数理解析研究所講究録
第 1826 巻 2013 年 163-167

163

On the Cyclicity of finite CM abelian varieties
Cristian Virdol

Graduate School of Mathematics
Kyushu University

virdol@imi.kyushu-u.ac.jp
July 17, 2012
AbstractLetbe an abelian variety over a number field $F$ of dimension$r\geq
1$ is an integer. Assume that
where, where $K$ is a
$CM$ -field such that $[K : \mathbb{Q}]=2r$ . Fora finite prime of $F$ , we denoteby
the residue field at . If has good reduction at , let be thereduction of at . Under GRH, we obtain ([V]) an asymptotic formulafor the number of primes of $F$ , with, for which
hasat most $2r-1$ cyclic components.
$A$$r,$$End_{\overline{F}}A\otimes \mathbb{Q}=K$$\wp$$\mathbb{F}_{\wp}$

$\wp$

$A$

$A$

$\wp$

$N_{F/\mathbb{Q}}\wp\leq x$

$\wp$

1

$\overline{A}$$\wp$$\overline{A}(\mathbb{F}_{\wp})$

The Main result

Consider an abelian variety defined over a number field $F$ , of conductor
and of dimension , where $r\geq 1$ is an integer. Let
be the set of finite placesof , and for a prime of $F$ , let
be the residue field at . Let
be theset of primesof good reduction for $A,$. For, we denote by
the reduction of at
We $have-$ thatfor any positive integer $m$satisfying. Hence$A$$\mathcal{N},$

$r$

$F$

$\Sigma_{F}$$\wp$$\mathbb{F}_{\wp}$$\wp$$\wp\in\Sigma_{F}$$A$$\overline{A}$$\wp\in \mathcal{P}_{A}$$\mathcal{P}_{A}$$(i.e. (N_{F/\mathbb{Q}}\wp, N_{F/\mathbb{Q}}\mathcal{N})=1)$$\wp.$$\overline{A}(\mathbb{F}_{\wp})\subseteq\overline{A}[m](\overline{\mathbb{F}}_{\wp})\subseteq(\mathbb{Z}/m\mathbb{Z})^{2r}$$|A(\mathbb{F}_{\wp})||m$$\overline{A}(\mathbb{F}_{\wp})\simeq \mathbb{Z}/m_{1}\mathbb{Z}\cross \mathbb{Z}/m_{2}\mathbb{Z}\cross\cdots\cross \mathbb{Z}/m_{s}\mathbb{Z}$

,

(1.1)

where $s\leq 2r,$, and $m_{i}|m_{i+1}$ for $1\leq i\leq s-1$ . Eachis calleda cyclic component of. If $s<2r$ , we say thathas at most $(2r-1)$
cyclic components (thus if $r=1$ this means thatis cyclic). For
define$m_{i}\in \mathbb{Z}_{\geq 2}$$\mathbb{Z}/m_{i}\mathbb{Z}$$\overline{A}(\mathbb{F}_{\wp})$$\overline{A}(\mathbb{F}_{\wp})$$\overline{A}(\mathbb{F}_{\wp})$$f_{A,F}(x)=|${$\wp\in \mathcal{P}_{A}|N_{F/\mathbb{Q}}\wp\leq x,\overline{A}(\mathbb{F}_{\wp})$$x\in \mathbb{R},$has at most $(2r-1)$ cyclic components}$|.[$Let $F(A[m])$ be the field obtained by adjoining to $F$ the $m$ -division points $A[m]$of $A.$
We obtain (this is the main result of [V]; whenand $r=1$ , i.e. whenis a $CM$ elliptic curve over , Theorem 1.1 is similar to Theorem 1.2 of [CM] :$F=\mathbb{Q}$$A$$\mathbb{Q}$$)$

164
Theorem 1.1. Let be an abelian variety over a number field $F$ of dimension, where $K$ is a $CM$-field, of conductor , such thatsatisfying $[K : \mathbb{Q}]=2r$ . Assume that the Generalized Riemann Hypothesis$(GRH)$ holds for the Dedekind zeta functions of the division fields for A. Thenwe have
$A$$r\geq 1$$End_{\overline{F}}A\otimes \mathbb{Q}=K$$\mathcal{N}$$f_{A,F}(x)=c_{A,F}lix+O_{A,F}(x^{\frac{5}{6}}(\log x)^{2}5)$where $lix$$:= \int_{2}^{x}\frac{l}{\log t}dt$,, and$c_{A,F}= \sum_{m=1}^{\infty}\frac{\mu(m)}{[F(A[m]):F]}.$

