International Mathematics Research Notices Advance Access originally published online on January 14, 2009
International Mathematics Research Notices (2009) 2009:912-952, doi:10.1093/imrn/rnn151 published on March 4, 2009
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Quadratic Exercises in Iwasawa Theory
Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095-1555 USA
Correspondence: Correspondence to be sent to: hida{at}math.ucla.edu
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Abstract |
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 TOP Abstract 1. Introduction2. Hilbert Modular Forms...3. Anticyclotomic Iwasawa Series4. Behaviour of p-Adic...References |
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The anticyclotomic main conjecture for CM fields was proven in 2006 under some restrictive conditions. In this paper, we remove the assumption on the conductor of the blanch character, and therefore, the conjecture is now proven to be true under very mild conditions.
Received for publication July 20, 2008. Revision received November 9, 2008. Accepted for publication November 13, 2008.
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1. Introduction |
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TOPAbstract1. Introduction2. Hilbert Modular Forms...3. Anticyclotomic Iwasawa Series4. Behaviour of p-Adic...References |
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The anticyclotomic main conjecture of p-ordinary CM fields M was proven in [14] under some hypothesis. In this paper, we propose the removal of the following hypothesis we made in [14]:
- (S) The prime-to-p part of the conductor of the branch character
is a product of primes of M of relative degree1 over the maximal totally real subfield F of M.
We shall remove this assumption by reducing the conjecture to the result in [14] under (S) by a simple argument of quadratic (automorphic) base change. The condition (S) is equivalent to having the automorphic induction of the character everywhere in the principal series (cf. [26]).
Write R (resp., O) for the integer ring of M (resp., F). The field F is totally real, and M is a totally imaginary quadratic extension of F (inside a fixed algebraic closure 
of 
). We fix an odd prime p>2 unramified in 
. The field M is called p-ordinary if there exists an abelian variety with complex multiplication by M having ordinary good reduction at p. This property can be detected by the CM type of the abelian variety. To describe this fact, fix an embedding 
. A CM-type 
is associated to an abelian variety with CM by M having ordinary good reduction at ip if the embeddings ip
for 
induce exactly a half p of the p-adic places of M. Such a CM type is called a p-ordinary CM type, and we fix a p-ordinary CM type and the set p of associated p-adic places. We identify p with a subset of prime factors of p in M. Fix 
inducing the generator of Gal(M/F). The disjoint union p 
cp is the set of all prime factors of p in M.
We study the arithmetic of the unique 
-extension M–
of M (unramified outside p and ) on which cc–1 = –1 for all 
–M = Gal(M–/M). We choose a complete discrete valuation ring W inside 
finite flat and unramified over 
. A Hecke character of 
is called anticyclotomic if (xc) = (x)–1. By class field theory, if is of finite order, we often regard as a character of 
, and then anticyclotomy of can be interpreted as (cc–1) = ()–1. We write the conductor of as 
for an ideal 
prime to p. Here for a multi-exponent 
, we write 
for 
.
Suppose that has finite order. Let M()/M be the class field cut out by ; in other words, induces the isomorphism 
. Consider the extension M–M()/M() which is called the anticyclotomic tower over M(). Let L/M–M() be the maximal p-abelian extension unramified outside p. Each 
Gal(L/M) acts on the normal subgroup X = XM 
Gal(L/M–M()) continuously by conjugation, and by the commutativity of X, this action factors through Gal(M()M–/M). We fix a splitting Gal(M()M–/M) = –M x Gtor() for the maximal torsion subgroup 
. Then we look into the –M–module: 
.
As is well known, X[] is a W[[–M]]-module of finite type, and if is anticyclotomic nontrivial over 
with p* = (–1)(p–1)/2p, it is proven to be a torsion W[[–M]]-module by a result of Fujiwara (cf. [10, Corollary 5.4] and [15, Theorem 5.33]) generalizing the fundamental work of Wiles [27] and Taylor–Wiles [25]. Thus, we can think of the characteristic element 
of the module X[]. As we have seen in [18] and [19], we have the anticyclotomic p-adic Hecke L-function 
(constructed by Katz), where 
is the completed p-adic integer ring of the maximal unramified extension of 
inside 
. We regard 
. Then the anticyclotomic main conjecture can be stated as follows.Anticyclotomic main conjecture. We have the identity: 
up to a unit in 
.
The main conjecture for imaginary quadratic fields (including the cyclotomic 
-extension) and its anticyclotomic version for imaginary quadratic fields have been proved by K. Rubin refining Kolyvagin's method of Euler systems, and after that, the anticyclotomic version was again treated by J. Tilouine (for imaginary quadratic cases) by a method similar to the one exploited in this paper, combined with the class number formula of the ring class fields. A partial result toward the general conjecture was studied in [14, 16, 18] and [19].
We fix a continuous anticyclotomic character 
of finite order. We shall prove the following theorem.
Theorem. Assume that p>3 and the following three conditions:
- the anticyclotomic character has order prime to p;
- the local character
is nontrivial over 
for all 
;
- the restriction * of to
is nontrivial.
Then the anticyclotomic main conjecture holds.
This theorem was proven in [14] under the aforementioned extra assumption (S) which we remove in this paper. In a forthcoming paper, resorting to a framed version of the "R = T" theorem in [3], we plan to remove assumption (2). Assumption (3) is at this moment difficult to remove (because it is fundamental for the Taylor–Wiles system to work in proving the necessary "R = T" theorems). Assumption (1) can be removed, but it is a technical endeavor; so, for simplicity, we assume it in this paper.
To give a short description of the idea of the proof, decompose 
so that 
is a product of inert prime factors, 
is a product of ramified prime factors 
and 
. Let 
. Write 
, where h(M) and h(F) are class numbers of M and F, respectively. Up to a p-adic unit, 
is the relative class number of the ray class group modulo 
, that is, the ratio of the order of the ray class group 
of M and that of 
of F.
