Xrthnty

In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension.

Ramification groups in lower numbering

Ramification groups are a refinement of the Galois group $G$ of a finite $L/K$ Galois extension of local fields. We shall write $w, mathcal O_L, mathfrak p$ for the valuation, the ring of integers and its maximal ideal for $L$ . As a consequence of Hensel's lemma, one can write $mathcal O_L = mathcal O_K[alpha]$ for some $alpha in L$ where $O_K$ is the ring of integers of $K$ .^[1] (This is stronger than the primitive element theorem.) Then, for each integer $i ge -1$ , we define $G_i$ to be the set of all $s in G$ that satisfies the following equivalent conditions.

(i) $s$ operates trivially on $mathcal O_L / mathfrak p^i+1.$

(ii) $w(s(x) - x) ge i+1$ for all $x in mathcal O_L$

(iii) $w(s(alpha) - alpha) ge i+1.$

The group $G_i$ is called $i$ -th ramification group. They form a decreasing filtration,

G_-1 = G supset G_0 supset G_1 supset dots *.

In fact, the $G_i$ are normal by (i) and trivial for sufficiently large $i$ by (iii). For the lowest indices, it is customary to call $G_0$ the inertia subgroup of $G$ because of its relation to splitting of prime ideals, while $G_1$ the wild inertia subgroup of $G$ . The quotient $G_0 / G_1$ is called the tame quotient.

The Galois group $G$ and its subgroups $G_i$ are studied by employing the above filtration or, more specifically, the corresponding quotients. In particular,

$G/G_0 = operatornameGal(l/k),$ where $l, k$ are the (finite) residue fields of $L, K$ .^[2]

$G_0 = 1 Leftrightarrow L/K$ is unramified.

$G_1 = 1 Leftrightarrow L/K$ is tamely ramified (i.e., the ramification index is prime to the residue characteristic.)

The study of ramification groups reduces to the totally ramified case since one has $G_i = (G_0)_i$ for $i ge 0$ .

One also defines the function $i_G(s) = w(s(alpha) - alpha), s in G$ . (ii) in the above shows $i_G$ is independent of choice of $alpha$ and, moreover, the study of the filtration $G_i$ is essentially equivalent to that of $i_G$ .^[3] $i_G$ satisfies the following: for $s, t in G$ ,

$i_G(s) ge i + 1 Leftrightarrow s in G_i.$

$i_G(t s t^-1) = i_G(s).$

$i_G(st) ge min i_G(s), i_G(t) .$

Fix a uniformizer $pi$ of $L$ . Then $s mapsto s(pi)/pi$ induces the injection $G_i/G_i+1 to U_L, i/U_L, i+1, i ge 0$ where $U_L, 0 = mathcalO_L^times, U_L, i = 1 + mathfrakp^i$ . (The map actually does not depend on the choice of the uniformizer.^[4]) It follows from this^[5]

$G_0/G_1$ is cyclic of order prime to $p$

$G_i/G_i+1$ is a product of cyclic groups of order $p$ .

In particular, $G_1$ is a p-group and $G_0$ is solvable.

The ramification groups can be used to compute the different $mathfrakD_L/K$ of the extension $L/K$ and that of subextensions:^[6]

G_i

If $H$ is a normal subgroup of $G$ , then, for $sigma in G$ , $i_G/H(sigma) = 1 over e_L/K sum_s mapsto sigma i_G(s)$ .^[7]

Combining this with the above one obtains: for a subextension $F/K$ corresponding to $H$ ,

v_F(mathfrakD_F/K) = 1 over e_L/F sum_s notin H i_G(s).

If $s in G_i, t in G_j, i, j ge 1$ , then $sts^-1t^-1 in G_i+j+1$ .^[8] In the terminology of Lazard, this can be understood to mean the Lie algebra $operatornamegr(G_1) = sum_i ge 1 G_i/G_i+1$ is abelian.

Example: the cyclotomic extension

The ramification groups for a cyclotomic extension $displaystyle K_n:=mathbf Q _p(zeta )/mathbf Q _p$ , where $zeta$ is a $p^n$ -th primitive root of unity, can be described explicitly:^[9]

displaystyle G_s=Gal(K_n/K_e),

where e is chosen such that $displaystyle p^e-1leq s<p^e$ .

Example: a quartic extension

Let K be the extension of $Q 2$ generated by $displaystyle x_1=sqrt 2+sqrt 2$ . The conjugates of x₁ are x₂= $displaystyle x_2=sqrt 2-sqrt 2 ,$ x₃ = −x₁, x₄ = −x₂.

A little computation shows that the quotient of any two of these is a unit. Hence they all generate the same ideal; call it $π$ . $sqrt 2$ generates $π$ ²; (2)= $π$ ⁴.

Now x₁ − x₃ = 2x₁, which is in $π$ ⁵.

and $displaystyle x_1-x_2=sqrt 4-2sqrt 2,,,$ which is in $π$ ³.

Various methods show that the Galois group of K is $C_4$ , cyclic of order 4. Also:

displaystyle G_0=G_1=G_2=C_4.

and $displaystyle G_3=G_4=(13)(24).$

$displaystyle w(mathfrak D_K/Q_2)=3+3+3+1+1=11,$ so that the different $displaystyle mathfrak D_K/Q_2=pi ^11$

x₁ satisfies x⁴ − 4x² + 2, which has discriminant 2048 = 2¹¹.

