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A note on the division values of ℘(u)

Published online by Cambridge University Press:  24 October 2008

J. W. S. Cassels
J. W. S. Cassels
Affiliation:
Trinity CollegeCambridge
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Extract

Let x = ℘′(u) and y = ½℘(u), where the Weierstrass ℘-function has invariants g2 = 4A, g3 = 4B and differentiation with respect to the parameter u is denoted by a dash, so that

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 45 , Issue 2 , April 1949 , pp. 167 - 172
Copyright
Copyright © Cambridge Philosophical Society 1949

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References

† Fricke, R., Elliptische Functionen, 2 (Leipzig, 1922), 184Google Scholar et seq., Weber, H., Lehrbuch der Algebra, 3 (2nd ed.Braunschweig, 1908), 196,Google ScholarJordan, C., Cours d'analyse, 3 (3rd ed.Paris, 1913), 190Google Scholar. These writers do not use ωm and øm but express everything in terms of ψm. I have changed the sign of ψm when m is even, so as always to have a positive first coefficient.

† van der Waerden, B. L., Moderne Algebra, 1 (New York, 1943), 75–7.Google Scholar

† Cf. B. L. van der Waerden, loc. cit.

† For they have degrees less than and respectively.

‡ Fractional or integral.

§ Lutz, Elizabeth, J. reine angew. Math. 177 (1937), 238–47Google Scholar, and summarized in C.R. Acad. Sci., Paris, 203 (1936), 20–2.Google Scholar That the denominator of x s is divisible by that of x 1 follows from

but if ss′ ≡ 1 (mod p k) we have and hence the stated result.

∥ Although, as stated earlier, the constant term of ψ3k(x) is not divisible by p qua function of A and B, it may be so when A and B have definite values.

† Loc. cit.

‡ Mahler, K., Quart. J. Math. 6 (1935), 74–7.CrossRefGoogle Scholar

§ Billing, G., Nova Acta Reg. Soc. Sci. Upsal. Ser. iv, xi, No. 1 (1938), 117.Google Scholar

∥ Weil, A., C.R. Acad. Sci., Paris, 203 (1936), 22.Google Scholar

F. Châtelet, ibid. 210 (1940), 70.

** Loc. cit. p. 594.