The idea of the proof is to reduce the conjecture to the case under (S) treated in [14] by quadratic base change to a well-chosen totally real quadratic extension F'/F and a further refinement, eliminating the assumption (S), of the method exploited in [18] and [16, Theorem 5.1], where we have proven 
in 
under (S). As is well known (e.g. [15, Lemma 5.31]), we can always find an algebraic Hecke character 
of M such that – = , where – = c–1 with c(x) = (xc). We choose well. Then one of the main ingredients of the proof is the congruence power series H() W[[–M]] of the CM-component of the universal nearly ordinary Hecke algebra h for GL(2)/F associated to the theta series of . In the joint works with Tilouine, we took h of (outside p) level 
for the conductor 
of and the relative discriminant d(M/F) of M/F. In this paper, as in [14] and [16, Section 2.10], we take the Hecke algebra of level 
which is a product of 
and d(M/F) (introducing a new type of Neben character determined by with – = ). Fujiwara formulated his results in [4] using such level groups. Another important ingredient is the following divisibility assertion (without assuming (S)) in Corollary 3.8 which will be proven refining the proof of a similar result in [16, Corollary 5.6] under (S):
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| (A) |
inverting p, applying the method of [16] to the p-primary parts, we prove the µ-invariant inequality
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| (1.1) |
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| (B) |
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| (C) |
, there exists a third quadratic extension M'/F in E/F. Note that M' is a CM field. In summary, we have |
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We can arrange F'/F so that
- (F1) for any prime
of F ramifying in M, the inertia group of 
in Gal(E/F) is given by Gal(E/M') (so any prime factor of 
in F' splits in E/F');
- (F2) for any prime
inert in M/F in the conductor 
of , the decomposition group of 
in Gal(E/F) is given by Gal(E/M') (so any prime factor of 
in F' splits in E/F');
- (F3) E/M ramifies at least one finite place outside p.
- (F2) for any prime
We consider the base change E = NE/M. Then by (F1–2), E has conductor whose prime factors all split in E/F' (so, (S) for E is satisfied). Thus by the main theorem of [14], we have 
up to units in 
. We have the restriction map Res:–E 
–M. By (F3), Res is a surjection. We can verify (see Section 4) that 
for 
and Res(L–E(E)) = L–M()L–M(
) up to units in 
. Thus
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. Note here that remains anticyclotomic (because has order 2). By (C): 
and 
, we conclude the individual identity: |
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. This finishes the proof of the theorem.
The proof of equation 1.1 again involves a base-change technique to quadratic extensions fitting into a diagram similar to the above (since the main result in [17] is again proved under (S)), but we need to choose the real quadratic extension F1/F more carefully than F'/F. We will postpone the exact specification of F1/F to Section 3.3, since the choice is subtle and rather technical.Notation. Here is a basic notation we use without explaining much. We write 
for the adele ring, and 
is the ring of finite adeles; so 
. We regard GL(2)as a linear algebraic group defined over Oand write G for 
. Write I for the set of all embeddings of F into 
, and define 
for the product of I copies of the upper half complex plane 
. A classical Hilbert modular form is a holomorphic function on 
with certain automorphy property (see equation 2.6), and an adelic Hilbert modular form (whose precise definition we will give later) is a function on the idele group 
.
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2. Hilbert Modular Forms and Hecke Algebras |
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TOPAbstract1. Introduction2. Hilbert Modular Forms...3. Anticyclotomic Iwasawa Series4. Behaviour of p-Adic...References |
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Let us give a short description of the adelic/classical Hilbert modular forms and their Hecke algebra of level
(cf. [16], [8, Sections 2.2–4], [13, Sections 4.2.8–4.2.12], and [15, Section 2.3]), limiting ourselves to the extent logically necessary to understand our proof of the key assertion (C) and the theorem. If the reader is familiar with the theory introduced in the above papers and books, he/she can skip this section.
2.1 Double-coset rings
We first recall formal Hecke rings of double cosets. For that, we fix a prime element 
of 
for every prime ideal 
of O. We consider the following open compact subgroup of 
:
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| (2.1) |
and 
. Then we introduce the following semigroup:
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| (2.2) |
is the projection of 
to 
for prime ideals 
. Writing T0 for the maximal diagonal torus of GL(2)/O and putting
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| (2.3) |
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| (2.4) |
with 
, the group U is assumed to be a subgroup of 
with 
for some multi-exponent (though we do not assume that 
is prime to p). Formal finite linear combinations 
cUU of double cosets of U in 
form a ring 
under convolution product (see [23, Chapter 3] or [12, 3.1.6]). The algebra is commutative and is isomorphic to the polynomial ring over the group algebra 
with variables 
for primes 
, 
corresponding to the double coset 
and 
(for primes 
) corresponding to 
. Here we have chosen a prime element 
in 
. The group element 
in 
corresponds to the double coset UuU (cf. [13, Lemma 5.2]).
2.2 Adelic Hilbert modular forms
The double-coset ring 
naturally acts on the space of modular forms on U whose definition we now recall. Recall that T0 is the diagonal torus of GL(2)/O; so 
. Since 
is canonically a quotient of 
for an ideal 
, a character 
can be considered as a character of 
. Writing 
, if 
– = –112 factors through 
for 
, then we can extend the character of 
to 
by putting 
for 
. In this sense, we hereafter assume that is defined modulo 
and regard as a character of 
. We choose a Hecke character 
with infinity type (1 – [
])I (for an integer []) such that +(z) = 1(z)2(z) for 
. We also write t+ for the restriction of + to the maximal torsion subgroup 
of 
(the strict ray class group modulo 
: 
).
Writing T2 for 
(the diagonal torus of G), the group of geometric characters X*(T2) is isomorphic to 
, so that the character 
has value 
at each diagonal matrix 
. Taking 
, we assume []I = 1 + 2 (identifying I with I), and we associate with a factor of automorphy,
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| (2.5) |
, 
, and |
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for 
. We define 
by the space of functions 
satisfying the following three conditions (e.g. [8, Section 2.2] and [13, Section 4.3.1]):- (S1) f(zxu) = (u)t+()f(x)J(u, i)–1 for all
and all u U · Ci and z is any element in the center 
representing an element in the maximal torsion subgroup 
of 
;
- (S2) choose
with u(i) = 
for 
, and put fx() = f(xu)J(u, i) for each 
(which only depends on ). Then fx is a holomorphic function on 
for all x;
- (S3) fx(
) for each x is rapidly decreasing as 
( = 
+ i) for all I uniformly.
- (S2) choose
It is easy to check (e.g. [12, 3.1.5]) that the function fx in (S2) satisfies the classical automorphy condition
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| (2.6) |
. Also by (S3), fx is rapidly decreasing toward all cusps of x (e.g. [12, (3.22)]); so it is a cusp form.- (+) Imposing that f has the central character + in addition to the action of
in (S1), we define the subspace 
of 
.
The symbols = (1, 2) and (
1, 2) here correspond to (2, 1) and (2, 1) in[13, Section 4.2.6, p. 171] and [15, Section 2.3] because of a different notational convention in [13] and [15].
We identify I with in 
. We have S = 0 unless 1 + 2 = [1 + 2]I for 
, because I – (1 + 2) is the infinity type of the central character of automorphic representations generated by S. We write simply [] for 
assuming S 
0. The SL(2)-weight of the central character of an irreducible automorphic representation 
generated by 
is given by k 1 – 2 + I (which specifies the infinity type of as a discrete series representation of 
). There is a geometric meaning of the weight : the Hodge weight of the motive attached to (cf. [2]) is given by {(1,, 2,), (2,, 1,)}, and thus, the requirement 1 – 2 
I is the regularity assumption for the motive (and is equivalent to the classical weight k 2I condition).
