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Divisibility of class numbers of imaginary quadratic function fields by a fixed odd number

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Abstract

In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field \(\mathbb{F}_{q}(x)\) whose class groups have elements of a fixed odd order. More precisely, for q, a power of an odd prime, and g a fixed odd positive integer ≥ 3, we show that for every ε > 0, there are \(\gg q^{L(\frac{1}{2}+\frac{3}{2(g+1)}-\epsilon)}\) polynomials \(f \in \mathbb{F}_{q}[x]\) with \(\deg f=L\), for which the class group of the quadratic extension \(\mathbb{F}_{q}(x, \sqrt{f})\) has an element of order g. This sharpens the previous lower bound \(q^{L(\frac{1}{2}+\frac{1}{g})}\) of Ram Murty. Our result is a function field analogue which is similar to a result of Soundararajan for number fields.

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Acknowledgements

The authors would like to thank Prof. Jeffrey Achter for suggestions and especially for bringing their attention to [1], [2] and [11]. The authors are indebted to Professors M Ram Murty and K Soundararajan for their comments on an earlier version of this paper. The authors would also like to thank the anonymous referee for carefully going through the manuscript and suggesting important changes for a better presentation.

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  1. Indian Statistical Institute, Stat-Math Unit, 203 Barrackpore Trunk Road, Kolkata, 700 108, India

    PRADIPTO BANERJEE

  2. Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai, 600 113, India

    SRINIVAS KOTYADA

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  1. PRADIPTO BANERJEE
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Correspondence to SRINIVAS KOTYADA.

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BANERJEE, P., KOTYADA, S. Divisibility of class numbers of imaginary quadratic function fields by a fixed odd number. Proc Math Sci 123, 1–18 (2013). https://doi.org/10.1007/s12044-012-0105-4

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