Bulletin of the London Mathematical Society最新文献

For a given group $�G$, Wolfgang Lück asked whether twisting a chain complex of finitely generated free $��C�\left[�G�\right]�$-modules with a finite-dimensional complex representation $�V$ of $�G$ before passing to the $�L^2$-completion has no other effect on $�L^2$-Betti numbers than a scaling by the factor $��{dim}_C�V�$. The purpose of the article is to answer this question affirmatively provided $�G$ is sofic.

This paper considers a specific autonomous Schwarzian differential equation given by

The paper studies the probability for a Galois group of a random polynomial to be $�A_n$. We focus on the so-called large box model, where we choose the coefficients of the polynomial independently and uniformly from $��\{�-�L�,�\text{…}�,�L�\}�$. The state-of-the-art upper bound is $��O�\left(�L^{�-�1�}�\right)�$, due to Bhargava. We conjecture a much stronger upper bound $�L^{�-�n�/�2�+�\epsilon �}$, and that this bound is essentially sharp. We prove strong lower bounds both on this probability and on the related probability of the discriminant being a square.

We show that over a perfect field, every non-semisimple finite tensor category with finitely generated cohomology embeds into a larger such category where the tensor product property does not hold for support varieties.

On a projective variety defined over a global field, any Brauer–Manin obstruction to the existence of rational points is captured by a finite subgroup of the Brauer group. We show that this subgroup can require arbitrarily many generators.

We call a non-trivial homology sphere a Kirby–Ramanujam sphere if it bounds both a homology plane and a Mazur or Poénaru manifold. In 1980, Kirby found the first example by proving that the boundary of the Ramanujam surface bounds a Mazur manifold and it has remained a single example since then. By tracing their initial step, we provide the first additional examples and we present three infinite families of Kirby–Ramanujam spheres. Also, we show that one of our families of Kirby–Ramanujam spheres is diffeomorphic to the splice of two certain families of Brieskorn spheres. Since this family of Kirby–Ramanujam spheres bound contractible 4-manifolds, they lie in the class of the trivial element in the homology cobordism group; however, both splice components are separately linearly independent in that group.

The description of (commutative and noncommutative) $�L_p$-isometries has been studied thoroughly since the seminal work of Banach. In the present paper, we provide a complete description for the limiting case, isometries on noncommutative $�L_0$-spaces, which extends the Banach–Stone theorem and Kadison's theorem for isometries of von Neumann algebras. The result is new even in the commutative setting.

We prove an arithmetic ‘path integral’ formula for the inverse $�p$-adic absolute values of Kubota–Leopoldt $�p$-adic $�L$-functions at roots of unity.

In this paper, we show the existence of a nontrivial weak solution for a nonlinear problem involving the fractional $�p$-Laplacian operator and a Berestycki–Lions type nonlinearity. This solution satisfies a Pohozaev identity. Moreover, we prove that any sufficiently smooth solution fulfills the Pohozaev identity.