Choose a prime element 
of 
for each prime 
of F. We extend 
to 
just by putting 
for 
. This is possible because 
for 
. Similarly, we extend 1 to 
. Then we define 
for 
. Let 
be the unipotent algebraic subgroup of GL(2)/O defined by 
For each 
, we decompose 
for finitely many u and t (see [23, Chapter 3] or [12, 3.1.6]) and define
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| (2.7) |
of cusp forms. This action for y with 
is independent of the choice of the extension of to 
. When 
, we may assume that 
, and in this case, t can be chosen so that 
(so 
is independent of single right cosets in the double coset). If we extend to 
by choosing another prime element 
and write the extension as ', then we have |
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'. Thus the sole difference is the root of unity 
. Since it depends on the choice of 
, we make the choice once and for all, and write 
for 
(if 
). By linearity, these action of double cosets extends to the ring action of the double-coset ring 
.
2.3 Fourier and q-expansion
To introduce rationality of modular forms, we recall Fourier expansion of adelic modular forms (cf. [8, Sections 2.3–4]). Recall the embedding 
, and identify 
with the image of i. Recall also the differental idele 
with 
and 
. Each member f of 
has its Fourier expansion
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| (2.8) |
is the additive character with eF(x) = exp(2iIx) for 
. Here y 
a(y, f) is a function defined on 
only depending on its finite part y(). The function a(y,f) is supported by the set 
of integral ideles.
Let F[] be the field fixed by 
, over which the character X*(T2) is rational. Write O[] for the integer ring of F[]. We also define O[, ] for the integer ring of the field F[, ] generated by the values of over F[]. For any F[, ]-algebra A inside 
, we define
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| (2.9) |
(U, ; A) as the space of A-rational global sections of a line bundle of a variety defined over A (e.g., [15, Chapter 4]); so we have, by the flat base-change theorem (e.g. [11, Lemma 1.10.2]),
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| (2.10) |
(U, ; A) 
EndA(S(U, ; A)) by the A-subalgebra generated by the Hecke operators of 
.
For any 
-algebras A, we define
|
| (2.11) |
a(y, f) extends to a function on 
with values in A. We define the q-expansion coefficients (at p) of f S(U, ; A) by
|
| (2.12) |
with values in the formal monoid algebra 
of the multiplicative semigroup F+ made up of totally positive elements, which is given by
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| (2.13) |
is the character given by 
.
We now define for any p-adically complete O[, ]-algebra A in 
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| (2.14) |
) gives rise to the geometric q-expansion at the infinity cusp of the classical modular form fx for 
(see [7, (1.5)] and [13, (4.63)]).
We choose a complete representative set {ci}i=1,...,h in finite ideles for the strict idele class group 
, where h is the strict class number of F. Let 
. Put 
, and consider 
as defined in (S2). The collection (fi)i=1,...,h determines f, because of the approximation theorem. Then f(cid–1) gives the q-expansion of fi at the Tate abelian variety with 
-polarization 
(
). By the q-expansion principle (e.g. [17, Section 4.1] or [15, 4.2.6]), the q-expansion f(y) determines f uniquely.
2.4 Hilbert modular Hecke algebras
We write T(y) for the Hecke operator acting on S(U, ; A) corresponding to the double coset 
for an integral idele y. We renormalize T(y) to have a p-integral operator 
: 
. Since this only affects T(y) with yp 1, 
if 
. However 
for primes 
. The renormalization is optimal to have the stability of the A-integral spaces under Hecke operators. We define 
for 
, which is equal to the central action of a prime element 
of 
times 
. We have the following formula of the action of 
and 
(e.g. [13], Section 4.2.10[):
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| (2.15) |
of U is the ideal maximal under the condition: 
. Thus 
(up to p-adic units) when 
is a factor of the level of U (even when 
; see [13, (4.65–66)]). Writing the level of U as 
, we assume
|
| (2.16) |
and 
preserve the space S(U, ; A) under this condition (see [13, Theorem 4.28]). We define the Hecke algebra h(U, ; A) (resp., 
) with coefficients in A by the A-subalgebra of the A-linear endomorphism algebra EndA(S(U, ; A)) (resp., 
) generated by the action of the finite group 
, 
and 
for all 
.
We have canonical projections
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β (
for all 
) taking canonical generators to the corresponding ones, which are compatible with inclusions |
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(U, ; A)}U (U running through open subgroups of 
containing 
), whose projective limit (when 1 – 2 I) gives rise to the universal Hecke algebra 
for a complete p-adic algebra A. This algebra is known to be independent of (as long as 1 – 2 I) and has canonical generators 
over A[[G]] (for 
), where 
is the prime-to-p part of 
. Here note that the operator 
is included in the action of G, because 
. We write hn.ord(U, ; A), 
and 
for the image of the (nearly) ordinary projector 
. The algebra hn.ord is by definition the universal nearly ordinary Hecke algebra over A[[G]] of level 
with "Neben character" . We also note here that this algebra 
is exactly the one h(+, ') employed in [18, p. 240] (when specialized to the CM component there) if A is a complete p-adic discrete valuation ring.
Let 
A = A[[]] for the maximal torsion-free quotient of G. We fix a splitting G = x Gtor for a finite group Gtor. If A is a complete p-adic valuation ring, then 
is a torsion-free A-algebra of finite rank and is A-free under some mild conditions on 
and ([13, 4.2.12]). Take a point P Spf()(A) = Homcont(, Ax). Regarding P as a character of G, we call P arithmetic if it is given locally by an algebraic character (P) X*(T2) with 1(P) – 2(P) I. Thus if P is arithmetic, P = P(P)–1 is a character of 
for some multi-exponent 0. Similarly, the restriction of P to 
is a p-adic Hecke character P+ induced by an arithmetic Hecke character of infinity-type (1 – [(P)])I. As long as P is arithmetic, we have a canonical specialization morphism,
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A-free. The specialization morphism takes the generators 
to 
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3. Anticyclotomic Iwasawa Series |
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TOPAbstract1. Introduction2. Hilbert Modular Forms...3. Anticyclotomic Iwasawa Series4. Behaviour of p-Adic...References |
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We fix a conductor
which is a nonzero R-ideal. We decompose 
so that 
consists of split primes over F, 
(resp., 
) consists of inert (resp., ramified) primes over F, 
and 
. In this section, we redo the computation done in [16, Section 5] allowing the case 
. In [16], we implicitly used the following inclusion diagram of semisimple extensions: |
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M) is embedded into M 
M by (, c) and M is embedded into M M diagonally. We replace this diagram by well-chosen field extensions, |
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without assuming (S).
