Slopes of Euclidean lattices, tensor product and group actions

Renaud Coulangeon (1) , Gabriele Nebe (2)

1 IMB - Institut de Mathématiques de Bordeaux
2 Lehrstuhl II für Mathematik

We study the behaviour of the minimal slope of Euclidean lattices under tensor product. A general conjecture predicts that $\mu_{min}(L \otimes M) = \mu_{min}(L)\mu_{min}(M)$ for all Euclidean lattices $L$ and $M$. We prove that this is the case under the additional assumptions that $L$ and $M$ are acted on multiplicity-free by their automorphism group, such that one of them has at most $2$ irreducible components.

Domaines: Théorie des nombres [math.NT]

Renaud Coulangeon, Gabriele Nebe. Slopes of Euclidean lattices, tensor product and group actions. Israel Journal of Mathematics, In press, ⟨10.1007/s11856-019-1944-9⟩. ⟨hal-01943136⟩