Ramification groups in upper numbering

If $u$ is a real number $ge -1$ , let $G_u$ denote $G_i$ where i the least integer $ge u$ . In other words, $s in G_u Leftrightarrow i_G(s) ge u+1.$ Define $phi$ by^[10]

phi(u) = int_0^u dt over (G_0 : G_t)

where, by convention, $(G_0 : G_t)$ is equal to $(G_-1 : G_0)^-1$ if $t = -1$ and is equal to $1$ for $-1 < t le 0$ .^[11] Then $phi(u) = u$ for $-1 le u le 0$ . It is immediate that $phi$ is continuous and strictly increasing, and thus has the continuous inverse function $psi$ defined on $[-1, infty)$ . Define
$G^v = G_psi(v)$ .
$G^v$ is then called the v-th ramification group in upper numbering. In other words, $G^phi(u) = G_u$ . Note $G^-1 = G, G^0 = G_0$ . The upper numbering is defined so as to be compatible with passage to quotients:^[12] if $H$ is normal in $G$ , then

(G/H)^v = G^v H / H

for all

v

(whereas lower numbering is compatible with passage to subgroups.)

Herbrand's theorem states that the ramification groups in the lower numbering satisfy $G_u H/H = (G/H)_v$ (for $v = phi_L/F(u)$ where $L/F$ is the subextension corresponding to $H$ ), and that the ramification groups in the upper numbering satisfy $G^u H/H = (G/H)^u$ .^[13]^[14] This allows one to define ramification groups in the upper numbering for infinite Galois extensions (such as the absolute Galois group of a local field) from the inverse system of ramification groups for finite subextensions.

The upper numbering for an abelian extension is important because of the Hasse–Arf theorem. It states that if $G$ is abelian, then the jumps in the filtration $G^v$ are integers; i.e., $G_i = G_i+1$ whenever $phi(i)$ is not an integer.^[15]

The upper numbering is compatible with the filtration of the norm residue group by the unit groups under the Artin isomorphism. The image of $G^n(L/K)$ under the isomorphism

G(L/K)^mathrmab leftrightarrow K^*/N_L/K(L^*)

is just^[16]

U^n_K / (U^n_K cap N_L/K(L^*)) .

Notes

^ Neukirch (1999) p.178

^ since $G/G_0$ is canonically isomorphic to the decomposition group.

^ Serre (1979) p.62

^ Conrad

^ Use $U_L, 0/U_L, 1 simeq l^times$ and $U_L, i/U_L, i+1 approx l^+$

^ Serre (1979) 4.1 Prop.4, p.64

^ Serre (1979) 4.1. Prop.3, p.63

^ Serre (1979) 4.2. Proposition 10.

^ Serre, Corps locaux. Ch. IV, §4, Proposition 18

^ Serre (1967) p.156

^ Neukirch (1999) p.179

^ Serre (1967) p.155

^ Neukirch (1999) p.180

^ Serre (1979) p.75

^ Neukirch (1999) p.355

^ Snaith (1994) pp.30-31

References

B. Conrad, Math 248A. Higher ramification groups

Fröhlich, A.; Taylor, M.J. (1991). Algebraic number theory. Cambridge studies in advanced mathematics. 27. Cambridge University Press. ISBN 0-521-36664-X. Zbl 0744.11001..mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em

Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.

Serre, Jean-Pierre (1967). "VI. Local class field theory". In Cassels, J.W.S.; Fröhlich, A. Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union. London: Academic Press. pp. 128–161. Zbl 0153.07403.

Serre, Jean-Pierre (1979). Local Fields. Graduate Texts in Mathematics. 67. Translated by Greenberg, Marvin Jay. Berlin, New York: Springer-Verlag. ISBN 0-387-90424-7. MR 0554237. Zbl 0423.12016.

Snaith, Victor P. (1994). Galois module structure. Fields Institute monographs. Providence, RI: American Mathematical Society. ISBN 0-8218-0264-X. Zbl 0830.11042.

[N178-1] Neukirch (1999) p.178

[2] since $G/G_0$ is canonically isomorphic to the decomposition group.

[S7962-3] Serre (1979) p.62

[4] Conrad

[5] Use $U_L, 0/U_L, 1 simeq l^times$ and $U_L, i/U_L, i+1 approx l^+$

[S64-6] Serre (1979) 4.1 Prop.4, p.64

[S63-7] Serre (1979) 4.1. Prop.3, p.63

[8] Serre (1979) 4.2. Proposition 10.

[9] Serre, Corps locaux. Ch. IV, §4, Proposition 18

[S67156-10] Serre (1967) p.156

[N179-11] Neukirch (1999) p.179

[S67155-12] Serre (1967) p.155

[N180-13] Neukirch (1999) p.180

[S75-14] Serre (1979) p.75

[N355-15] Neukirch (1999) p.355

[Sn3031-16] Snaith (1994) pp.30-31

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Xrthnty

Ramification group

Contents

Ramification groups in lower numbering

Example: the cyclotomic extension

Example: a quartic extension

Ramification groups in upper numbering

See also

Notes

References

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