The cuspidal automorphic induction () of is supercuspidal at prime factors 
of 
, but by definition, the local Galois representation at 
associated to () is the induced representation of the character 
of the quadratic extension 
. This fact (the condition (ind) below) still allows us to easily determine the exact Euler factor at 
of the p-adic Rankin product studied in [7]. The local computation at primes where () is nonsupercuspidal (so principal) has been done in [16, Section 5], but for the sake of completeness, we will repeat some details.
We consider 
for the ray class group 
of M modulo 
. We split 
for a finite group 
and a torsion-free subgroup 
. Since the projection: 
induces an isomorphism 
, we identify 
with 1 and write it as M, which has a natural action of Gal(M/F). We define +M = H0(Gal(M/F), ) and –M = M/+M. Write –:Z –M and 
:Z for the two projections. Take a character 
, and regard it as a character of Z through the projection: Z 
. The Katz measure 
on 
associated to the p-adic CM type p as in [18, Theorem II] induces the anticyclotomic -branch µ– by
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) for this measure dµ– regarding it as an element of the algebra – = W[[–M]] made up of measures with values in W.
We look into the arithmetic of the unique 
-extension M– of M on which we have cc–1 = –1 for all Gal(M–/M) for complex conjugation c. The extension M–/M is called the anticyclotomic tower over M. Writing 
for the ray class field over M modulo 
, we identify 
with 
via the Artin reciprocity law. Then 
and Gal(M–/M) = –M. We then define M by the fixed field of 
in 
; so Gal(M/M) = . Since is a character of , factors through Gal(M–M/M). Let L/M–M be the maximal p-abelian extension unramified outside p. Each Gal(L/M) acts on the normal subgroup X = Gal(L/M–M) continuously by conjugation, and by the commutativity of X, this action factors through Gal(MM–/M). Then we look into the –M-module: 
.
As is well known, X[] is a –-module of finite type, and in many cases, it is torsion by a result of Fujiwara (cf. [4], [10, Corollary 5.4] and [15, Section 5.3]) generalizing the fundamental work of Wiles [27] and Taylor–Wiles [25]. If one assumes the -Leopoldt conjecture for abelian extensions of M, we know that X[] is a torsion module over – unconditionally (see [19, Theorem 1.2.2]). If X[] is a torsion –-module, we can think of the characteristic element 
of the module X[]. If X[] is not of torsion over –, we simply put 
. A character of is called anticyclotomic if (cc–1) = –1().
We are going to prove the following theorem in this section.
Theorem 3.1
Let, then the anticyclotomic p-adic Hecke L-function L–M(
in
Regarding as a Galois character, we define –() = (cc–1–1) for 
. Then – is anticyclotomic. By enlarging 
if necessary, we can find a character such that = – for any given anticyclotomic (e.g. [11, p. 339] or [15, Lemma 5.31]). Thus we may always assume that = –.
It is proven in [18] and [19] that LM(–) is a factor of 
in 
. Thus the improvement concerns the p-factor of L–M(–), which has been shown to be trivial in [17] under the assumption (S) except for the rare cases of positive µ by a trivial reason, but it can often be nontrivial without the assumption (S) as such examples are given in [17] at the end. The main point of this section is to give a new proof of the assertion (A), reducing it to the vanishing of the µ-invariant of the p-adic Hecke L-functions in [17, Theorem I] (which still assumes (S) but as we have already explained, by a quadratic base change we can reduce things to the nonvanishing result over a quadratic CM extension K of M where (S) is satisfied). We will restate the assertion (A) as Corollary 3.8. The proof is similar to the argument in [18, 19], and [16], but the use of µ(L–K) = 0 is a new point. We first deduce a refinement of the result in [18, Section 7] using a unique Hecke eigenform (in a given automorphic representation) of minimal level at nonsupercuspidal places and new at supercuspidal places. The minimal level is possibly a proper factor of the conductor of the representation.
Here we describe how to reduce Theorem 3.1 to Corollary 3.8. Since the result is known for 
by the works of Rubin and Tilouine, we may assume that 
. Put = W[[M]]. By definition, for the universal Galois character 
sending 
to () and M to the group element M , the Pontryagin dual of the adjoint Selmer group 
defined in [12, 5.2] is isomorphic to the direct sum of 
and 
for the ray class groups 
and 
modulo 
, respectively, for M and F (see [19, Proposition 3.32] and also [15, Theorem 5.33]). Thus the characteristic power series of the Selmer group is given by 
.
To relate this power series 
( = –) to congruence among automorphic forms, put 
and 
, and we identify 
and 
. Note that 
. Recall the maximal diagonal torus T0 GL(2)/O. Thus, restricted to 
gives rise to the character of 
. We then extend to a character F of 
by 
. Then we define the level ideal 
by 
and consider the Hecke algebra 
. It is easy to see that there is a unique W[[]]-algebra homomorphism 
:hn.ord such that the associated Galois representation 
([8, 2.8]) is 
. Here is the maximal torsion-free quotient of G introduced in Section 2. Note that the restriction of to the inertia group 
at each prime 
is the diagonal representation 
with values in GL2(W). For supercuspidal primes 
, 
for the unique prime 
of M above 
(in this case, 
is absolutely irreducible). We write H() for the congruence power series H() of (see [8, Section 2.9], where H() is written as ()). Writing 
for the local ring of hn.ord through which factors, the divisibility: 
follows from the surjectivity onto 
of the natural morphism from the universal nearly ordinary deformation ring Rn.ord of 
(without deforming 
for each 
prime to p and the restriction of the determinant character to the torsion part of 
). See [19, Sections 3.3 and 6.2] for details of this implication. The surjectivity is obvious from our construction of 
because it is generated by 
for primes 
outside 
and by the diagonal entries of restricted to 
for 
. Thus we prove the assertion (A): 
as Corollary 3.8, which will be proved in the rest of this section. As a final remark, if we write 
for the quotient of 
which parameterizes all p-adic modular Galois representations congruent to 
with a given (compatible) determinant character 
, we have 
for the maximal torsion-free quotient + of 
(cf. [12, Theorem 5.44]). This implies H() W[[–M]].
3.1 Adjoint square L-values as Petersson metric
Let 
. Let S be a finite set of finite places of F. Let be a cuspidal automorphic representation of 
which are everywhere principal at finite places outside S, supercuspidal at all places of S, and in holomorphic discrete series at archimedean places. Since is associated to holomorphic automorphic forms on 
, is rational over the Hecke field generated by eigenvalues of the primitive Hecke eigenform in . We have = () 
for representations () of 
and of 
. We further decompose
|
|
of 
with two characters 
. By the rationality of , these characters have values in 
. Write + for the central character of (), and its local component 
for 
is given by 
. For 
, we put 
and 
. Thus + = 12 for the product 
over all finite places 
. Write 
as a character of 
. For any 
, we write dS OxS for the projection, and we put d(S) = d/dS.
