HARUZO HIDA. on the inertia I p for the p-adic cyclotomic character N ordered from top to bottom as a 1 a 2 0 a d. Thus. a 2 0 N. ( N a 1.

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1 L-INVARIANT OF -ADIC L-FUNCTIONS HARUZO HIDA 1. Lecture 1 Let Q C be the field of all algebraic numbers. We fix a rime >2and a -adic absolute value on Q. Then C is the comletion of Q under. We write W = { x K x < 1 } for the -adic integer ring of sufficiently large extension K/Q inside C. We write Q for the field of all numbers in C algebraic over Q. Start with a strictly comatible system {ρ l } of semi-simle Galois reresentations ρ l : Gal(Q/Q GL d (E l for rimes l of the coefficient field E Q. We assume that ρ does not contain the trivial reresentation as a subquotient. We write S for the finite set of ramification of ρ and ρ l is unramified outside S {,l}, where l is the residual characteristic of l. We write = { ξ O E ξ < 1 } and often write W := O E,, where O E is the integer ring of E. Often we just write ρ for ρ which acts on V = E. d For simlicity, we assume that S. Let E l (X = det(1 ρ q (Frob l VIl X E[X] (assuming q l. We always assume that ρ is ordinary in the following sense: ρ restricted to Gal(Q /Q is uer triangular with diagonal characters N a j on the inertia I for the -adic cyclotomic character N ordered from to to bottom as a 1 a 2 0 a d. Thus ( N a 1 0 N ρ I = a N a d In other words, we have a decreasing filtration F i+1 V F i V stable under Gal(Q /Q such that the Tate twists gr i (V ( i :=(F i V/F i+1 V ( i is unramified. Define H (X = i det(1 Frob gr i (V ( i i X= (1 α j X. j=1 Then it is believed to be E (X =H (X if S and E (X H (X otherwise. In any case, ord (α j Z. Let us define { α j if ord (α j 1, β j = α 1 j if ord (α j 0 Date: July 30, The first lecture (90 minutes at TIFR on July 30, 2008, Mumbai. The author is artially suorted by the NSF grant: DMS , DMS and DMS

2 L-INVARIANT OF -ADIC L-FUNCTIONS 2 and ut e = {j β j = }. E(ρ = (1 β j 1 and E + (ρ = j=1 j=1,β j (1 β j 1. Then the comlex L-function is defined by L(s, ρ = l E l(l s 1 whose value at 1 is critical. We suose to have an algebraicity result (basically conjectured by Deligne that for a well defined eriod c + (ρ(1 C such that L(s,ρ ε Q for all finite order c + (ρ(1 characters ε : Z µ (Q. Then we should have Conjecture 1.1 (L-invariant. There exist a ower series Φ an (X W [[X]] and a - adic L-function L an (s, ρ =Φ an ρ (γ 1 s 1 interolating L(1,ρ ε for -ower order character ε such that Φ an ρ (ε(γ 1 E(ρ ε L(1,ρ ε with the modifying -factor c + (ρ(1 E(ρ as above (utting E(ρ ε =1if ε 1. The L-function L an (s, ρ has zero of order e + ord s=1 L(s, ρ for a nonzero constant L an (ρ C (called the analytic L-invariant, we have L an lim s 1 (s, ρ (s 1 = e Lan (ρe + (ρ L(1,ρ c + (ρ(1, where lim s 1 isa-adic limit, c + (ρ(1 is the transcendental factor of the critical comlex L-value L(1,ρ and E + (ρ is the roduct of nonvanishing modifying -factors. When e>0, we call that L an (s, ρ has an excetional zero at s = 1. Here is an examle. Start with a Dirichlet character χ :(Z/N Z Q with χ( 1 = 1. Then c(ρ + (1 = (2πi. If we suose χ = ( for a square free ositive ( integer D, the modifying Euler factor vanishes at s = 1 if the Legendre symbol =1 ( = in O Q[ ] with = { x O Q[ ] x < 1 }. By a work of Kubota Leooldt and Iwasawa, we have a -adic analytic L-function L an (s, χ =Φan (γ 1 s 1 for a ower series Φ an (X Λ=W [[X]] and γ =1+ such that for E(χN m =(1 χ( m 1 L an (m, χ =Φ an (γ 1 m 1 = E(χN m L(1 m, χ E(χN m L(m, χ (2πi m for all ositive integer m as long as n m n < 1 for all n rime to. If we have an excetional zero at 1, it aears that we lose the exact connection of the -adic L-value and the corresonding comlex L-value. However, the conjecture says we can recover the comlex L-value via an aroriate derivative of the -adic L- function as long as we can comute L an (ρ. We may regard χ as a Galois character Gal(Q[µ N ]/Q =(Z/N Z {±1}, χ and we remark that χ(frob = 1 to have the excetional zero. For our later use, for the class number h of Q[ ], we write the generator of h as ϖ; so, h =(ϖ for ϖ Q[ ]. Though we formulated conjecture for S, ifρ is ordinary semi-stable, we have the same henomena and can formulate the conjecture. Here is such an examle. Start with an ellitic curve E /Q, which yields a comatible system ρ E := {T l E} given by the l-adic Tate module T l E. Suose that E has slit multilicative reduction at. In this case, H (X =(1 X(1 X and E (X =(1 X, E(ρ E =0 and E + (ρ = 1. Then by the solution of the Shimura-Taniyama conjecture by Wiles

