Consider ƒ € N and let pr : Ga — GLa(Ó) be the associated Galois representa- tion (after fixing a lattice in #”). Let ỉứ;: Gq —> GLa(F) be its reduction modulo
Xr. Since pz is unramified away from S = {2, ¢}, it factors through Gas, the Galois group of the maximal Galois extension of Q unramified away from S. Let Gx,s be the image of Gx under the map Gx ~ Gq — Gag. Then we have a commutative diagram
0 >GK >GaQ › Gal(K/Q)——>0
|} of
0——>Œy„g——>Gq,g—— Gal(K/Q)——>0
in which the rows are exact. We will be considering deformations of the represenation
Pr : Gas — Gho(F) and py x := Prleg.s- From now on we assume that both 9; and ỉ; „ are absolutely irreducible. Let (Rg, pg) and (Rx, px) denote the universal couples of ứ; and py x respectively, which exist by Theorem 8.2.1. We will denote
Mr and mr, by mg and mx respectively. Let A be a density zero set of primes
of Q and g € N, g =a f. Then py : Gas —> GLe(Q) is a deformation of ỉ;, and pglc, is a deformation of Ps x (this follows from Lemma 8.1.7). As in the proof of Proposition 8.1.13 we identify Tm, and Tw, with appropriate subalgebras
of ]] ge, gap O and of Hye N. g=uf Oằ Tespectively. Let T denote the O-subalgebra
of T generated by the operators T,, for p # 2, @ and let T’ denote the O-subalgebra
of T’ generated by the operators 7; (p split p # £) and by 772 (p inert, p 4 £). We put My := Tn my and tị := T n my. Let Ly denote the subset of NV consisting of
those eigenforms which are congruent to ƒ except possibly at 2 or ý. Similarly let 2; be the subset of N consisting of those eigenforms which are weakly congruent to
f except possibly at 2 or ý. We have 3; C &. Note that we can identify T,, (resp.
Tạ, ) with a subalgebra of ll;s>, Ó (resp. Ilex O) via the map T; +> (b,(p)) gen;
(resp. T, (0;(p));ex,), where g = yr bg(n)g”. Consider the representations
0:= ll;s>, Pg: Gas > GL (Tycx, 0), and ỉ := ứ|e„,s. Choose bases for each p,
so that 6, = py for all g, g’ € Uy, and so that p,(c) = [! _¡] for all g € >;, where c
is the complex conjugation.
Lemma 8.2.2. The image of the representation p is contained in GL2(Ta,).
Proof. [DDT97], Lemma 3.27. O
We claim that ứ'(Œz,s) is contained in the image of GLạ(T, ) inside GLa(Ta,).raf
my
To prove it, let @ denote the map Ty — Ta ; induced by TT. In fact, if we
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identify Ta, with a subalgebra of [] O and Tạ, with a subalgebra of [] 962, O,g€5/ then  is just the restriction to Tạ, of the natural projection Toes, O -ằ In O.
It is easy to see that o(Liy ) is local. It is clearly complete and Noetherian, and its residue field is F. Hence o(Tyy) is an object of C. Consider pf’ : Ớxys — GLằ (Tye, 0), ð(ỉ) = (04(2) ges, è.e., we have gop! = ỉ'. For 7 € Gx,9 we denote
by [7] the conjugacy class of 7 in Gx.g. Note that Œx,s is topologically generated
by the set |J;csseo„,pazzsltFrobpl. Moreover, if p is inert, then Frobp = Frobs,
where p = pOx, while if p = pp is split, then the choice of the prime $8 of the ring of integers of the maximal Galois extension of Q unramified away from Š which lies over p, determines whether Frob, = Frob, or Frob, = Frobg. In any case, switching from one choice to the other results in conjugating the fixed decomposition group of the prime ?# by an element of Go s, which doesn’t alter the trace of p’. Hence for a split p, we have trp’(Frob,) = trp’(Frobp) = trp’(Frob,). Note that for p split trứ(Frob,) = Tp € Tạ, and for p inert trứ(Erob2) = (trứ'Œtrob,g))? + det p/(Frob,) = T? — p*-? € Tạ, . Thus trỉ(Gx,s) C Tạ, which implies that trứ(Gx,s) = tr (doZ(Gr,s)) C ú(T, ). By a theorem of Mazur ([Maz97], Corollary
6, page 256)) this implies that (after possibly changing the basis of p’), we have
ỉứ{GŒx,s) C GLa(2(T,)). Then by Lemma 4.1 in [dSL97], ứ is a deformation of
py and ỉ : Ges — GLo(o(Tiy, )) is a deformation of ỉ;„. Hence there are unique O-algebra homomorphisms ¢g : Rg — Tạ, and éx : ly — d(T), such that
69° pg © p, and ox o px & p’. In fact as pale, is a deformation of p; x, there is a unique O-algebra homomorphism Ú : Rx —› Rg, such that Wo pr % pale,-. Hence
we get the following diagram
(8.3) Re—— Ro
o(Tyy) —— Tạ,
where ¿ denotes the embedding o(Tyy )C Tạ. As the composite
10 p': Gx — GLo(o(Ty)) + GLo(Ta,)m
is a deformation of ỉ; x, there is a unique O-algebra homomorphism a: Ry —> Tạ, "
such that œòy © Lop’. Since dx opK & p' we get LodgopK & Lop’, hence ,oĨy =a
by uniqueness of a. On the other hand as stated in the paragraph before diagram (8.3), Wo px & pale,x, thus đo 9 9 pK óo 9 pala,. Since dg 9 pg & p, we have
$0 9 Polex â Pla, = +40 ứ. Hence Âg° Wo px 140’, which implies as before that
óo 9Ú =a. So, uoóy = óoo0, and thus the above diagram commutes. Moreover as
tr p(Frob,) = Tp for p # 2, and óo(tr pg(Frob,)) = tr (¢gopq(Frob,)) = tr p(Frob,),
we see that all T, for p # 2, £ are in the image of Rg. So, óo is surjective. Similarly óx
is surjective. Our goal now is to prove surjectivity of ý which will imply surjectivity
of ¿.
