In this paper, we give a description of Deligne's periods $c^\pm$ for tensor product of pure motives $M \otimes M'$ over $\mathbb{Q}$ in terms of the period invariants attached to $M$ and $M'$ by Yoshida. The period relations proved by the author and Raghuram in an earlier paper follow from the results of this paper.Comment: 6 pages, published in Comptes rendus - Math\'ematique in January 2015. Formerly appendix to arXiv1312.595

Investigation of divisibility properties of natural numbers is one of the most important themes in the theory of numbers. Various tools have been developed over the centuries to discover and study the various patterns in the sequence of natural numbers in the context of divisibility. In the present paper, we study the divisibility of natural numbers using the framework of a growing complex network. In particular, using tools from the field of statistical inference, we show that the network is scale-free but has a non-stationary degree distribution. Along with this, we report a new kind of similarity pattern for the local clustering, which we call “stretching similarity”, in this network. We also show that the various characteristics like average degree, global clustering coefficient and assortativity coefficient of the network vary smoothly with the size of the network. Using analytical arguments we estimate the asymptotic behavior of global clustering and average degree which is validated using numerical analysis.

Abstract. In this paper, we prove some period relations for the ratio of Deligne's periods for certain tensor product motives. These period relations give a motivic interpretation for certain algebraicity results for ratios of successive critical values for Rankin-Selberg L-functions for GL n × GL n proved by Günter Harder and the second author.

We prove a strong multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces i.e. if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length.

We prove spectral analogues of the classical strong multiplicity one theorem for newforms. Let Γ 1 and Γ 2 be uniform lattices in a semisimple group G. Suppose all but finitely many irreducible unitary representations (resp. spherical) of G occur with equal multiplicities in L 2 (Γ 1 \G) and L 2 (Γ 2 \G). Then L 2 (Γ 1 \G) ∼ = L 2 (Γ 2 \G) as G -modules (resp. the spherical spectra of L 2 (Γ 1 \G) and L 2 (Γ 2 \G) are equal).