3 L-INVARIANT OF -ADIC L-FUNCTIONS 3 et al, this L-function has a -adic analogue constructed by Mazur such that we have Φ an E (X Λ with Φan E (ε(γ 1 = E(ρ E ε G(ε 1 L(1,E,ε Ω E for all -ower order character ε : Z W ; in other words, L an (s, E =Φ an E (γ 1 s 1. Here Ω E is the eriod of the Néron differential of E. Thus if Frob has eigenvalue 1 on T l E, the excetional zero aears at s = 1 as in the case of Dirichlet character. The Frob has eigenvalue 1 if and only if E has multilicative reduction mod. The roblem of L-invariant is to comute exlicitly the L-invariant L an (ρ. The L-invariant in the cases where ρ = χ = ( as above and ρ = ρe for E with slit multilicative reduction is comuted in the 1970s to 90s, and the results are Theorem 1.2. Let the notation and the assumtion be as above. (1 L an (χ = log (q = log (q ord (q for q C h given by q = ϖ/ϖ ( h =(ϖ and the class number h of Q[ ] (Gross Koblitz and Ferrero Greenberg; (2 For E slit multilicative at, writing E(C =C /qz for the Tate eriod q Q, we have L an (ρ E = log (q ord (q. This was conjectured by Mazur Tate Teitelbaum and later roven by Greenberg Stevens. Here log is the Iwasawa logarithm and x = ord(x Arithmetic L-invariant. Starting with an ordinary -adic Galois reresentation ρ : Gal(Q/Q G(W for a slit reductive grou G /Z, there is a systematic way to create many Galois reresentations whose eigenvalues of Frob contain 1. Indeed, let ρ acts on the Lie algebra s of the derived grou of G by the adjoint action. If G = GL(n, s = {x M n (W Tr(x =0} and the adjoint action is by conjugation. Write this Galois reresentation as Ad G (ρ (or just Ad(ρ. If G = GL(n, since M n (W =Ad(ρ {scalar matrices}, the action of Ad(ρ(Frob onad(w has eigenvalue 1 with multilicity n 1. If G is symlectic or orthogonal, Ad(ρ is often critical at s =1. Now require that ρ F : Gal(Q/F GL 2 (W be a totally odd -ordinary Galois reresentation of dimension 2 over a totally real field F. We make Ad(ρ F and consider the induced reresentation ρ := Ind Q F Ad(ρ F whose eigenvalues of Frob has 1 with multilicity e for the number e of rime factors of in F. Returning to a general ordinary reresentation ρ = ρ, we describe an arithmetic way of constructing -adic L-function due to Iwasawa and others. We can define Galois cohomologically the Selmer grou Sel(ρ Z Q /Z H 1 (Gal(Q/Q,ρ Q /Z for the Z -extension Q /Q inside Q(µ. The Galois grou Γ = Gal(Q /Q acts on H 1 (Gal(Q/Q,ρ Q /Z and hence on Sel(ρ, making it as a discrete module over the grou algebra W [[Γ]] = lim n W [Γ/Γ n ]. Identifying Γ with 1 + Z by the cyclotomic character, we may regard γ Γ. Then W [[Γ]] = Λbyγ 1+X. By the classification theory of comact Λ-modules, the Pontryagin dual Sel (ρ has a Λ-linear ma into f Ω Λ/fΛ with finite kernel and cokernel for a finite set Ω Λ. The ower series Φ ρ = f Ω f(x is uniquely determined u to unit multile. We then define L (s, ρ =Φ ρ (γ 1 s 1. Greenberg gave a recie of defining L(ρ for this L (s, ρ and verified in 1994 the conjecture for this L (s, ρ excet for the nonvanishing of L(ρ

