Tabulation of Cubic Function Fields with Imaginary and Unusual Hessian

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Abstract

We give a general method for tabulating all cubic function fields over http://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-79456-1_24/MediaObjects/978-3-540-79456-1_24_IEq1_HTML.png whose discriminant D has odd degree, or even degree such that the leading coefficient of − 3D is a non-square in http://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-79456-1_24/MediaObjects/978-3-540-79456-1_24_IEq2_HTML.png , up to a given bound on \(|D| = q^{\deg(D)}\). The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields. We present numerical data for cubic function fields over http://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-79456-1_24/MediaObjects/978-3-540-79456-1_24_IEq4_HTML.png and over http://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-79456-1_24/MediaObjects/978-3-540-79456-1_24_IEq5_HTML.png with \(\deg(D) \leq 7\) and \(\deg(D)\) odd in both cases.