-called SOP 4 ). The advantage of those conditions is witnessed by proving that the class of groups satisfies one of those conditions.

A basis of identities for an algebra is irredundant if each of its proper subsets fails to be a basis for the algebra. The first known examples of finite involution semigroups with infinite irredundant bases are exhibited. These involution semigroups satisfy several counterintuitive properties: their semigroup reducts do not have irredundant bases, they share reducts with some other finitely based involution semigroups, and they are direct products of finitely based involution semigroups.