The two-variable pp-adic Birch–Swinnerton-Dyer conjecture

Let r=rkE(K)r=\operatorname{rk}E(K), let LΛ\mathbf{L}\in\mathbf{\Lambda} be the two-variable pp-adic LL-function, and let I\mathbf{I} be the augmentation ideal of Λ\mathbf{\Lambda}. Let Rp(E,K)R_p(E,K) be the two-variable pp-adic regulator, let c(χtriv)c(\chi_{\mathrm{triv}}) be the interpolation factor for the trivial character, let \Sha(E/K)\Sha(E/K) be the Shafarevich–Tate group, and let cvc_v be the Tamagawa factors.

Two-variable pp-adic Birch–Swinnerton-Dyer conjecture. The element L\mathbf{L} lies in Ir\mathbf{I}^r and satisfies

\mathbf{L}\equiv c(\chi_{\mathrm{triv}})\\,\\#\Sha(E/K)\\,\prod_v c_v\cdot R_p(E,K)\pmod{\mathbf{I}^{r+1}}.

This is the two-variable leading-term formula predicted by the pp-adic Birch–Swinnerton-Dyer principle. The supplied text gives no resolution.

Sources & referencesView supporting material

Primary source

Barry Mazur and Karl Rubin, “Elliptic curves and class field theory”, arXiv:math/0304235 (2003).

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