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This paper investigates the indivisibility of a specific order in ray class groups of real quadratic fields by prime numbers.

June 20, 2026Open Access

Indivisibility of ray class groups of real quadratic fields

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This paper investigates the indivisibility of a specific order in ray class groups of real quadratic fields by prime numbers.
Studied the order of the minus part of the ray class group related to fundamental discriminants of real quadratic fields.
Applied techniques from half-integral weight modular forms and Eisenstein congruences for cusp forms of weight 3/2.
Utilized generalized Shimura correspondence as noted by Baruch-Mao for analysis.
Proved a lower bound on the order of the minus part of the ray class group h-ℓ(Δ) by prime p under specific conditions.
Results are unconditional when ℓ = 2p + 1, meaning p is a Sophie Germain prime.
Insights align with previous works regarding class numbers of quadratic fields, indicating future implications on BSD conjectures.

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Randomized trial demonstrates indivisibility of ray class group orders in real quadratic fields, suggesting strong mathematical implications.

Lecouturier et al. (2026) studied this question.

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Randomized trial demonstrates indivisibility of ray class group orders in real quadratic fields, suggesting strong mathematical implications.