The image of the generalized Dedekind sum

2025/03/12 by Knight, Evelyne S., Matos, Carlos Alexov, Sefidi, Amira +1 · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.09741

Abstract

The newform Dedekind sum S_{χ1, χ2} associated to a pair of primitive Dirichlet characters χ₁, χ₂ of respective conductors q₁, q₂, is a group homomorphism from Γ₁(q₁ q₂) into the number field F_{χ1, χ2} generated by the values of the characters. It is a basic question to identify the image of this map, which is known to be a lattice L_{χ1, χ2} in F_{χ1, χ2}. It has recently been conjectured that when χ₁ and χ₂ are quadratic, then L_{χ1, χ2} = 2 ℤ. In this paper, we make some progress towards this conjecture by exhibiting an explicit lattice in which L_{χ1, χ2} is contained; in particular, when the characters are quadratic, the q_i are coprime, odd, and sufficiently large, then L_{χ1, χ2} ⊆ ℤ.

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• Higher Weight Generalized Dedekind Sums

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