Journal of the London Mathematical Society-Second Series最新文献

We study the geometry and arithmetic of the curves $��C�:�y^3�=�x^4�+�a�x^2�+�b�$ and their associated Prym abelian surfaces $�P$. We prove a Torelli-type theorem in this context and give a geometric proof of the fact that $�P$ has quaternionic multiplication by the quaternion order of discriminant 6. This allows us to describe the Galois action on the geometric endomorphism algebra of $�P$. As an application, we classify the torsion subgroups of the Mordell–Weil groups $��P�\left(�Q�\right)�$, as both abelian groups and $��\mathrm{End}�\left(�P�\right)�$-modules.

Let $�p$ be a prime. We prove bounds on short Dirichlet character sums evaluated at a class of homogeneous polynomials in arbitrary dimensions. In every dimension, this bound is nontrivial for sums over boxes with side lengths as short as $�p^{�1�/�4�+�\kappa �}$ for any $��\kappa �>�0�$. Our methods capitalize on the relationship between characters mod $�p$ and characters over finite field extensions as well as bounds on the multiplicative energy of sets in products of finite fields.

We investigate Riemannian manifolds $��(�M^n�,�g�)�$ whose curvature operator of the second kind $�\mathring{R}$ satisfies the condition

The Carrollian fluid equations arise from the equations for relativistic fluids in the limit as the speed of light vanishes, and have recently experienced a surge of interest in the theoretical physics community in the context of asymptotic symmetries and flat-space holography. In this paper, we initiate the rigorous systematic analysis of these equations by studying them in one space dimension in the $�C^1$ setting. We begin by proposing a notion of isentropic Carrollian equations, and use this to reduce the Carrollian equations to a $��2�\times �2�$ system of conservation laws. Using the scheme of Lax, we then classify when $�C^1$ solutions to the isentropic Carrollian equations exist globally, or blow up in finite time. Our analysis assumes a Carrollian analogue of a constitutive relation for the Carrollian energy density, with exponent in the range $��\gamma �\in �(�1�,�3�]�$.

In the setting of length PI spaces satisfying a suitable deformation property, it is known that each isoperimetric set has an open representative. In this paper, we construct an example of a length PI space (without the deformation property) where an isoperimetric set does not have any representative whose topological interior is nonempty. Moreover, we provide a sufficient condition for the validity of the deformation property, consisting in an upper Laplacian bound for the squared distance functions from a point. Our result applies to essentially nonbranching $��\mathrm{MCP}�(�K�,�N�)�$ spaces, thus in particular to essentially nonbranching $��\mathrm{CD}�(�K�,�N�)�$ spaces and to many Carnot groups and sub-Riemannian manifolds. As a consequence, every isoperimetric set in an essentially nonbranching $��\mathrm{MCP}�(�K�,�N�)�$ space has an open representative, which is also bounded whenever a uniform lower bound on the volumes of unit balls is assumed.

We study representations of finite groups on Stanley–Reisner rings of simplicial complexes and on lattice points in lattice polytopes. The framework of translative group actions allows us to use the theory of proper colorings of simplicial complexes without requiring an explicit coloring to be given. We prove that the equivariant Hilbert series of a Cohen–Macaulay simplicial complex under a translative group action admits a rational expression whose numerator is a positive integer combination of irreducible characters. This implies an analogous rational expression for the equivariant Ehrhart series of a lattice polytope with a unimodular triangulation that is invariant under a translative group action. As an application, we study the equivariant Ehrhart series of alcoved polytopes in the sense of Lam and Postnikov and derive explicit results in the case of order polytopes and of Lipschitz poset polytopes.

In this work, we classify the thick subcategories of the bounded derived category of dg modules over a Koszul complex on any list of elements in a regular ring. This simultaneously recovers a theorem of Stevenson when the list of elements is a regular sequence and the classification of thick subcategories for an exterior algebra over a field (via the BGG correspondence). One of the major ingredients is a classification of thick tensor submodules of perfect curved dg modules over a graded commutative noetherian ring concentrated in even degrees, recovering a theorem of Hopkins and Neeman. We give several consequences of the classification result over a Koszul complex, one being that the lattice of thick subcategories of the bounded derived category is fixed by Grothendieck duality.