The map ý : Rx — Rg is local (by the virtue of residue fields of Rg and Rx being the same), hence induces an O/A-linear homomorphism on the cotangent spaces
mx /(mz,ARK) > mọ/(mỗ, Rg), which we will call woe. (Note that as O > Rx is local, the image of À is contained in mx; however, the ideal ARx Z mx. Similarly for Rg.) Let C denote the cokernel of /„ : ma /(mz,ARK) — mọ/(m,ÀRq). As the above map is just a map of F-vector spaces, the sequence of dual maps is exact:
0 — Homp(C, F) + Homp(mg/(m3,, ARg), F) “+ Homp(mx/(m%, ARK), F).
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where 5: œ >œo„;. We will prove that Wi, is injective, which will imply that C=0.
8.2.3 Zariski tangent space of the deformation functor
For a profinite group G and the universal couple (R¥’, op’) of a representation ứ:G — GL;(F), there is an action of G on the space of 2x2 matrices with entries in
F given by g-M = p(g)Mp(g)7!. This G-module will be denoted ad(5). We will write
R instead of RY to shorten notation. The F-vector space Homg(mz/(mậ, AR), F)
is called the Zariski tangent space of the deformation functor representaed by R.
Lemma 8.2.3. If 7 is absolutely irreducible, then
Homg(ma/(m, \R), F) % H*(G,ad(p)),
where the cohomology group stands for continuous group cohomology and ad(p) is a discrete G-module.
Proof. [Hid00], Lemma 2.29. Oo
When G = Gq (or G = Gx) and R = Rg (or R = Rx), we will denote the isomorphism from Lemma 8.2.3 by tg (or tx, respectively).
Proposition 8.2.4. The following diagram is commutative:
(8.4) Homr(mo/(mộ, \Rg), F) —2—> Homp(mx/(m3,, ARx), F)
‘al: ft
H' (Gg, ad(Ð)) = > H1(G, ad(p))
Proof. We first unravel the isomorphism tr from Lemma 8.2.3. There is a natural sequence of bijections:
(8.5)
Homp(mp/(m, AR), F) % Homp_aig(B/(m2, AR), Fle]) Homo_az(R, Fle]),
where Fle} denotes the dual numbers. To explain this we first note that the exact sequence of F-vector spaces
0 — mg/(m2 + AR) > R/(m, +R) > F- 0
has a natural splitting sp : F #/(mệ + ÀR) coming from the structure morphism
O — R. This provides us with an isomorphism of F-vector spaces
(8.6) R/(m?, + \R) = P ©emp/(m2 + AR).
Let pr denote the natural epimophism R/(m? + AR) -ằ R/mpg = F. Then the map (8.6) is given by (pr,idp — so pr). The inverse to the map (8.6) is given
by sz +i, where i is the embedding of mg/(m2 + AR) into R/(mộ +ÀR). To see that this map is the inverse of (8.6), one only needs to show that the composite ể@nosg: F —ơ R/(m2,+AR) — F is the identity. This follows from the commutativity
of the following diagram
© ———*> R/(m2, + AR)
F = +F | =
The upper triangle is clearly commutative, and so is the square since all maps in C induce identity on the residue fields. This implies that the lower triangle commutes
as well, which is the asserted claim.
