Low dimensional strongly perfect lattices. III: Dual strongly perfect lattices of dimension 14

Low dimensional strongly perfect lattices IV: The dual strongly perfect lattices of dimension 16

We classify the dual strongly perfect lattices in dimension 16. There are four pairs of such lattices, the famous Barnes-Wall lattice $Λ_{16}$, the extremal 5-modular lattice $N_{16}$, the odd Barnes-Wall lattice $O_{16}$ and its dual, and one pair of new lattices $Γ_{16}$ and its dual. The latter pair belongs to a new infinite series of dual strongly perfect lattices, the sandwiched Barnes-Wall lattices, described by the authors in a previous paper. An updated table of all known strongly perfect lattices up to dimension 26 is available in the catalogue of lattices.

Perfect numbers and Fibonacci primes (III)

In this article, we consider the Diophantine equation $σ_{2}(n)-n^2=An+B$ with $A=P^2\pm2$. For some $B$, we show that except for finitely many computable solutions in the range $n\leq(|A|+|B|)^{3}$, all the solutions are expressible in terms of Lucas sequences. Meanwhile, we obtain some results relating to other linear recurrent sequences.

An integral model of the perfectoid modular curve

We construct an integral model of the perfectoid modular curve. Studying this object, we prove some vanishing results for the coherent cohomology at perfectoid level. We use a local duality theorem at finite level to compute duals for the coherent cohomology of the integral perfectoid curve. Specializing to the structural sheaf, we can describe the dual of the completed cohomology as the inverse limit of the integral cusp forms of weight $2$ and trace maps.

具有完美剩余域的二维局部环的算术对偶性

Arithmetic duality for two-dimensional local rings with perfect residue field

本研究针对具有完美剩余域的二维局部环，通过为算术上同调群构建代数群结构，对Saito的算术对偶性理论作出了改进。

On the GIT stratification of prehomogeneous vector spaces III

We determine all orbits of the prehomogeneous vector space $(\mathrm{GL}_5\times \mathrm{GL}_4,\wedge^2\mathrm{Aff}^5 \otimes \mathrm{Aff}^4)$ rationally over an arbitrary perfect field in this paper.

The weight distributions of a class of cyclic codes III

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases. In this paper we solve one more special case. The problem of finding the weight distribution is transformed into a problem of evaluating certain character sums over finite fields, which in turn can be solved by using the Jacobi sums directly.