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Introduction to Skinner-Urban's Work on the Iwasawa Main Conjecture For Introduction to Skinner-Urban’s Work on the Iwasawa Main Conjecture for GL2 Xin Wan Columbia University [email protected] 1 Introduction These notes are an introduction to the recent work of Christopher Skinner and Eric Urban [11] proving (one divisibility of) the Iwasawa main conjecture for GL2=Q (see Theorem 1). We give the necessary background materials and explain the proofs. We focus on the main ideas instead of the details and therefore will sometimes be brief and even imprecise. These notes are organized as follows. In Section 2 we formulate various Iwasawa main conjectures for modular forms. We also explain an old result of Ribet to illustrate the rough idea of the strategy behind the later proofs. In Sections 3 and 4 we introduce the notions of automorphic forms and Eisenstein series on the unitary group GU(2;2). Section 5 is devoted to explaining the Galois argument. Sections 6-9 give the tools used in the computation for the Fourier and Fourier-Jacobi coefficients of various Eisenstein series, which is a crucial ingredient in the argument. In Section 10 we give an example of a theorem of the author generalizing the Skinner-Urban work. 2 Main conjectures We introduce the objects required to state the Iwasawa main conjectures for GL2. 2.1 Families of Characters ¯ Let p be an odd prime. Choose i : C ' Cp. Let GQ = Gal(Q=Q) and Q¥ ⊂ Q(mp¥ ) be the cyclotomic Zp-extension of Q. Let GQ = Gal(Q¥=Q). Let LQ := Zp[[GQ]]. We also define LA = LQ;A = LQ ⊗Zp A for A a Zp-algebra. Let Y = Y : G ! L × be the composition of G G with G ,! L ×. Let e be a character of ×n × which is the Q Q Q Q Q Q Q Q Q AQ composition of YQ with the reciprocity map of class field theory (normalized using geometric Frobenius elements). Take g 2 G to be the topological generator such that e(g) = 1 + p where e is the cyclotomic character giving the × canonical isomorphism Gal(Q(mp¥ )=Q) ' Zp . For each z 2 mp¥ and integer k we let yk;z be the finite order character of ×n × that is the composition of Y with the map L × ! × that maps g to z(1+ p)k. We also write y for y . Q AQ Q Q Cp z 0;z We let w be the Teichimuller character. 2.2 Characteristic Ideals and Fitting ideals Let A be a Noetherian normal demain and X a finite A-module. The characteristic ideal charA(X) ⊂ A is defined to be zero if X is not torsion and (X) = fx 2 Aj x ≥ (X ); P ⊂ Ag: charA ordP lengthAP P for all height one primes Now take any presentation Ar ! As ! X ! 0 of X. The Fitting ideal is defined by the ideal of A generated by all the determinants of the s × s minors of the matrix representing the first arrow. Remark 1. Fitting ideals respect any base change while characteristic ideals do not in general. 2.3 Selmer Groups for Modular Forms ¥ n × Let f = ∑n=1 anq 2 Sk(N;y0), k ≥ 2, be a cuspidal eigenform with character y0 of (Z=NZ) and let L=Qp be a finite extension containing all Fourier coefficients an of f . Let OL be the ring of integers of L. Assume that f is ordinary, which means that ap is a unit in OL. Let r = r f : GQ ! AutLVf be the usual two dimensional Galois representation + + associated to f . Then it is well known (by [14], for example) that there is a GQp -stable L-line Vf ⊂ Vf such that Vf =Vf + + is unramified. We fix a GQ-stable OL lattice Tf ⊂ Vf and let Tf = Tf \Vf . Definition 1. (Selmer Groups) Let S be a finite set of primes. S (T ) = fH1( ;T ⊗ ∗ ( −1)) ! H1(I ;(T =T +) ⊗ ∗ ( −1)) SelL f : ker Q f OL LOL Y p f f OL LOL Y × H1(I ;T ⊗ ∗ ( −1))g ∏ ` f LOL Y `6=p;`62S where L ∗ = Hom (L ; = ) is the Pontryagin dual and L ∗ (Y −1) means that the Galois action is given by the A Zp A Qp Zp OL character Y −1. Let S S XL (Tf ) := HomZp (SelL(Tf );Qp=Zp) and S S char ( f ) = char (X (Tf )): f ;Q LQ;OL L 2.4 p-adic L-functions Let 0 < n < k −2 be an integer and z 6= 1 a pt−1th root of unity. Let S be a finite set of primes. We define the algebraic part