Geodesic
The quadratic reciprocity law and the elementary theta function
Glasgow mathematical journal, Tome 26 (1985), pp. 19-30

Voir la notice de l'article provenant de la source Cambridge University Press

This note points out a new aspect of the well-known relationship between the subjects mentioned in the title. The following result and its generalization in totally real algebraic number fields is central to the discussion. Let denote the Legendre symbol for relatively prime numbers a and b Ɛ Z and a substitution of the modular subgroup Γ0(4). Then, if γ>0 and b≡1 mod 2,withandAccording to (1), the Legendre symbol behaves somewhat like a modular function (apart from the known behaviour under and ). (1) follows (see below) from the functional equationwithprovided thatHere we used and always will use the abbreviationand Ɛδ means the absolutely least residue of δ mod 4. In the proof, Hecke [4] assumed γ>0 (see also Shimura [5]). 
Eichler, Martin. The quadratic reciprocity law and the elementary theta function. Glasgow mathematical journal, Tome 26 (1985), pp. 19-30. doi: 10.1017/S0017089500006042
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[1] 1.Eichler, M., Einführung in die Theorie der algebraischen Zahlen und funktionen (Birkhäuser 1963), 55–57, Introduction to Algebraic numbers and functions (Academic Press, 1966), 41–43. Google Scholar | DOI

[2] 2.Freitag, E., Siegelsche Modulfunktionen, (Springer, 1982), 2. Anhang, Satz A 2.4. Google Scholar

[3] 3.Hecke, E., Mathematische Werke (Vandenhoek und Ruprecht, 1959), No 13. Google Scholar

[4] 4.Hecke, E., Mathematische Werke (Vandenhoek und Ruprecht, 1959) No 42, 940. Google Scholar

[5] 5.Shimura, G., On modular forms of half integral weight, Ann. Math. 97 (1973), 443. Google Scholar | DOI

[6] 6.Siegel, C. L., Gesammelte Abhandlungen, Bd. III. (Springer, 1966). Google Scholar | DOI

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