Resultants of CM division polynomials and reciprocity laws

Abstract: We explain how to calculate the resultants of division polynomials of endomorphisms on CM elliptic curves and how this provides a uniform simple algebraic approach to cubic and biquadratic reciprocity, quadratic reciprocity over imaginary quadratic fields, and Rubin and Silverberg’s determination of the Grössencharackter of a CM elliptic curve. Although at its heart this circle of ideas goes back to Eisenstein, we need to make use of the relatively recent general consideration of division polynomials of general isogenies by Mazur and Tate.

Seminarraum 05.002, Spiegelgasse 5