Congruences among automorphic forms on unitarygroups and the Bloch-Kato conjecture

Krzysztof Klosin
University of Utah, Salt Lake City, USA

7-8 January 2008

Krzysztof Klosin

Congruences among automorphic forms

The Bloch-Kato philosophy

M= motive
NNNqqqqq
qqqx qq

NNN
NNNNN'( )Sel(M )The Bloch-Kato Conjecture (1990) predicts a relationship between
the two bottom objects.
L M

Examples:Analytic class number formula
BSD conjecture
Krzysztof Klosin

Congruences among automorphic forms

Ribet (1976) - Converse to Herbrand's Theorem

Let  : (Z=pZ) ! p 1 be an even Dirichlet character.
Theorem (Ribet, 1976)
If p j L( 1; ), then p j # ClQ(p ) ( 1 ! 1 ).

Krzysztof Klosin

Congruences among automorphic forms

Ribet (1976) - Converse to Herbrand's Theorem

Proof.

 gives rise to an Eisenstein series E .
2 If p j L( 1; ), then E  cusp form f mod p .

3 Thus the mod p Galois representation  attached to f isf1

4

reducible but not semisimple (since f is a cusp form).
So it gives rise to a non-trivial extension
0 ! F ! f ! F(!) ! 0;
which corresponds to a non-trivial element of the 1 ! 1 -part of ClQ(p ) .

Krzysztof Klosin

Congruences among automorphic forms

Ribet's approach revisited=algebraic group, =automorphic representation ofM (A).
2 Realize M as a Levi in some algebraic group G=Q .
3 Lift  to =aut. repr. of G (A). The Galois repr. attached to will be reducible, semisimple.
4 Assuming that p divides the relevant L-value(s) of  ,construct an aut. repr. 0 of G (A) such that
1M=Q

its Hecke eigenvalues are congruent to those of  mod p
the p-adic Galois representation of  is irreducible.
05Get a non-trivial extension, hence elements in Selmer group.

Krzysztof Klosin

Congruences among automorphic forms

Examples of such results

K. Ribet (M = GL1, G = GL2,  = E) (1976)
A. Wiles (same as Ribet, MC of Iwasawa Theory) (1990)
Skinner-Urban (M = RK =Q GL2, G = U(2; 2),  =Eisenstein series) (2006)
T. Berger (M = RK =Q GL1, G = RK =Q GL2, =Eisensteinseries) (2005)
J. Brown (M = GL2, G = Sp4,  = Saito-Kurokawa lift)(2005)
K. (M = RK =Q GL2, G = U(2; 2),  = CAP representation)(2006/07)

Key diculty: Construct 0 with irreducible Galois representation,\congruent" to . Very di erent techniques are used.
Krzysztof Klosin

Congruences among automorphic forms

Main result (rough version)=imaginary quadratic eld of discriminant DK
f =modular form of weight k
1, level DK
f =p -adic Galois representation attached to f .
KTheorem (Main theorem (rough version))
nalg
2n0 _
p j L (Sym f ; k ) =) p j # Sel(K ; ad f ) :Krzysztof Klosin

Congruences among automorphic forms

SetupQ = imaginary quadratic, DK =discriminant
K = Dirichlet character associated to K =Q
K=G= U(2; 2) = fM 2 RK =Q GL4 j MJ M = J g, J =t

H = fZ 2 M2(C) j i (Z Z ) > 0g
If g = CA DB 2 G (R), thentgZ= (AZ + B )(CZ + D ) 1 :Sk := fF : H ! C j F (gZ ) = det(CZ + D )k F (Z ); g 2G (Z); F = holomorphic, cuspidalg
Krzysztof Klosin

Congruences among automorphic forms

2I

2 .

I

Maass lifting

Theorem (Kojima, Gritsenko, Krieg (1982-1991))
There exists aC-linear liftSk

1 (DK ; K ) ! Sk ;which is Hecke-equivariant if

f

7! Ff

# ClK = 1.

If # ClK > 1, no way to de ne Hecke operators at split primes !
need an adelic theory.

