The rationality of Stark-Heegner points over genus fields of real quadratic fields
We study the algebraicity of Stark-Heegner points on a modular elliptic curve E. These objects are /?-adic points on E given by the values of certain /?-adic integrals, but they are conjecturally defined over ring class fields of a real quadratic field K. The present article gives some evidence for this algebraicity conjecture by showing that linear combinations of Stark-Heegner points weighted by certain genus characters of K are defined over the predicted quadratic extensions of K. The non- vanishing of these combinations is also related to the appropriate twisted Hasse-Weil L-series of E over K, in the spirit of the Gross-Zagier formula for classical Heegner points.
This paper considers a method to realize a two-valued input three-valued output function f:{0, 1}/sup n//spl rarr/{0, 1, 2} by using a cascade of four-input cells. Decomposition of a group function is used to find a canonical form. We show that the Walsh spectrum specifies the canonical form, and the number of non-zero coefficients in the spectrum is proportional to the number of cells in the cascade. Finally, we show a design algorithm for a p-valued function f:{0, 1}/sup n//spl rarr/{0, 1, ..., p-1}. The designed cascades are reversible and conservative.