In the space of automorphic forms in , there is a unique normalized Hecke eigenform f = f of minimal level satisfying the following conditions (see [6, Corollary 2.2]).
- (L1) The level
is given by 
for the conductor 
of the character – of 
and the conductor 
of 
(for 
).
- (L2) Note that
is a character of 
whose restriction to U0(C()) for the conductor C() of induces the "Neben" character 
. Then 
satisfies f(xu) = (u)f(x).
- (L3) The cusp form f gives rise (in the manner described in (S3)) to holomorphic cusp forms of weight
.
- (L2) Note that
In short, f is a cusp form in 
. It is easy to see that 
= –12 has conductor 
, and that v 2 is a constant multiple of f for the new vector v of (note here that may not be automorphic, but is an admissible irreducible representation of 
; so the theory of new vectors still applies). Since the conductor C() of is given by the product of the conductors of 1 and 2, the minimal level 
is a factor of the conductor C() and is often a proper divisor of C().
By (L2), the Fourier coefficient a(y,f) satisfies a(uy, f) = 1(u(S))a(y, f) for 
(
). In particular, the function: 
only depends on the fractional ideal yO. Thus writing 
for the ideal 
, we defined in [7] the self-Rankin product by
|
|
. We have a shift: s s – [] – 1, because in order to normalize the L-function, we used in [7, (4.6)] the unitarization 
in place of to define the Rankin product. The weight u of the unitarization satisfies [u] = 1 and 
. Note that (cf. [7, (4.2a)])
|
| (3.1) |
We are going to define Petersson metric on the space of cusp forms satisfying (L1–3). For that, we write
|
|
|
| (3.2) |
|
|
of F, 
for the Dedekind zeta function 
F(s) of F, and 
(k = 1 – 2 + I and w = I – 2) is the Eisenstein series of level 
defined above of [7, (4.8e)] for the identity characters (1, 1) in place of (–1–1, 
) there.
By the residue formula at s = 1 of 
(e.g. (RES2) in [9, p. 173]), we find
|
| (3.3) |
is the regulator of F.
Since f corresponds to v 2 for the new vector v = –12 of the principal series representation (S) of minimal level in its twist class { } ( running over all finite-order characters of 
), by making product 
, the effect of tensoring 2 disappears. Thus, we may compute the Euler factor of D(s, f, f) as if f were a new vector of the minimal-level representation. Then for each prime factor 
, the Euler 
-factor of 
is given by
|
|
by [5, Lemma 12.2]. We now look into the local factor at 
. We suppose that- (ind) the local representation
of 
(for each place 
), associated to 
by the Langlands functoriality, is of the form 
for a quadratic extension 
.
The above condition is satisfied always for any odd prime in S (see [26]). Since 
(
) is supercuspidal, 
is irreducible, for 
inducing the generator of 
on 
, 
is not equal to 
. Take
|
|
-module by f = f–1.
If 
is unramified, for the inertia subgroup 
of 
, we find
|
|
to be the Frobenius map 
of 
and 
. Thus 
. Since |
|
. Writing 1 (resp., 1 for the identity map of 
(resp., 
), 
is generated by v = 1 –1 and interchanges the two components; so we have v = –v. Thus, the corresponding Euler factor of 
at 
is given by 
. If 
is a ramified quadratic extension, we see easily that 
is still irreducible, and hence 
, and the Euler factor is trivial.
Split 
for the collection Sur of 
such that 
is unramified. Then
- at
, the zeta function 
has the single Euler factor 
, and the zeta function 
has its square 
at 
, because 
contributes one more factor 
;
- at
, 
has the trivial Euler factor 1, and the zeta function 
has 
;
- at
, 
has the trivial Euler factor 1, and the zeta function 
has the factor 
.
The Euler factors outside 
are the same by the standard computation. Therefore, under (ind), the left-hand-side of equation 3.3 is given by
|
| (3.4) |
|
| (3.5) |
(e.g. [9, Section 2.3]). Here we have written 
for 
, and F(s) = 
(s) for the -function (s) = 
0e–tts–1dt. This formula is consistent with the one given in [18, Theorem 7.1] (but is much simpler).
3.2 Primitive p-adic Rankin product
Let 
and 
be integral ideals of F prime to p. We shall use the notation introduced in Section 2. Thus, for a p-adically complete valuation ring 
, 
and 
are the universal nearly ordinary Hecke algebra with level 
and 
, respectively. The character = (1, 2, t+) is made of the characters of j of 
(for an ideal 
) of finite order and for the restriction t+ to 
(the torsion part of 
) of a Hecke character + extending 12. Similarly we regard as a character of 
for an ideal 
); so, – = –112 and – are well defined (finite order) character of 
and 
, respectively. In particular we have 
and 
, where 
is the prime-to-p part of the conductor 
of –. We assume that
|
| (3.6) |
|
| (3.7) |
Let 
and 
be -algebra homomorphisms for integral domains and ' finite torsion-free over . Write 
(thus, 
is the level at supercuspidal places for ). Let 
be the Galois representation associated to (so 
for almost all primes 
), where Q() is the quotient field of . Consider its restriction 
to 
for a prime factor 
of 
. We suppose to have a quadratic extension 
such that
- (SC)
is isomorphic to an irreducible induced representation 
for a Galois character 
at each prime factor 
.
Since we only deal with automorphic induction from a quadratic extension of F in our application, this condition is always satisfied (and as we mentioned already, it holds for any odd prime factor of 
by [26]).
For each arithmetic point 
, let 
be the normalized Hecke eigenform of minimal level belonging to at P. In other words, for 
, we have a(y, fP) = P(T(y)) for all integral idele y with yp = 1. In the automorphic representation generated by fP, we can find a unique automorphic form fordP with a(y, fordP) = (T(y)) for all y, which we call the (nearly) ordinary projection of fP. Similarly, using ', we define 
for each arithmetic point 
. Recall that we have two characters (P,1, P,2) of 
associated to P. Recall 
. The 
-component (for a prime 
) of the automorphic representation (P) generated by the nearly ordinary form fP is necessarily principal or special, because (T(p)) is a p-adic unit. For simplicity, we assume that
- (PR) for each prime
, the 
-components of (P) and ('Q) are principal.
Since we only deal with automorphic induction from a p-ordinary CM quadratic extension of F in our application, this condition is always satisfied. This condition combined with 
in equation 3.6 implies that all local factors of ('Q) at finite places are in principal series.
The central character P+ of fP coincides with P,1P,2 on 
and has infinity-type (1 – [(P)])I. We suppose the following.
|
| (3.8) |
is unramified outside p. As seen in [7, 7.F], we can find an automorphic form gQ|–1 on 
whose Fourier coefficients are given by a(y, gQ|–1) = a(y, gQ)–1(yO), where 
if 
is not prime to 
. The above condition implies, as explained in the previous subsection, |
|
|
|
for the above product when 
and define
|
| (3.9) |
= (P) and ' = (Q) if confusion is unlikely.