4 L-INVARIANT OF -ADIC L-FUNCTIONS 4 (under some restrictive conditions. For the adjoint square Ad(ρ F for ρ F associated to a Hilbert modular form, the conjecture (excet for the nonvanishing of L(ρ was again roven in my aer in Israel journal (in 2000 under the milder conditions, for examle, ρ F =(ρ F mod m W is absolutely irreducible over Gal(Q/F [µ ] and the -distinguishedness condition for ρ F Gal(F /F for all (which we recall later. If there exists an analytic -adic L-function L an (s, ρ =Φ an ρ (γ 1 s 1 interolating comlex L-values, the main conjecture of Iwasawa s theory confirms Φ ρ =Φ an ρ u to unit multile. Suose now that ρ F is associated to a Hilbert modular Hecke eigenform of weight k 2 over a totally real field F. Following Greenberg s recie, we try to comute L(Ad(ρ F = L(Ind Q F Ad(ρ F exlicitly. By ordinarity, we have ρ F Gal(Q /F = ( 0 α with distinct diagonal characters factoring through I Gal(F [µ ]/F for the inertia grou I for all. We consider the universal nearly ordinary deformation ρ : Gal(Q/Q GL 2 (R overk with the ro-artinian local universal K-algebra R. This means that for any Artinian local K-algebra A with maximal ideal m A and any Galois reresentation ρ A : Gal(Q/F GL 2 (A such that (1 unramified outside ramified rimes for ρ F ; (2 ρ A Gal(Q /F = ( 0 α A, with α A, α mod m A such that the diagonal characters factor through I Gal(F [µ ]/F for all ; (3 det(ρ A = det ρ F ; (4 ρ A ρ F mod m A, there exists a unique K-algebra homomorhism ϕ : R A such that ϕ ρ = ρ A. Write Γ = Z for the -rofinite art of the inertia subgrou I of Gal(F [µ ]/F. Choose a generator γ of Γ and identify W [[Γ ]] with W [[X ]] by γ 1+X. Since ρ Gal(Q /F = ( 0 δ, δ α 1 : I R factors through Γ and induces an algebra structure on R over W [[X ]]. Thus R is an algebra over K[[X ]]. If we write ϕ ρ : R K for the morhism with ϕ ρ ρ = ρ F, by our construction, Ker(ϕ ρ (X =(X. We state a conjecture: Conjecture 1.3 (Deformation. We have R = K[[X ]]. The conjecture is verified by Kisin when F = Q via Serre s modularity conjecture roven by Khare/Wintenberger/Kisin. For general F Q, this follows from the work of Skinner-Wiles, Kisin, Lin Chen and Fujiwara generalizing the fundamental work by Wiles and Taylor-Wiles if Im(ρ E mod is nonsoluble. Here is my theorem: Theorem 1.4. Assume Conjecture 1.3. Then for the local Artin symbol [, F ]= Frob, we have ( L(Ind Q F Ad(ρ δ ([, F ] F = det X, log (γ α ([, F ] 1, X=0 where γ is the generator of Γ by which we identify the grou algebra W [[Γ ]] with W [[X ]]. Here are some examles showing usefulness of this theorem: Take a totally imaginary quadratic extension M/F in which all rime factors in F slits as PP. Take asetσ={p } so that Σ Σ is the set of all rime factors of in M. Write h for the

5 L-INVARIANT OF -ADIC L-FUNCTIONS 5 class number of M and choose ϖ(p M so that P h =(ϖ(p for P Σ. For any Galois character ψ : Gal(Q/M W of M with ψ(σ ψ c (σ for ψ c (σ =ψ(cσc 1 and a comlex conjugation c Gal(Q/F, we have Ad(Ind F M ψ =χ Ind F M ψ 1 c for ( χ = M/F, and we can easily show L(χ =L(Ad(Ind F M ψ. The arithmetic -adic L-function L (s, χ for χ = constructed à la Iwasawa has an excetional zero ( M/F of order e for e = Σ. Since we can comute exlicitely the universal deformation ρ of ρ = Ind F M ψ, we get from the theorem Corollary 1.5. We have L(χ = det(log (N P (ϖ(p(1 c P,P Σ u to a simle constant, P Σ ord(n P(ϖ(P (1 c where N P is the local norm N MP /Q. If E /F is an ellitic curve with slit multilicative reduction at all, we write E /F (F = F /q Z for the Tate eriod q F. Then we have directly from the theorem the following Corollary 1.6 (L-invariant. The L-invariant Ad(ρ E is given by L(Ind Q F Ad(ρ E = where N is the local norm N F/Q. log (N (q ord (N (q, A more general case is treated in my aer aeared in IMRN 2007 Vol. 2007, Article ID rnm102, doi: /imrn/rnm102 (osted in my web age and the roof of the case treated in this note is given in my book Hilbert modular forms and Iwasawa theory from Oxford University Press. The roof of the factorization of the L-invariant of the above corollary is automorhic in the book (a Hecke algebra argument and is Galois cohomological in the IMRN aer (and in my aers quoted in the IMRN aer, all osted in my web age.