The first bijection in the sequence (8.5) comes from the F-vector space decom- position (8.6). Indeed, any O-algebra map from R to Fle] must factor through R/(m2,ÀAR). Also, any such map being an O-algebra map must send F in the de- composition R/(m?,, \R) = F@ma/(mậ, AR) to F in the decomposition Fe] = F@Fe
via the identity map, and being local it must send the maximal ideal part into the
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maximal ideal Fe of Fle]. Hence any such map being identity on the F-factor is com- pletely determined by what it does to my/(m%, AR). On this factor the map can be any F-vector space map. Identifying Fe with F we get the first bijection in (8.5). The second bijection in (8.5) stems from the fact that the maximal ideal of F[e] has square zero. To summarize, an element a € Homg(ma/(mậ, AR), F), is mapped (under the above sequence of bijections) to the element (idg @€- a) o (pr, idrp — sRO pg) OTR
of Homo_iz(J, F[e]), where mz : — R/(m%,R) is the natural projection, and idp 6 €-a@ sends (r,m) € F @mp/(mệ, AR) to the element r + €-a(m) € Fe]. For any element @ of Homo_aig(R, Fle]), the representation pg := Bo pTM :
G — GL;(F[e]) is a deformation of p called an infinitesimal deformation. It is unique up to strict equivalence. We write pg(g)p(g)~' = 1+ ecg(g) € Fle]. One then shows that cg is a cocycle in H'(G,ad(p)). Two such cocycles differ by a coboundary exactly when the two infinitesimal deformations giving rise to those cocycles are strictly equivalent. We conclude that tp is the composite of the maps
œ > (idg G€- œ) o (pr, idrp — sgS ún) OTR and Gr cg. To prove that the diagram (8.4) is commutative we need to show that resotg = tx o /5,. We will prove that both squares in the diagram (8.7) below are commutative.
(8.7) Homr(mo/(mộ, AR), F) —”#— Homg(mz /(mậ., Rx), F)
Homo-aig(Ro, F[s|) “ + Homo_ale(Rx, F|a|) | |
pe iors
H*(Gq, ad(2)) = > H!(Gk, ad())
Here the top left vertical arrow is the map
œ r> (idp B€- a) o (pg,idrg — $Q © pg) 9 TQ,
while the top right vertical arrow is the map
at (idp @c- a) o (px, ida, — SK OPK) OTK.
As asserted before the composite of the two left vertical arrows is the map tg, and the composite of the two right vertical arrows is the map tx. We first prove the commutativity of the upper square. To do so, we begin by noting that the following diagrams are commutative:
(8.8) Rx “A+ Rx /(mi + ARK)
| |
Ro 7. Ro/(m3, + \Rq)
(8.9) Re/(m> +ARK) —* F TS Rie /(m2. + ARK)
+ le |e
Ra/(m2, + ARg) — +F 2+ Ro/(md + ARQ)
The commutativity of diagram (8.8) is clear, while the left square in diagram (8.9) commutes because w induces identity on the residue fields and the right one does because w is an F-algebra homomorphism. Thus we conclude that the following diagram commutes:
(8.10)
7 ;idR„.~8K£©9PK)
Rr Ky Ra Im + Rye) ee E59 XE my /(m?. + ARK)
ằ| + |&s-
Rg T— Ro/(mã + rARg) a ›xE@ mọ/(m + ARg)
From this diagram we get (pg, idrg — sạ ©pq) 9q ow = (idr Wet) ° (pK, ldg„ — 8K © PK)OTK. To prove the commutativity of the upper square of diagram (8.7), we need to show that for any a € Homr(mo/(mà+ÀRa), F), we have /*((idg@@-€)©(pạ, idpg — 8Q°Pg) Tg) = (idp@Ÿ,(œ)-e)o(px, idn„ —SK°pK)OmK. However, the left-hand side
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*((Idp@đứ-€)â (nạ, idg¿ — sọopg)9g) = (idp đa-€) 0 (pạ, ldg„ — sạâpQ) âq âÚ =
(idp Ba-€)o (idp @ WH) 9 (px, ide, — s90) by the commutativity of diagram (8.10), which in turn equals (idp 6 ~i,(a)-€) 0 (px, idr, —sx°pK)onmK. This proves that the upper square of diagram (8.7) commutes.
We will now show that the lower square in diagram (8.7) commutes. For this we need to show that for any @ € Homog_aig(Rq/(mg+ARQ), Fle), we have cx o*(8) = res o cg. (Recall that Homo_aiz(Rọ, Fle]) = Homo_aiz(Rọ /(mộ +ÀR), F{e|).) Note that cx ow*(8) = cK o Bow = cK poy), ie., the infinitesimal deformation Goo px :
Gx — Glo(Fle]), satisfies the identity (đe â ứx)(g)8”}{g) = 1+ cx (gow)(g)e for any g € Gx. Recall that ứo|ứ„ is a deformation of J; x, and ứg|ứ„ â â 0x. So, on
the one hand we have (đo 0Q)(g)ỉ; (9) = l1+ecaza(ứ)c. On the other hand replacing
òx with pelc, leads to a cohomologous cocycle, hence we conclude that cx (gow) = co,z|lo„ as elements of H*(Ga,ad(f;s)). Thus cx o J*(ỉ) = res (cg,g) = res o cq(), and the commutativity of the lower square in diagram (8.7) is proved. This finishes the proof of Proposition 8.2.4. L]