On quadratic reciprocity over function fields (Q1919925)

The law of quadratic reciprocity for number fields may be obtained from the functional equation of the theta function [see \textit{E. Hecke}, Lectures on the theory of algebraic numbers, Springer Verlag (1981; Zbl 0504.12001)]. Artin proved the analogue of the reciprocity law in the case of function fields [\textit{E. Artin}, Math. Z. 19, 153-206 (1924; JFM 50.0107.01); The collected papers of Emil Artin, edited by S. Lang and J. Tate, Addison-Wesley (1965; Zbl 0146.00101), 1-54]. In this paper the author defines an analogue of the theta function in the function-field case and, as in the number-field case, obtains the reciprocity law through the Gauss sums associated with the theta series.