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We classify newforms with rational Fourier coefficients and complex multiplication for fixed weight up to twisting. Under the extended Riemann hypothesis for odd real Dirichlet characters, these newforms are finite in number. We produce tables for weights 3 and 4, where finiteness holds unconditionally.

This paper gives upper and lower bounds for the degree of the field of definition of a singular K3 surface, generalising a recent result by Shimada. We use work of ShiodaMitani and Shioda-Inose and classical theory of complex multiplication.

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Let k be a field of characteristic 2. We give a geometric proof that there are no smooth quartic surfaces S ⊂ P 3 k with more than 64 lines (predating work of Degtyarev which improves this bound to 60). We also exhibit a smooth quartic containing 60 lines which thus attains the record in characteristic 2. After this paper was written, A. Degtyarev stated in [2], partly based on machine-aided calculations from [3], that the bound of Theorem 1.1 can be improved to 60 in characteristic 2. The record is attained by a quartic with S 5-action which we shall exhibit explicitly in Section 8. We point out that unlike in other characteristics (by work of us and Veniani [9], [10], [19]), there exist non-smooth quartic K3 surfaces with more lines than in the smooth case, in fact with as many as 68 lines in characteristics 2 (see Remark 2.3), indicating how special this situation is. We emphasize that originally we were expecting the bound from characteristics = 2, 3 to go up in characteristic 2, since just like in characteristic 3, there may be quasi-elliptic fibrations and the flecnodal divisor may degenerate. With this in mind, our previous best bound ended up at 84 in [10, Prop. 1.3]. In