Cuspidal divisor class groups of non-split Cartan modular curves

2016/05/02 by Pierfrancesco Carlucci, Carlucci, Pierfrancesco
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1605.00375

35 pages

arxiv created 2016/05/02 · arxiv updated 2016/05/03

Abstract

I find an explicit description of modular units in terms of Siegel functions for the modular curves X⁺_ns(p^k) associated to the normalizer of a non-split Cartan subgroup of level p^k where p\not=2,3 is a prime. The Cuspidal Divisor Class Group \mathfrakC⁺_ns(p^k) on X⁺_ns(p^k) is explicitly described as a module over the group ring R = ℤ[(ℤ/p^kℤ)^*/\± 1\]. In this paper I give a formula involving generalized Bernoulli numbers B_2,χ for |\mathfrakC⁺_ns(p^k)|.

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