Hereand $F.$$\mu(\cdot)$is the Mobius function, and the implicit $O_{A,F}$ -constant depends on$A$

Odds and ends

2$:=$ Gal. Let be an abelian variety
If $F$ is a number field, we denotebe the set ofover $F$ of dimension $r\geq 1$ , and of conductor . We denote by$A,$. For $m\geq 1$
primesof good reduction for
an integer, let $A[m]$ be the $m$ -division points of in . Then
$A$$(\overline{F}/F)$$G_{F}$$\mathcal{N}$$\mathcal{P}_{A}$$(i.e. (N_{F/\mathbb{Q}}\wp, N_{F/\mathbb{Q}}\mathcal{N})=1)$$\wp\in\Sigma_{F}$$A$$\overline{F}$$A[m]\simeq(\mathbb{Z}/m\mathbb{Z})^{2r}.$

If $F(A[m])$ is the field obtained by adjoining to $F$ the elements of $A[m]$ , thenwe have a natural injection$\Phi_{m}$For$l$: Gal $(F(A[m])/F)\hookrightarrow$ Aut$(A[m])\simeq GL_{2r}(\mathbb{Z}/m\mathbb{Z})$.

a rational prime, define$T_{l}(A)= \lim_{arrow}A[l^{n}].$The Galois group$G_{F}$

acts on$\tau_{\iota}(A)\simeq \mathbb{Z}_{l}^{2r},$

where$\mathbb{Z}_{l}$is the -adic completion of$l$$\rho_{A,l}$:$G_{F}arrow$$\mathbb{Z}$Autat , and we obtain a representation$l$$(T_{l}(A))\simeq GL_{2r}(\mathbb{Z}_{\iota})$,, let. If
be the Artin symbol
which is unramified outside, and let be a rational prime satisfying $(l, N_{F/\mathbb{Q}}\wp)=1$ . We denoteof inby $P_{A,\wp}(X)=X^{2r}+a_{2r-1,A}(\wp)X^{2r-1}+\ldots+a_{1,A}(\wp)X+N_{F/\mathbb{Q}}\wp^{r}\in \mathbb{Z}[X]$ theon $T_{l}(A)$ . Then $P_{A,\wp}(X)$ is independent ofcharacteristic polynomial of, where is the reduction of at , and theOne can identify $T_{l}(A)$ withof ononaction ofis the same as the action of the Frobenius$lN_{F/\mathbb{Q}}\mathcal{N}$$G_{F}$$\wp$$\wp\in \mathcal{P}_{A}$$\sigma_{\wp}$$l$$l.$$\sigma_{\wp}$$T_{l}(\overline{A})$$\overline{A}$

$A$

$\wp$

$\overline{A}$$\sigma_{\wp}$$T_{l}(\overline{A})$.

$\tau_{\iota}(A)$$\pi_{\wp}$165
We say that an abelian variety defined over a number field $F$ of dimension
is $CM$ (or has many complex multiplications) if, where$K$ is a $CM$-field satisfying $[K : \mathbb{Q}]=2r$ . We denote by the maximal totallyreal number field contained in $K$ , and let
be the ring of integers of andlet
be the ring of integers of $K$ . Let, be the set ofembeddings of $K$ into , whereis the complex conjugate of . Then we
have $[K:\mathcal{F}]=2$ , and $K=\mathcal{F}(\sqrt{-D})$ for some totally positive
$A$

$r$

$End_{\overline{F}}(A)\otimes \mathbb{Q}=K$$\mathcal{F}$$O_{\mathcal{F}}$$O_{K}$$\mathcal{F}$$\phi_{r},\overline{\phi}_{1},$$\phi_{1},$$\ldots,\overline{\phi}_{r}$$\ldots,$$\mathbb{C}$$\overline{\phi}_{i}$$\phi_{i}$$D\in O_{\mathcal{F}}.$

Lemma 2.1. (Ribet $[RJ)$ Let be a $CM$ abelian variety defined over a number
field , of dimension , of conductor , and let $m$ be a positive integer. Then
1.
$A$

