A geometric proof of Kummer's reciprocity law for seventh powers

Abstract

We provide a “geometric” proof of the reciprocity law for seventh powers, making use the arithmetic of the curve y = x+1/4, which is a hyperelliptic curve of genus 3. This proof is mainly the content of the Ph.D thesis of one of the authors([1]). We follow the ideas of D. Grant([2]), who gave a proof of a quintic reciprocity law based on the arithmetic of a hyperelliptic curve of genus 2. Our main tools are, like in D. Grant’s paper, the formal group in the origin of the Jacobian of the curve and the theorems of complex multiplication. However, there is a significant jump from the genus 2 to the genus 3 setting. For instance, in order to prove of complementary laws, we have to compute some 7-torsion points of the Jacobian; in our case, this computation become difficult due to the high dimension of the Riemann space involved. Along the proof of the complementary laws we produce some interesting units, similar to the classical elliptic units: they appear as the values on torsion points of some rational functions of the Jacobian.