Note that for 
,
|
|
further complicates our notation, we can do away with it just replacing gQ by g'Q, since the local component 
of the automorphic representation generated by g'Q satisfies '1,Q = 1,P, and hence without losing much generality, we may assume a slightly stronger condition,
|
| (3.10) |
For each holomorphic Hecke eigenform f, we write M(f) for the rank 2 motive attached to f (see [2]), 
for its dual, f for the 
-adic Galois representation of M(f), and 
for the contragredient of f. Here 
is the p-adic place of the Hecke field of f induced by 
. Thus L(s, M(f)) coincides with the standard L-function of the automorphic representation generated by f, and the Hodge weight of M(fP) is given by {(1,, 2,), (2,, 1,)} for each embedding 
. We have 
(
; see [15, 2.3.8]).
Lemma 3.2
Suppose 3.6 and 3.8. Write, and assume that at primes
, the
-factor
of the automorphic representation generated by fP is supercuspidal and is an automorphic induction of a character of a quadratic extension of
. Then for primes
, the Euler
-factor of
is equal to the Euler
-factor of
given by
where V is the space of the
-adic Galois representation of the tensor product:
and VI = H0(I, V) for the inertia group
at
.
Proof
As already explained, we may assume equation 3.10 instead of equation 3.8. Letbe a prime. By abusing the notation, we write
(resp.,
) for the
-factor of the representation generated by fP (resp., gQ). By the work of Carayol, R. Taylor, and Blasius–Rogawski combined with a recent work of Blasius [1], the restriction of
to the decomposition group at
is isomorphic to
(regarding
as Galois characters by local class field theory). The same fact is true for gQ. If
but
, then VI is one-dimensional on which Frob
acts by
because
is ramified unless i = j = 1 (
and
). If
, both
and
are unramified principal series. By
:3.10, we have an identity
on the inertia group, which is unramified. Therefore V is unramified at
. At the same time, the L-function has full Euler factor at
.
Now assume that
is a prime factor of
. Then
for a character
of index 2. Write
for a quadratic extension
. By the supercuspidality assumption (SC),
is absolutely irreducible, and hence the character
with
. If
is ramified,
for the inertia group I
does not have any I-invariant. In particular, the two Euler factors we are comparing are both trivial. Suppose that
is unramified; so I
of I have an empty intersection, because if they have a nontrivial intersection,
I,
, where
does not have any nontrivial I-invariant subspace. Thus, the two Euler factors we are comparing are both trivial and again identical.
We continue to assume (SC) that the 
-component 
of the automorphic representation generated by fP is supercuspidal for all primes 
. We would like to compute 
for 
for an idele N = N(P) with 
and 
(whose prime-to-p factor is 
). We continue to abuse notation and write, at a prime 
, 
as 
(thus 
is the character of 
inducing the original 
on 
). We write 
(which is a unitary character). In the Whittaker model 
of 
(realized in the space of functions on 
), we have a unique function 
on 
whose Mellin transform gives rise to the local L-function of 
. In particular, we have (cf. [7, (4.10b)])
|
|
is the complex conjugate of 
belonging to the representation space 
, and 
is the epsilon factor of the representation 
as in [7, (4.10c)] (so 
). Then 
is in 
and gives rise to the 
-component of the global Whittaker model of the representation generated by fP. Similarly, for a prime factor 
, in the Whittaker model of 
we have a unique function 
on 
whose Mellin transform gives rise to the local L-function of 
, and we have |
|
is the complex conjugate of 
, and 
is the epsilon factor of the representation 
as in [7, (4.10c)] (so 
).
The above formula then implies
|
|
and 
. Here note that 
if the prime 
is outside 
. We conclude from the above computation the following formula:
|
| (3.11) |
for all 
. This shows
|
| (3.12) |
Theorem 3.3
Suppose (SC), (PR), equations 3.6, and 3.7. There exists a unique elementin the field of fractions of
satisfying the following interpolation property: Let (P, Q)
- (W)
.
Then
is finite at (P, Q) and we have
where, writing k(P) =
Here
and
. Moreover, for the congruence power series H(
.
The expression of p-Euler factors and root numbers is simpler than the one given in [7, Theorem 5.1], because automorphic representation of gQ is everywhere principal at finite places (by equation 3.6). The shape of the constant W(P,Q) appears to be slightly different from [7, Theorem 5.2]. Firstly, the present factor (–1)k(P)+k(Q) is written as (Q+P+)(–1) in [7]. Secondly, in [7], it is assumed that –1Q,1 and –1P,1 are both induced by a global character 'P and 'P unramified outside p. Thus the factor ('Q,'P,)(–1) appears there. This factor is equal to (Q,1,pP,1,p)(–1) = p(–1), which is trivial because of the condition (W). We do not need to assume the individual extensibility of Q,1 and P,1. This extensibility is assumed in order to have a global Hecke eigenform f°P = fuP 'P. However this assumption is redundant, because all computation we have done in [7] can be done locally using the local Whittaker model. Also, C(P,Q) in the above theorem is slightly different from the one in [7, Theorem 5.2], because (fP, fP) = D[(P)]+1(f°P, f°P) for f°P appearing in the formula of [7, Theorem 5.2].
Proof
We start with a slightly more general circumstance. We shall use the symbol introduced in [7]. Supposeand
, and take normalized Hecke eigenforms
and
. Suppose
by
. Then
for
. We put
. Then we see
(wu) =
. Since
. We write simply
for the central character of
, which is the Hecke character
. Then we have
. We then define
for
. Here B is the algebraic subgroup of G made of matrices of the form
. We extend
outside
just by 0. Similarly, we define
by
For each
-subalgebra
, we write
. Note that
for
is left-invariant under
. Then we compute
for the measure
defined in [7, p. 340]. We have
where
is defined in equation 3.9. Define
by the stabilizer in
of
. We now choose an invariant measure
, so that
whenever
and the two integrals are absolutely convergent. There exists a unique invariant measure
,
and the right-hand-side is left C
where
(3.13) Note that E(zw, s) = (
. By definition, E(
; in particular, it is invariant under
,
. Thus
has eigenvalue
. The averaged Eisenstein series
satisfies
, where a runs over complete representative set for
and
is the central character of gc. Defining the PGL2 modular variety
, by averaging equation 3.13, we find
Writing
(3.14) and r =
by
where k =
is given by
. Then changing variable
, we can rewrite equation 3.14 as
where
(3.15) for
, and
is the normalized Petersson inner product on
. This formula is (essentially) equivalent to the formula in [7, (4.9)] (although we have more general forms f and g with character
Let E be the Eisenstein measure of level
defined in [7, Section 8], where
is written as L. We take an idele L with
and
. Similarly, we take ideles J and N replacing in the above formula
by
and
, respectively, and L by the corresponding J and N, respectively.
