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Elliptic curves and class field theory. (English) Zbl 1036.11023
Li, Ta Tsien (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20–28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press (ISBN 7-04-008690-5/3-vol. set). 185-195 (2002).
In this article, the authors focus on the following variation of the Mordell-Weil problem: Fixing an elliptic curve \(E\) defined over \({\mathbb Q},\) an imaginary quadratic field \(K,\) and a prime number \(p,\) to study the variation of the Mordell-Weil group of \(E\) over finite subfields of the (unique) \({\mathbb Z}_{p}^{2}\)-extension \(K_{\infty}\) of \(K.\)
The object of this paper is to sketch a “package” of conjectures (due to, or built on ideas of many authors) on this piece of the arithmetic of elliptic curves. There are three parts in the picture:
1) The universal norm theory, which arises from purely arithmetic considerations, but provides analytic invariants:
If \(K \subset F \subset K_{\infty},\) the universal norm module \(U(F)\) is the projective limit \(U(F):= {\mathbb Q}_{p} \otimes \;\displaystyle\lim_{\buildrel\longleftarrow \over{L}}\;(E(L) \otimes {\mathbb Z}_{p}),\) \(L\) running through the finite subextensions of \(F.\) Mazur’s conjecture on the rank of \(U(F)\) for a \({\mathbb Z}_{p}\)-extension \(F/K\) (1983 ICM) has been largely settled by recent work of Kato for the cyclotomic \({\mathbb Z}_{p} \)-extension \(\Bigl( U(K^{\text{cyc}}_{\infty}) = 0 \Bigl)\) and Vatsal and Cornut for the anticyclotomic \({\mathbb Z}_{p}\)-extension \(( U(K^{\text{anti}}_{\infty})\) is free of rank one over \(\Lambda_{\text{anti}}).\) The complex conjugation \(\tau\) gives \({\mathcal U}:= U(K^{ \text{anti}}_{\infty})\) a natural semi-linear \(\tau\)-module structure, so that \({\mathcal U}\) is completely determined by its “sign”. Moreover, the canonical cyclotomic \(p\)-adic height pairing gives \({\mathcal U}\) a canonical hermitian structure. Finally, \({\mathcal U}\) contains an important submodule, the so-called Heegner submodule \({\mathcal H}.\) Conjectures 5 and 6 of the paper take care of the sign and the height pairing, while conjecture 7 relates \(\Gamma_{\text{cyc}} \oplus_{{\mathbb Z}_{p}}\;\text{char} ({\mathcal U}/{\mathcal H})^{2}\) to the so-called Heegner \({\mathcal L}\)-function.
2) The analytic theory, which constructs and studies the relevant \(L\)-functions, both classical and \(p\)-adic: the “two-variable” \(p\)-adic \(L\)-function for \(E/K\) is an element \(L \in \Lambda = {\mathbb Q}_{p} \oplus_{{\mathbb Z}_{p}} {\mathbb Z}_{p} [[ G(K_{\infty}/K))]]\) which interpolates special values of the classical Hasse-Weil function of twists of \(E.\) Conjecture 9 (or \(\Lambda\)-adic Gross-Zagier conjecture) relates the image of \(L\) in \(\Gamma_{\text{cyc}} \oplus_{{\mathbb Z}_{p}}\;\Lambda_{\text{anti}}\) to the Heegner \({\mathcal L}\)-function. The two-variable \(p\)-adic regulator \(R_{p} (E, K)\) is defined as the discriminant of a certain \(p\)-adic height pairing and it admits a certain decomposition \(R_{p} (E, K) = \oplus^{r}_{j = 0} R_{p}(E, K)^{r-j,j},\) \(r =\operatorname{rank} (E).\) Conjecture 11 (maximal nondegeneracy) gives a range where \(R_{p}(E, K)^{r-j, j} \not= 0.\) It implies the sign conjecture 5.
3) The arithmetic theory, which studies Selmer modules over Iwasawa algebras: one feature of the so-called “control theorem” is that, for \(K \subset F \subset K_{\infty},\) \(U(F) \displaystyle\buildrel \sim \over\to \text{Hom}_{\Lambda_{F}}\;(S_{p} (E/F), \Lambda_{F}),\) where \(S_{p} (E/F) =\;\text{Hom}(\text{Sel}_{p} (E/F, {\mathbb Q}_{p}/{\mathbb Z}_{p})) \otimes {\mathbb Q}_{p}\) (other notations are obvious). Conjecture 13 (or two-variable Main Conjecture) states that the two-variable \(p\)-adic \(L\)-function generates the ideal \(\text{char}_{\Lambda} (S_{p}(E/K_{\infty}))\) of \(\Lambda.\) It can be also restricted to a cyclotomic and an anticyclotomic main conjecture. Finally, the two-variable \(p\)-adic BSD conjecture gives a precise expression of the \(L\)-function mod \(I^{r+1},\) where \(I\) is the augmentation ideal of \(\Lambda.\)
At the end of the paper, the authors “suggest the beginnings of a new algebraic structure”, the so-called orthogonal \(\Lambda\)-modules “to organize [all previous] conjectures” (p. 186).
For the entire collection see [Zbl 0993.00022].

11G05 Elliptic curves over global fields
11R23 Iwasawa theory
11R37 Class field theory
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