Krzysztof Klosin

Congruences among automorphic forms

Adelic Maass lifting

Theorem (K. 2007)
If 2 # ClK then there exists a C-linear map-Sk1 (DK ; K )#ClK ! A0 (G (A))

which is Hecke equivariant (for all local Hecke algebras).

The p-adic Galois representation attached to a Hecke eigenformlying in the Maass space is reducible and semisimple.

Krzysztof Klosin

Congruences among automorphic forms

CAP representations! f on GL2(A)Ff ! f = CAP associated to f on G (A), i.e.,
f ;p = IndGM((QQpp)) (BCK =Q f )p
ffor almost every p, where M = RK =Q GL2 =Levi.

Jacquet-Shalika (1981), Moeglin-Waldspurger (1990): There
are no CAP representations on GLn .

Krzysztof Klosin

Congruences among automorphic forms

CAP ideal

Sk = SkM  SkNM

O = ring of integers in a nite extension of Qp ,TNM
= the O-Hecke algebra of Sk ,T = quotient of T acting on SkNM,
:T TNM canonical epimorphism.

De nition (CAP ideal)
Set If = CAP ideal = ideal of TNM generated by (Ann(Ff )).

TNM=If measures congruences between Ff and non-Maass forms.
Krzysztof Klosin

Congruences among automorphic formsConjectures

Conjecture (On the size of the CAP ideal)
Let #OK j k , f 2 Sk 1 (DK ; K ) a newform, p > k a prime notdividing some explicit constants. Thenvalp (#TNM =If )  valp (#O=Lalg (Sym2 f ; k )):Conjecture (On the existence of Galois representations)
If F is a hermitian cuspidal eigenform, there exists a continuous
representation

F : GK ! GL4 (Qp )having the expected properties.

Krzysztof Klosin

Congruences among automorphic forms

Consequence a la Bloch-KatoLetf : Gal(Q=Q) ! GL2 (Qp ) be the Galois representation

attached to f by Deligne;
0T = ad f jGK ( 1),W =T
Frac(O)=O.

Corollary
Assume the above two conjectures. Under some mild hypotheseson f one has

valp (# Sel(K ; W )_ )  valp (#O=Lalg (Sym2 f ; k )):Krzysztof Klosin

Congruences among automorphic forms

Main resultTheorem (K. 2007)
Suppose # ClK = 1 (# ClK =odd - work in progress). Ifjf is ordinary at p with f GK absolutely irreducible

and there exists a Hecke characterto p with1 (z ) = z t z t and0valp @2Y
j =1of K of conductor prime

1alg (BC(f ); j + t + k =2; )A = 0;Lthen the conjecture on the size of the CAP ideal is true.

Krzysztof Klosin

Congruences among automorphic forms

Beyond U(2; 2) (work in progress...)Ikeda lifts: lifting modular forms on GL2 to modular forms on
U(n; n) (f 7! Ff ).
Conjecture (On the size of the CAP ideal of an Ikeda lift toU(n; n))
Let n be even, f 2 Sk 1 (DK ; K ) a newform, p > k a prime notdividing some explicit constants. Then0valp (#TNM =If )  valp @#O=nYj =2alg (Sym2 f ; k + j

L12; jK )A :Similar conjecture can be phrased for Ikeda lifts to SU(n; n).
Krzysztof Klosin

Congruences among automorphic forms

Constructing the non-Maass cusp form

F with F  F .

Ingredients used in the proof:
Properties of Siegel Eisenstein series due to Shimura;
Inner product relations between Ff and other hermitian
modular forms;
Hida theory + deformation theory of Galois representations.

Krzysztof Klosin

Congruences among automorphic formsfThank you.
For preprints visit www.math.utah.edu/klosin

Krzysztof Klosin

Congruences among automorphic forms