$F$

$\mathcal{N}$$r$$\phi(m)^{2}\ll[F(A[m]):F],$

where $\phi(m)$ is the Euler function,
2. the extension $F(A[m])/F$ is mmified only at places dividing$m\mathcal{N}.$

Lemma 2.2. (Shimum $[SHJ)$ Let
be a $CM$ abelian variety defined over anumber field , of dimension , and of conductor . Then for all, thechamcteristic polynomial $P_{A,\wp}(X)$ has roots,
whereis the complex conjugate of, and, for all$i=1,$. Moreover one can assume that, and that
, we have.
for any $i=1,$
$A$

$F$

$\mathcal{N}$$r$$\pi_{1}(\wp),$$\overline{\pi}_{i}(\wp)$$\ldots,$$\pi_{i}(\wp)$$r$$\wp\in \mathcal{P}_{A}$$\pi_{r}(\wp),\overline{\pi}_{1}(\wp),$$\ldots,$$\ldots,\overline{\pi}_{r}(\wp)$$\pi_{i}(\wp)\overline{\pi}_{i}(\wp)=N_{F/\mathbb{Q}}\wp$$\pi_{1}(\wp)\in End_{\overline{F}}(A)\subseteq O_{K}$$\ldots,$$\pi_{i}(\wp)=\phi_{i}(\pi_{1}(\wp))$$r$

On can prove the following results (see [V]):Lemma 2.3. Let be an abelian variety over a number field $F$ , of conductor. Let, and let be the mtional prime below . Let $q\neq p$ be a rational
prime. Thencontains $a(q, \ldots, q)$ -type subgroup ( appears $2r$ -times), $i.e.$a subgroup isomorphic to, iff splits completely in $F(A[q])$ .
$A$$\mathcal{N}$$\wp\in \mathcal{P}_{A}$

$p$

$\wp$

$\overline{A}(\mathbb{F}_{\wp})$$q$$\mathbb{Z}/q\mathbb{Z}\cross\ldots\cross \mathbb{Z}/q\mathbb{Z}$$\wp$

Lemma 2.4. Let be a $CM$ abelian variety defined over a number field
dimension , and of conductor . Let $m$ be a positive integer. Then$A$$\mathcal{N}$$r$$F$, of$\wp\in \mathcal{P}_{A},$with $(N_{F/\mathbb{Q}}\wp, m)=1$ , splits completely in $F(A[m])$ iff
whereappears in Lemma 2.2.

$\frac{\pi_{1}(\wp)-1}{m}\in End_{\overline{F}}(A)$,$\pi_{1}(\wp)$

Lemma 2.5. Let be an abelian variety over a number field $F$ , of conductor, and let be the mtional pnme below . Thencontains
at most $(2r-1)$ -cyclic components iff does not split completely in $F(A[q])$ for
any rational prime $q\neq p.$$A$$\mathcal{N}.$$\cdot Let$$\wp\in \mathcal{P}_{A}$$p$$\overline{A}(\mathbb{F}_{\wp})$

$\wp$

$\wp$

Lemma 2.6. With the same notations as above, for any
we have that$m\in \mathbb{N}^{*}$

and any$S_{m}:=|\{\wp\in\Sigma_{F}|N_{F/\mathbb{Q}}\wp\leq x, N_{F/\mathbb{Q}}\wp=(\alpha m+1)^{2}+D\beta^{2}m^{2},$$f$or some $\alpha+\sqrt{-D}\beta\in O_{K}$ , where$\ll\frac{x^{\frac{3}{2}}}{m^{3}}+1.$$\alpha,$$\beta\in \mathcal{F}$}$|$$x\in \mathbb{R},$166
3Chebotarev

Consider $L/F$ a Galois extension of number fields, with Galois group . We, and by
thethe degree and the discriminant of
anddenote byprimes
lie
below
set
which
be
theofrational.
Letofdiscriminant
places of which ramify in $L/F.$$G$$d_{L}$$n_{L}$$\mathcal{P}(L/F)$$F/\mathbb{Q}$$d_{F}$$L/\mathbb{Q}$

$p$

$F$

Lemma 3.1. (Serve $[SEJ)$ If $L/F$ is Galois extension of number fields, then$\log d_{L}\leq|G|\log d_{F}+n_{L}(1-\frac{1}{|G|})\sum_{p\in \mathcal{P}(L/F)}\log p+n_{L}\log|G|.$