The algebra homomorphism
induces, by the W-duality,
, where
is a subspace of p-adic modular forms of level
(see [8, 2.6]). We then consider the p-adic convolution as in [7, Section 9, p. 382],
where [L/J] is the operator defined in [7, Section 7.B] and all the ingredients of the above formula are as in [7, p. 383]. An important point here is that we use the congruence power series H(
) defined with respect to
instead of h(
We write the minimal level of fordP as
for
. Then we define
. The integer
is given by the exponent of
in
or 1 whichever is larger. We now compute
. We shall give the argument only when j = [
. We write
and put
. As before, m = L
and
. Put r(P, Q) =
,
By [7, Corollary 6.3], we have, for r = r(P, Q),
(3.16) Then by equation 3.15, we get
where
(3.17) Now we compute the Petersson inner product (fordP|
The computation we have done in [7, p. 357] in the proof of Lemma 5.3(vi) is valid without any change for each
(3.18) , since at p-adic places, fP in [7] has the Neben type we introduced in this paper also for places outside p. The difference is that we compute the inner product in terms of (fP, fP) not (f°P, f°P) as in [7, Lemma 5.3(vi)], where f°P is the primitive form associated to fuP
A key point of the proof of Lemma 5.3(vi) is the formula writing down ford,uP
(3.19) of local operators
given as follows: If the prime
is a factor of
, then
is the identity operator. If
is prime to
(
is spherical), then
, where
with f|g(x) = f(xg) for
. Writing U for the level group of fP and
, we note
. This shows
where
,
, and (·, ·)U is the Petersson metric on
. Similarly, we have
By equations 3.11 and 3.19, we conclude from [7, Lemma 5.3(vi)] that
for
(3.20) running over the prime factors of p.
We now give a brief description of the computation of the extra Euler factors: E(P,Q) and W(P,Q). Again the computation is the same as in [7, Lemma 5.3(iii)–(v)], because the level structure and the Neben character at p-adic places are the same as in [7] for fP and gQ and these factors only depend on p-adic places. Then we get the Euler p-factor E(P,Q) and W(P,Q) as in the theorem from [7, Lemma 5.3].
Remark 3.4
We assumed condition 3.7 to make the proof of the theorem simpler. We now remove this condition. Letbe the set of all prime factors
of
outside
such that
on
. Thus we assume that
. Then in the proof of Lemma 3.2, the inertia group at
fixes a two-dimensional subspace of
, one corresponding to
and the other coming from
. The Euler factor corresponding to the latter does not appear in the Rankin product process; so we get an imprimitive L-function, whose missing Euler factors are
Thus, the final result is identical to Theorem 3.3 if we multiply E(P,Q) by E'(P, Q) in the statement of the theorem. In our application,
and
for two ordinary CM fields M/F and M1/F. If the prime-to-p conductor of
are made of primes split in M/F,
is the set of primes ramifying commonly in M/F and M1/F outside
. Then E'(P, Q) is the specialization of
at (P, Q), and
is not divisible by the prime element of W (that is, the µ-invariant of E' vanishes). Actually, we can choose
and
so that
, and under this choice, we may assume that
.
3.3 Comparison of p-adic L-functions
For each character 
, we have the extension 
sending 
to () for the group element M inside the group algebra . Regarding 
as a character of 
, the induced representation 
is modular nearly ordinary at p, and hence, for a suitably chosen dependent on , by the universality of the nearly p-ordinary Hecke algebra 
defined in Section 2.4, we have a unique algebra homomorphism = :h such that 
for the universal nearly ordinary modular Galois representation Hecke with coefficients in h, where 
is the prime-to-p part of 
for the relative discriminant d(M/F) of M/F, and for the conductor 
of the anticyclotomic projection –. Thus for each arithmetic point 
(in the sense of [8, 2.7]), we have a classical Hecke eigenform 
of weight (P), which is a (nonstandard) theta series of the Galois character 
introduced in [16, 5.3]. We write the automorphic induction of the complex Hecke character associated (via global class field theory) to the Galois character 
as 
(which was written as (P) before). Then 
is the normalized Hecke eigenform in 
minimal at nonsupercuspidal places and new at supercuspidal places. Hereafter, we use the same symbol 
for the complex Hecke character associated to the Galois character 
.
To give an explicit description of 
and 
, decompose the conductor 
of = – into the product 
as in the introductory section so that 
, 
, 
is a product of inert primes in M/F, and 
is the product of primes ramified in M/F. Since the case of 
has been dealt with in [16], we assume that 
. Then the automorphic representation 
of weight (P) has minimal prime-to-p level 
given by 
, where 
is the prime-to-p part of 
. The set S of super cuspidal places for 
is made up of primes 
of O appearing in 
. By [8, 7.1], the Hecke character 
has infinity-type
|
|
, we have a unique normalized Hecke eigenform 
of minimal level.
The prime-to-p level of the cusp form 
is 
as above, and it satisfies (L1–3) in Section 3.1 for = P given as follows. To describe the local component of , we use local class field theory and identify local characters of 
with the corresponding characters of the inertia group 
(resp., 
) for 
) for each prime 
of R. Here is the description,
|
| (3.21) |
and 
are distinct primes in M.
Write 
, and write t+ for the restriction of + = 12 to 
, which is independent of P (because it factors through the torsion part of 
). Since is of minimal level, the congruence module C0(; ) is a well-defined -module of the form /H() (see [8, 2.9], and recall here = –). As we already remarked, we can choose H() in – = W[[–M]] (see [12, Theorem 5.44]). The element H() is called the congruence power series of (identifying – with a power series ring over W of 
variables).
We now choose well a totally real quadratic extension F1/F, and put K = F1M. Then in the composite K = F1M, there are three quadratic extensions M, F1, and M1 of F inside K,
|
|
and 
for all primes in K over 
and 
for all primes 
in K. In other words,- M1/F is a CM field;
- for primes
in F, the decomposition group of 
in Gal(K/F)is given by Gal(K/M1).
Since we impose how places decompose in F1/F only at the finite set 
of places of F, there will be infinitely many choices of F1. The field M1/F and hence K/F1 are a p-ordinary CM field in which all primes in S and over p split. Our choice of F1 and M1 could be different from the choice (F', M') we made in the introductory section; so we use different symbols.
