2018/10/17 by Tonghai Yang, Yang, Tonghai, Hongbo Yin +3
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1810.07381
arxiv created 2018/10/17 · openalex publication_date 2018/10/17 · arxiv updated 2018/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the paper, we show that λ(z1) -λ(z2), λ(z1) and 1-λ(z1) are all Borcherds products in X(2) × X(2). We then use the big CM value formula of Bruinier, Kudla, and Yang to give explicit factorization formulas for the norms of λ(\fracd+√ d2), 1-λ(\fracd+√ d2), and λ(\fracd1+√(d1)2) -λ(\fracd2+√(d2)2), with the latter under the condition (d1, d2)=1. Finally, we use these results to show that λ(\fracd+√ d2) is always an algebraic integer and can be easily used to construct units in the ray class field of ℚ(√(d)) of modulus 2. In the process, we also give explicit formulas for a whole family of local Whittaker functions, which are of independent interest.