2025/12/23 by Shi-Chao Chen, Chong Wu, Chen, Shi-Chao +1
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Primary 11R29 #Secondary 11R11
paper · doi:10.48550/arxiv.2512.20023
openalex publication_date 2025/12/23 · openalex created_date 2025/12/25 · openalex updated_date 2026/07/28
Let n≥1, r≥0 and s≥0 be integers satisfying 4+r+3 s≤3n+1. Given linear polynomials fi(x)=mi x+ni for 1 ≤ i ≤ r+s, where the coefficients mi , ni are positive integers satisfying certain conditions, we prove that there exist infinitely many fundamental discriminants D>0 such that the 3-rank of the class group of each quadratic fields ℚ(√(f1(D))), …, ℚ(√(fr(D))) and ℚ(√-fr+1(D)), …, ℚ(√-fr+s(D)) is simultaneously less than n. Moreover, for any positive integer k, there exist positive integers a, d such that the 3-rank of the class group of each quadratic fields ℚ(√(a+g1(d))), …,ℚ(√(a+gk(d))) is simultaneously less than n for polynomials g1(x), g2(x), …, gk(x) that take integer values at the integers and have no constant terms.