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].

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].

Reviewer: Thong Nguyen Quang Do (BesanĂ§on)

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\textit{B. Mazur} and \textit{K. Rubin}, in: Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20--28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press; Singapore: World Scientific/distributor. 185--195 (2002; Zbl 1036.11023)

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