nt.number theory - Galoisian sets of prime numbers

Sunday, 8 April 2012

nt.number theory - Galoisian sets of prime numbers

In order to add my bit to the already rich content on this site, here is a
nice family of galoisian extensions $K_l|{bf Q}$ with group ${rm GL}_2({bf F}_l)$ (indexed by primes $lneq5$ ) for which a ``reciprocity law'' can be
written down explicitly. I've come across this family recently while writing
an expository article.

Let $E$ be the elliptic curve (over $bf Q$ ) of conductor $11$ defined by
$y^2+y=x^3-x^2$ , with associated modular form

$eta_{1^2,11^2}=qprod_{k>0}(1-q^k)^2(1-q^{11k})^2=sum_{n>0}c_nq^n.$
Let

$K_l={bf Q}(E[l])$ , which is thus galoisian over

$bf Q$ and unramified at
every prime

$pneq11,l$ .

One can deduce from cor.1 on p.308 of Serre (Inventiones 1972) that for every
prime $lneq5$ , the representation

$rho_{E,l}:{rm Gal}(K_l|{bf Q})rightarrow{rm GL}_2({bf F}_l)$
we get upon choosing an

${bf F}_l$ -basis of

$E[l]$ is an isomorphism;
cf. the online notes on Serre's conjecture by Ribet and Stein. Shimura did
this for

$lin[9,97]$ (Crelle 1966).

Suppose henceforth that $l$ is a prime $neq5$ and that $p$ is a prime
$neq11,l$ . The characteristic polynomial of $rho_{E,l}({rm Frob}_p)in{rm GL}_2({bf F}_l)$ is

$T^2-bar c_pT+bar pin{bf F}_l[X].$
The prime

$p$ splits completely in

$K_l$ if and only if

${rm Frob}_p=1$ in

${rm Gal}(K_l|{bf Q})$ , which happens if and only if

$rho_{E,l}({rm Frob}_p)=pmatrix{1&0cr0&1}.$
If so, then

$p,c_pequiv1,2pmod l$ but not conversely, for the matrix

$displaystylepmatrix{1&1cr0&1}$ also has the characteristic polynomial

$T^2-bar2T+bar1$ . But these congruences on

$p,c_p$ do rule out an awful lot
of primes as not splitting completely in

$K_l$ .

In summary, we have the following ``reciprocity law" for $K_l$ :

$hbox{($p$ splits completely in $K_l$)} quadLeftrightarrowquad E_p[l]subset E_p({bf F}_p),$
where

$E_p$ is the reduction of

$E$ modulo

$p$ . Indeed, reduction modulo

$p$
identifies

$E[l]$ with

$E_p[l]$ and the action of

${rm Frob}_p$ on the former
space with the action of the canonical generator

$varphi_pin{rm Gal}(bar{bf F}_p|{bf F}_p)$ on the latter space. To say that

$varphi_p$
acts trivially on

$E_p[l]$ is the same as saying that

$E_p[l]$ is contained in
the

${bf F}_p$ -rational points of

$E_p$ . The analogy with the multiplicative
group

$mu$ is perfect:

$hbox{($pneq l$ splits completely in ${bf Q}(mu[l])$)} quadLeftrightarrowquad mu_p[l]subset mu_p({bf F}_p)$
(

$Leftrightarrow l|p-1Leftrightarrow pequiv1pmod l$ ), where

$mu_p$ is not the

$p$ -torsion of

$mu$ but the reduction of

$mu$ modulo

$p$ .

I requested Tim Dokchitser to compute the first ten $p$ which split completely
in $K_7$ , and his instantaneous response was 4831, 22051, 78583, 125441,
129641, 147617, 153287, 173573, 195581, and 199501.

It is true that all this (except the list of these ten primes) was known before Serre's conjecture was proved
(2006--9) or even formulated (1973--87), but I find this example a very good
illustration of the kind of reciprocity laws it provides.

I hope you enjoyed it as much as I did.

AnswerDesk

Sunday, 8 April 2012