On the Picard number and the extension degree of period matrices of complex tori

2025/03/27 by Auffarth, Robert, Franco, Jorge Duque
#11G10 #14K20 #14K22 #32G20 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.21642

Abstract

The rank ρ of the Néron-Severi group of a complex torus X of dimension g satisfies 0≤ρ≤ g²=h^1,1. The degree \mathfrakd of the extension field generated over ℚ by the entries of a period matrix of X imposes constraints on its Picard number ρ and, consequently, on the structure of X. In this paper, we show that when \mathfrakd is 2, 3, or 4, the Picard number ρ is necessarily large. Moreover, for an abelian variety X of dimension g with \mathfrakd=3, we establish a structure-type result: X must be isogenous to E^g, where E is an elliptic curve without complex multiplication. In this case, the Picard number satisfies ρ(X)=(g(g+1))/(2). As a byproduct, we obtain that if \mathfrakd is odd, then ρ(X)≤(g(g+1))/(2).

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