Write R1 for the integer ring of M1. We choose a conductor 
(an R1-ideal prime to p) made of primes split in M1/F. Then in the same manner as above, we define the groups 
for M1 in place of 
for M. Choose a character 
of conductor 
. Let 
, and choose a p-ordinary CM-type ' of M1. We then put 
and define the character 
for in the same way of the construction of 
. The Hecke character 
has infinity-type
|
|
relative to M1/F, we have a unique normalized Hecke eigenform 
of minimal level. Thus, we get an algebra homomorphism 
which gives rise to the family of minimal modular forms 
for each arithmetic points Q Spec(').
The prime-to-p level of the cusp form 
is 
as above, and it satisfies the corresponding conditions (L1–3) in Section 3.1 for S = 
and (1, 2) = (1(Q), 2(Q)) (after replacing M/F by M1/F). Since we need the explicit form of the Neben character = Q of 
later, we repeat its description, although it is the same as the one given for 
(replacing the data concerning P by those of 
), and the description is indeed simpler, since 
does not have supercuspidal places. To write down the Neben character = Q of 
, as before for , we use local class field theory and identify local characters of 
with the corresponding characters of the inertia group 
(resp., 
) for 
) for each prime 
of R1. Here is the description,
|
| (3.22) |
and 
are distinct primes in M1.
By Theorem 3.3 and Remark 3.4, we have the (imprimitive) p-adic Rankin product 
with missing Euler factor 
as in Remark 3.4 for two characters 
and 
. Writing 
, we have 
.
Let 
and 
. We define a p-adic L-function 
for 
and 
by
|
|
We follow the argument in [7], [18], and [8] to show the following identity of p-adic L-functions.
Theorem 3.5
Let(resp.,
) with values in Wx. Suppose the following two conditions:
is prime to any inert or ramified prime of M1/F;
is nonempty at each prime factor
of
.
Then we have, for a power series
,
where
. The power series
is equal to (
.
In our earlier papers [18, Theorem 8.1] and [16, Theorem 5.5], we have taken M = M1; so this theorem is a version of the result there taking different CM fields M and M1 (well chosen). We have an extra error factor 
in 
which does not appear in [16, Theorem 5.5]. As noticed by R. Gillard and described in [17] at the end, if we take p to be a prime appearing in this factor, the anticyclotomic p-adic Hecke L-function L–M(–) often has a positive µ-invariant. The above theorem is not empty because of the following.
Lemma 3.6
Assumption (ii) of the above theorem can be achieved by choosing (F1,and given semisimple quadratic extensions
for
, we may impose to have
for all
and
to find such a pair (F1,
Proof
For prime factorsof d(M/F)d(M1/F), we may choose both
and
to be the trivial characters; so we need to choose
for a factor
of
unramified in K/F. We can choose F1 so that
is as specified for
, and that all prime factors in
split over F split also in M1. For such split prime factors
of F, we can identify
and
and we impose
on
. For
, we choose
to be unramified. For prime factors
of
nonsplit in M,
splits into
in M1/F1 by our choice of F1 already made. Then we choose
on
, and we impose
. Then by our definition,
on
for all
, and
. By adding ramification to F1 outside S1 and
if necessary, we may assume that R1x = Ox. Since
is trivial over Rx
Ox = R1x, we can extend
to a finite-order character of
, which can be extended to a finite-order character of
without adding any more ramification, as desired.
Proof of Theorem 3.5. The proof of the theorem is identical to the proof given in [8, Section 7.4], (and of [16, Theorem 5.5]), since the factors C(P,Q), W(P,Q), S(P), and E(P,Q) appearing in Theorem 3.3 are identical to the one appearing in [7, Theorem 5.2] except for the power of the discriminant D, which is compensated by the difference of (f°P, f°P) appearing in [7, Theorem 5.2] from (fP, fP) in Theorem 3.3. Since we do not lose any Euler factors in equation 3.5 (thanks to our minimal-level structure and assumption 3.6 assuring principality everywhere for gQ) and the epsilon factor of 
is the product of the individual one associated to 
and 
, we can write the numerator of the Rankin product as 
. This finishes the proof of Theorem 3.5.
Lemma 3.7
Choosing (F1,projected to W[[
Proof
By our choice of F1 and M1, the conductor ofis made of primes splits over F1; so the condition (S) with respect to K/F1 is satisfied by
. In particular, all primes ramifying in M/F split in M1/F. Thus we do not need to multiply the Katz p-adic L-function
by the extra Euler factor E'(P, Q) in Remark 3.4 to get
, and hence we need to prove the vanishing of µ of the anticyclotomic projection
, which is studied in [17].
In [17, Theorem I], under the split prime-to-p conductor assumption (S) (with respect to K/F1) of
(which holds by our choice of F1/F as we have seen), a set of three conditions (M1–3) equivalent to the nonvanishing of
is stated. One of them is the modulo p identity of Hecke characters of F1:
- (M3)
, where
is the maximal ideal of W, ?1 is the restriction to the Hecke character to
, and
We further restrict these characters to
. Since we may assume
by the above lemma, we get
over
, which is not the case if p>3. Indeed,
is ramified at p and
is unramified at p, since p is unramified in
by our choice of M and F1. Thus
holds under
(and such a choice is possible by the above lemma).
This implies the following corollary.
Corollary 3.8
We havein
Proof
Let µ be the µ-invariant of. Then
with
µ
µ(H(
Since 
as explained in the introductory section, this implies Theorem 3.1.
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4. Behaviour of p-Adic L-Functions under Base Change |
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TOPAbstract1. Introduction2. Hilbert Modular Forms...3. Anticyclotomic Iwasawa Series4. Behaviour of p-Adic...References |
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The following theorem is the only ingredient left to be proven in the proof of the anticyclotomic main conjecture given in the introductory section.
Theorem 4.1
Assume (F1–3) for E, M', F in the introductory section. Then we haveand Res(L–E(
, where
.
Proof
We work in the setting and with the symbols we defined in the introductory section. Let X(R) be the CM abelian variety defined overwith
(see [24] and [22] for such a choice). Let RE be the integer ring of E. Then X(RE) = X(R)
is written as c) to construct the p-adic L-function L–M(
for a well-chosen purely imaginary element 0
To show the identity on the Galois side, we write LE
Let L
since E/M ramifies at some places outside p by (F3). Put
Let
of
, we can let Gal(E/M) act on XEM and XEM[
. Then XM[
as W[[
By [15], Theorem 5.33 and Lemma 5.23(2)], XE[
is a principal ideal generated by the characteristic power series
. Thus we find
Since the reflexive closure of
is generated by
, we conclude the identity of the principal ideals,
as desired, and moreover
is principal and is generated by
(though this can be also proven by the result of Fujiwara [4] following the argument proving Theorem 5.33 in [15], Chapter 5], removing some of the simplifying assumptions (h1–4) made there).
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Acknowledgments |
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The author is partially supported by the NSF grants DMS 0244401, DMS 0456252, and DMS 0753991.
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References |
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