Bulletin of the London Mathematical Society最新文献

For an elliptic curve with complex multiplication over a number field, the $�p^{\infty }$-Selmer rank is even for all $�p$. Česnavičius proved this using the fact that $�E$ admits a $�p$-isogeny whenever $�p$ splits in the complex multiplication field, and invoking known cases of the $�p$-parity conjecture. We give a direct proof, and generalise the result to abelian varieties.

We consider the inverse problem of determining coefficients appearing in semilinear elliptic equations stated on Riemannian manifolds with boundary given the knowledge of the associated Dirichlet-to-Neumann map. We begin with a negative answer to this problem. Owing to this obstruction, we consider a new formulation of our inverse problem in terms of a rigidity problem. Precisely, we consider cases where the Dirichlet-to-Neumann map of a semilinear equation coincides with the one of a linear equation and ask whether this implies that the equation must indeed be linear. We give a positive answer to this rigidity problem under some assumptions imposed on the Riemannian manifold and the semilinear term under consideration.

Let $�R$ be a commutative Noetherian ring. For a natural number $�k$, we prove that the class of modules of projective dimension bounded by $�k$ is of finite type if and only if $�R$ satisfies Serre's condition $��\left(�S_k�\right)�$. In particular, this answers positively a question of Bazzoni and Herbera in the specific setting of a Gorenstein ring. Applying similar techniques, we also show that the $�k$-dimensional version of the Govorov–Lazard theorem holds if and only if $�R$ satisfies the ‘almost’ Serre condition $��\left(�C_{�k�+�1�}�\right)�$.

Let $��P�\left(�x�\right)�\in �Z�\left[�x�\right]�$ be a polynomial with at least two distinct complex roots. We prove that the number of solutions $���(�x_1�,�\cdots �,�x_k�,�y_1�,�\cdots �,�y_k�)��\in �{�\left[�N�\right]�}^{�2�k�}�$ to the equation

Let $�V$ be a finite-dimensional vector space over the complex numbers and let $��G�⩽�SL�\left(�V�\right)�$ be a finite group. We describe the class group of a minimal model (i.e., $�Q$-factorial terminalization) of the linear quotient $��V�/�G�$. We prove that such a class group is completely controlled by the junior elements contained in $�G$.

Small noise problems are quite important for all types of stochastic differential equations. In this paper, we focus on rough differential equations driven by scaled fractional Brownian rough path with Hurst parameter $��H�\in �(�1�/�4�,�1�/�2�]�$. We prove a moderate deviation principle for this equation as the scale parameter tends to zero.

The existence of bound states for the magnetic Laplacian in unbounded domains can be quite challenging in the case of a homogeneous magnetic field. We provide an affirmative answer for almost flat corners and slightly curved half-planes when the total curvature of the boundary is positive.

We show the existence of a constant $��c�>�0�$ such that, for all positive integers $�n$, there exist integers $��1�\le �a_1�<�\cdots �<�a_k�\le �n�$ such that there are at least $��c�n^2�$ distinct integers of the form $��{\sum }_{�i�=�u�}^v�a_i�$ with $��1�\le �u�\le �v�\le �k�$. This answers a question of Erdős and Graham. We also prove a non-trivial upper bound on the maximum number of distinct integers of this form and address several open problems.

In the classical Geometry of Numbers, the calculation of successive minima may be quite difficult, even in $�R^2$ using the norm coming from a distance function associated to a set. In the literature, there seem to be hardly any analogues when $�R$ is replaced by the algebraic closure of a function field in one variable and one uses a norm arising from the absolute height. Here, we calculate a one-parameter family of examples that naturally arose in our recent paper on bounded heights. We also comment on whether the minima are attained.

We obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta-function averaged over a subfamily of zeros of the zeta function that is expected to be arbitrarily close to full density inside the set of all zeros. For $��k�⩽�1�/�2�$, our bounds for the $��2�k�$-th moments are expected to be almost optimal. Assuming a conjecture about the maximum size of the argument of the zeta function on the critical line, we obtain upper bounds for these negative moments of the same strength while summing over a larger subfamily of zeta zeros.