In 1925, Hecke found two different quadratic fields having the same $L$-functions attached to certain ray class groups. In this paper, we show that if $K/\mathbb{Q}$ is a Galois extension whose Galois group is isoclinic to an extraspecial group, then there are many elementary abelian extensions inside $K$ whose $L$-functions coincide.

In this paper we prove the asymptotic stability of solitons for a discrete nonlinear Schrödinger equation near the anti-continuous limit. Our novel insight is that the analysis of linearized operator, usually non-symmetric, can be reduced to a study of simple self-adjoint operator almost like the free discrete Laplacian restricted on the space of odd functions.

In this paper, we determine the Batalin-Vilkovisky algebra structure on the Hochschild cohomology of self-injective Nakayama algebras with the diagonalizable Nakayama automorphism over an algebraically closed field $K$. Moreover, in the case that the characteristic of $K$ divides the order of the Nakayama automorphism, we compute the Batalin-Vilkovisky algebra structure on cohomology of Hochschild complex related to the Nakayama automorphism.