On the conjecture of Birch and Swinnerton-Dyer

Supersingular primes and p-adic L-functions

“…This is what Coates and Wiles do in the ordinary case [1,2], using the basic Lubin-Tate formal group. This in turn leads to an understanding of the p-adic interpolation properties of those L-values.…”

“…§6 generalizes this to higher-order derivations of a two-variable formal power series, showing how the p-character and the local grossencharacter act together. Finally, Theorems 7.4 and 7.6 are local computations with logarithmic derivatives analogous to those done by Coates and Wiles in [1,2] for primes of ordinary reduction, hopefully of use to those who may wish to obtain explicit results on the p-adic growth properties of L-values, for example. We prove the relevant properties of the logarithmic differentiation homomorphisms used for these computations.…”

“…This is a very useful fact, since all Lubin-Tate formal groups over O p are isomorphic and one can choose among them one well suited for computations. This idea is illustrated in [1,2].…”

“…, (see [1]), Λ ρ u 2 = Λ ρ u 1 if and only if f is a unit in Λ ρ , and this is so if and only if f (0, 0) is a unit at p. Since f (κ(γ 1 ) − 1, κ(γ 2 ) − 1) ≡ f (0, 0) mod π, we see that this is the case if and only if the quotient of δ(u 1 ), δ(u 2 ) is a unit at p.…”

“…One may obtain an analogous p-adic relationship. Details of these facts may be found in [6], which draws from [1,2]. To get L-values, one uses the Robert elliptic units, which are defined by picking a suitable theta function Θ.…”