Let$C$

be a conjugacy class in$\pi_{C}(x, L/F)$ $:=|$

wherethat$\sigma_{\wp}${$G$. For a positive real number , let$x$$\wp\in\Sigma_{F}|N_{F/\mathbb{Q}}\wp\leq x,$

is a Frobenius element at

$\wp$

$\wp$

unramified in $L/F,$$\sigma_{\wp}\in C$}$|,$. The Chebotarev density theorem says$\pi_{C}(x, L/F)\sim\frac{|C|}{|G|}1ix\sim\underline{|C|}\underline{x}$$|G|\log x$’and moreover:Lemma 3.2. (Serre $[SEJ)$ Let $L/F$ be a Galois extension of number fields. Ifthe Dedekind zeta function of satisfies the $GRH$, then$L$,$| \pi_{C}(x, L/F)-\frac{|C|}{|G|}lix|\ll|C|x^{\frac{1}{2}}(\log x+\frac{\log|d_{L}|}{|G|})$

where the implied

4$O$-constant depends only on $F.$

Sketch of the proof of Theorem 1.1

Using \S 2 one obtains (see \S 4 of [V]), for $y=y(x)$ any real number withthat$y\leq 2x^{\frac{1}{2}},$$f_{A,F}(x)= \sum_{2m\leq 2x^{1}}\mu(m)\pi_{1}(x, F(A[m])/F)$$= \sum_{m\leq y}\mu(m)\pi_{1}(x, F(A[m])/F)+\sum_{:y<m\leq 2x}\mu(m)\pi_{1}(x, F(A[m])/F)$

$=$

main

$+$

error.

Using \S 2 and Chebotarev, under GRH, one obtains (see \S 4 of [V])main$= \sum_{m\leq y}\frac{\mu(m)}{n(m)}$li$x+ \sum_{m\leq y}O(x^{1}\sigma\log(mN_{F/\mathbb{Q}}\mathcal{N}x))$

(4.1)

167
$= \sum_{m\leq y}\frac{\mu(m)}{n(m)}$li$x+O(yx^{\frac{1}{2}}\log(N_{F/\mathbb{Q}}\mathcal{N}x))$,(4.2)

where $n(m)$ $:=[F(A[m]) : F]$ , anderror$\sum$$\ll$$\frac{x^{\frac{3}{2}}}{m^{3}}\ll\frac{x^{\frac{3}{2}}}{y^{2}}.$$y<m\leq 2_{X^{\Sigma}}^{1}$$m$square-free

For$x^{\frac{1}{3}}$$y:=\overline{(\log(N_{F/\mathbb{Q}}\mathcal{N}x))^{\frac{1}{3}}},$

from \S 2 one gets (see \S 4 of [V])$\sum_{m>y}\frac{\mu(m)}{n(m)}$li $x\ll$$\frac{(\log\log m)^{2}}{m^{2}}1ix\ll\frac{(\log\log y)^{2}}{y}1ix\ll x^{\frac{5}{6}}.$$\sum_{m>y}$$m$square-freeHence$f_{A,F}(x)= \sum_{m=1}^{\infty}\frac{\mu(m)}{n(m)}$li$x+O(x^{\frac{5}{6}}(\log(N_{F/\mathbb{Q}}\mathcal{N}x))^{\frac{2}{3}})$.

$\blacksquare$

References[CM] A. C. Cojocaru and M. R. Murty, Cyclicity of elliptic curves modulo
and elliptic curve analogues of Linniks problem, Math. Ann. 330 (2004)$p$601-625.

[M] D. Mumford, Abelian varieties, Tata Institute of Fundamental ResearchStudies in Mathematics, No. 5. Published for the Tata Institute of Fundamental Research, Bombay, byUniversity Press.
$Oxford^{\backslash }$[R] K. A. Ribet, Division points of abelian varieties with complex multiplication,
Mem. Soc. Math. de France 2e serie 2 (1980), 75-94.
[SE] J. -P. Serre, Quelques applications du theoreme de densite de Chebotarev,Inst. Hautes Etudes Sci. Publ. Math., no. 54, 1981, pp. 123-201.

[SH] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton University Press, 1971.

[SI] J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves,
Graduate Texts in Mathematics, vol. 151. Springer, New York (1994).
[V] C. Virdol, Cyclicity of finite CM abelian varieties, submitted.