of a special L-value for f by: pt(n+1)n!LS ( f ;y−1wn;n + 1) S −1 n −t z Lalg( f ;y w ;n + 1) := ap( f ) m z n −1 n sgn((−1) ) (−2pi) t(yz w )W f 2 k−1 ± where ap( f ) is the p-adic unit root of x −apx+ p y0 = 0, t(y) is a Gauss sum for y and W f are Hida’s canonical periods of f . (There is also a formula for z = 1 which is more complicated which we omit here). The p-adic L-function is a certain element S 2 L characterized by the following interpolation property. Let f : L ! (z) be L f ;Q Q;OL n;z OL OL n the OL homomorphism sending g to z(1 + p) . Then: ( S ) = LS ( f ; −1 n;n + ); ≤ n ≤ k − : fn;z L f ;Q alg yz w 1 0 2 This was constructed in [1], and also [6]. 2.5 The Main Conjecture The Iwasawa main conjecture for f is the following Conjecture 1. The module XS (T ) is a finite torsion L -module and charS is generated by S . L f Q;OL f ;Q L f ;Q The main result that we are going to prove in this lecture series is: Theorem 1. (Kato, Skinner-Urban) Suppose f has trivial character, weight 2 and good ordinary reduction at p. Sup- pose also that: • The residual representation r¯ f is irreducible. • For some p 6= `jjN, r¯ f is ramified at `. Then the Iwasawa main conjecture is true in LQ;OL ⊗Zp Qp. If, moreover, there exists an OL-basis of Tf with respect to which the image of r f contains SL2(Zp), then the equality holds in LQ;OL . The last condition is put by Kato [4] who proved “⊇”. It is satisfied, for example, by the p-adic Tate modules of semistable elliptic curves if p ≥ 11. We will focus on Skinner-Urban’s proof for “⊆”. The technical condition (ii) can be removed by working with forms over totally real fields and a base change trick . 2 2.6 Two and Three-Variable Main Conjectures More p-adic characters: Let K =Q be an imaginary quadratic extension such that p splits as v0v¯0, where v0 is deter- 2 mined by our chosen isomorphism i : C ' Cp. By class field theory there is a unique Zp-extension of K unramified ¯ outside p, which we denote by K¥. Let GK := Gal(K =K ) and GK := Gal(K¥=K ). There is an action of com- ± ¯ plex conjugation c on GK . We write GK for the subgroup on which c acts by ±1. For any Zp-algebra A ⊂ Qp we 1 1 ± ± ± ± 2 ± 2 ± define: LK ;A = A[[GK ]], LK ;A = A[[GK ]] and generators g of GK by requiring recKp ((1 + p) ;(1 + p) ) = g . Here recKp is the reciprocity map of class field theory. (Note that Kp ' Qp × Qp.) Let YK be the composition × ± ± × × ± GK ! GK ,! LK ;A. We define YK similarly. We also define characters eK ;eK of K nAK by composing eK ;eK with the reciprocity map. Now let f be a cuspidal eigenform of weight k ≥ 2. We define the set of arithmetic points by: a + + k−2 − − ± ¯ ¥ X f ;K := ff : OL homomorphism LK ;OL ! Qp : f(g ) = z (1 + p) ;f(g ) = z ;z 2 mp g: 2 a = 2−k −1 = ( ◦ )( 2−k k−2: ) f For f X f ;K , let qf : w c f xf with xf : f YK e e c f and let qf be the conductor of qf . In particular we show that for any finite set S of primes containing all the bad primes (all the primes where f or K is S ramified), we have the two variable p-adic L-function L f ;K 2 LK ;OL such that: 2 k−2 S ord (Nm(f )) ((k − 2)!) g(qf )Nm(fq d) L ( f ;qf ;k − 1) ( S ) = u a ( f ) p qf : f K f L f ;K f p 2k−2 + − (2pi) W f W f a for any sufficiently ramified f 2 X f ;K , where d is the different of K and u f is a p-adic unit depending on f . We − remark that if z = 1 then our special L-value is just the product of the special L-values for f and f ⊗ cK twisted by some y such that y ◦ Nm = qf . Here cK is the quadratic character for K =Q. S S S We can also define Selmer groups Sel f ;K and Xf ;K , char f ;K in the exact same way as in the one-variable case. We have the two-variable main conjecture: S S S Conjecture 2.X f ;K is a finite torsion LK ;OL -module. Furthermore charK ; f is principal and generated by L f ;K . Additionally f can be embedded in a Hida family of ordinary cuspidal eigenforms f (we discuss these in the next sec- S tion in more detail).
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