CoCalc Shared Fileswww / papers / anti-cyclotomic_height_pairing / anti-cyclotomic_height.tex
Author: William A. Stein
1\magnification=1200
2Dear William,
3\bigskip
4Thanks loads for the Sha computations, and I think I have an idea about why
5the bizarre
6behavior, but let me ponder a bit more this weekend. I also promised some
7precision regarding
8the other project I was hoping to interest you in; that is, the search  for
9evidence
10for, or against, what I call the "sign conjecture".   Here are some words
12conjecture is, in  a particular instance, equivalent to the assertion that the
13$p$-adic anti-cyclotomic height pairing is {\it not} identically zero which
14brings me to the
15situation I suggest working in.
16\bigskip
17Fix $E$  an elliptic curve over ${\bf Q}$ of conductor $N$, and of
18Mordell-Weil rank $2$ (or any
19even positive number).  Fix $p$ a prime number with respect to which $E$
20has good ordinary
21reduction.  Let $K$ range through quadratic imaginary fields with the
22following properties:
23
24\smallskip
25{\bf (a)} The discriminant of $K$ is prime to $Np$  (this can be
26significantly weakened).
27\smallskip
28{\bf (b)}The ideal $p{\cal O}_K$ splits to the product of two ideals which
29I will call $\pi$ and
30${\bar \pi}$.
31 \smallskip
32{\bf (c)} The field $K$ satisfies the Heegner condition" with respect to
33$N$, which means
34that there is an ideal ${\cal N} \subset {\cal O}_K$  such that ${\cal 35O}_K/{\cal N}$  is a cyclic
36group of order $N$.
37 \smallskip
38{\bf (d)}  The Mordell-Weil group of $E$ over $K$ (which is necessarily of
39odd rank) is of rank
40one more than the rank of $E({\bf Q})$.  Equivalently, the Mordell-Weil
41rank of the twist of $E$
42by the quadratic character determining $K$ has Mordell-Weil rank one over
43${\bf Q}$.
44\bigskip
45
46  Now what I would want to know is whether or not the $p$-adic
47anti-cyclotomic height pairing
48$$E(K) \times E(K) \to {\bf Q}_p$$
49is identically trivial, or not. If $P,Q$ are points in $E(K)$, let this
50pairing be denoted $<P,Q>$.  I
51should also say that this pairing is equivariant for complex conjugation,
52where the action of
53complex conjugation on the 
54${\bf Q}_p$" is multiplication by $-1$; so $$<{\bar P},{\bar Q}> = - 55<P,Q>.$$ In particular, the
56only possibly nontrivial piece of this pairing comes from taking the height
57of some point that is
58not an eigen-point for complex conjugation, and, in particular, not defined
59over  ${\bf Q}$.
60\bigskip
61Let me give you the recipe for computing $<P,Q>$.  Put
62$S =\{\pi, {\bar 63\pi} 64\}$ and for some of the notation I will use, let us just consult page 14
65(and surround) of
66Perrin-Riou's notes.  In particular, she defines a subgroup of finite index
67$E_S^{(1)}(K)\subset E(K)$  (this is defined by imposing conditions on the
68reduction of points to
69$S$) and gives a somewhat naive definition for the computation of the
70pairing $<P,P>$ restricted
71to points $P\in E_S^{(1)}(K)$.  First consider the anti-cylotomic mapping
72$${\tilde \rho}: {\rm ideals \ rel \ prime\ to\ } S \longrightarrow 73{\bf Z}_p$$
74defined as follows.  Note that if we are given a principal  ideal
75relatively prime to $S$,  $(\alpha) 76\subset {\cal O}_K$, then $\alpha$ is well-defined modulo an $e$-th root of
77$1$ where $e$ is
78usually $2$, and in exceptional situations is $4$ or $6$.  Therefore,
79taking an appropriate power
80of any ideal $I$ relatively prime to $S$ to make it principal, and then to
81obliterate this finite
82amibiguity, we render that power $I^{\nu}$  generated by a canonical
83element, call it
84$\alpha_\nu$, which is a unit mod $\pi$ and mod ${\bar \pi}$.  Fix $$\lambda: 85{\cal O}_\pi^* \to {\bf Z_p}$$ a logarithm"  (we can assume that it is an
86isomorphism between
87${\cal O}_\pi^*$ mod its torsion with ${\bf Z_p}$) and $${\bar \lambda}: 88{\cal O}_{\bar\pi}^* \to {\bf Z_p}$$ its complex  conjugate.  Now put
89$${\tilde \rho}(I): = {1\over \nu}\cdot (\lambda(\alpha_\nu) - {\bar 90\lambda}(\alpha_\nu)) 91\in {1\over \nu}\cdot {\bf Z_p} \subset {\bf Q_p}.$$
92
93\bigskip
94
95OK, now write $P$ in coordinates", as in Perrin-Riou's manuscript.
96Specifically, we write
97$$P = ({x(p)\over e(P)^2}, {y(p)\over e(P)^3})$$ where one has a bit of
99numerators and denominators" are really ideals rather than numbers, but
100in any event, we
101then put  (as P.-R. does on page 14):
102
103$$H_2(P) = {1\over 2}{\tilde \rho}(e(P)^2\cdot x(p)),$$ noting that our
104definition of  the
105subgroup $E_S^{(1)}(K)\subset E(K)$ is fashioned to guarantee that the
106ideal $e(P)^2\cdot x(p)$ is
107prime to $S$.
108
109\bigskip
110
111Now comes the standard Tate trick"; that is, we average, to define:
112$$<P,P>:= {\rm lim}_{n \to {\cal 1}}{ H_2(p^n\cdot P)\over p^{2n}}.$$
113
114\bigskip
115
116  Is it feasible to make some computations along the above lines?
117
118    \bigskip
119
120Best,
121\bigskip
122
123Barry
124