Periods and congruences of various lifts

Miyama Conference Periods and congruences of various lifts KATSURADA Hidenori (Muroran I. T.) October 2010

1. Introduction G 12 (z) := Γ(12) 2(2π) 12 (c,d) Z 2 \{(0,0)} (cz + d) 12 the Eisenstein series of weight 12 with respect to SL 2 (Z) (z) := e(z) n=1 Ramanujan s delta fucntion G 12 (z) = ζ( 11) + (z) = n=1 (1 e(nz)) 24 e(z) = exp(2πiz) n=1 c (n)e(nz) σ 11 (n)e(nz) c (n) σ 11 (n) mod 691, ζ( 11) = 691 12 Question: Can one characterize prime ideals (congrurnce primes) giving congruence between different kinds of Siegel (or Hermite) modular forms in terms of certain L-values?

The aim of my talk is to give an affirmative answer to this question in some cases and to show that such congruence primes can be characterized in terms of L-values appearing in the periods of modular forms. Period : the Petersson norm of a cusp form Congruence : Congruence between Hecke eigenvalues of lifted Hecke eigenforms and those of non-lifted Hecke eigenforms in a (fixed) space of cusp forms Γ : a modular group (elliptic modular group, symplectic group, unitary group etc.) η : a character of Γ S l (Γ, η) : the space of cusp forms of weight l and character η for Γ

Γ : a cogruence subgroup of SL 2 (Z), k Z/2 For a Hecke eigenform g S k (Γ, χ), let ĝ be a certain lift of g to S l (Γ, η) for some another modular group Γ, that is, let ĝ S l (Γ, η) be a Hecke eigenform whose certain L-function is expressed in terms of certain L-functions related with g. Ex. ĝ = Doi-Naganuma lift, Saito-Kurokawa lift, Duke-Imamoglu-Ikeda lift (D-I-I lift), Hermitian Ikeda lift. Problem A. Express the ratio g. ĝ, ĝ g, g in terms of certain L-values related with Problem B. Characterize primes giving congruence between ĝ and another Hecke eigenform in S l (Γ, η) not coming from the lifting in terms of the L- values in Problem A. Problem C. Costruct a non-trivial element of a certain Bloch-Kato selmer group by using the congruence in Problem B.

The aim of today s talk : (1) Answer to Problem A for the D-I-I lift (Ikeda s conjecture on the period of the D-I-I lift) (2) Answer to Problem B for the D-I-I lift (3) Answer to Problem A for the Hermitian Ikeda lift (Ikeda s conjecture on the period of the Hermitian Ikeda lift) (4) (Conjectural) Answer to Problem B for the Hermitian Ikeda lift

2. Ikeda s conjecture on the period of the D-I-I lift L n := { half-integral matrices of degree n over Z} L n >0 := {A L n A > 0} Γ n = Sp(n,Z) := {g GL 2n (Z)) t gj n g = J n } J n = Γ : a congruence subgp. of Γ n, η : a character of Γ ( On 1 n 1 n O n F, G := [Γ n : {±1}Γ] 1 Γ \H n F(Z)G(Z)(det(Im(Z))) l n dxdy ) for F, G S l (Γ, η). L n := { b g Γ n gγ n (a g C)} (the Hecke ring) g GSp(n,Q) + F(Z) : a Hecke eigenform, F T ( = λ F (T)F ) for any T L n D 0 D GL n (Q) M n (Z), T D := Γ n 0 t D 1 Γ n L(s, F,St) := GL n (Z)DGL n (Z) (standard L-function of F) λ F (T D )(det D) s n ζ(s + n) n i=1 ζ(2s 2i + n)

Rem. L(s, F,St) has an Euler product of the following form: (α i,p C ) L(s, F,St) = p {(1 p s ) n i=1 (1 α i,p p s )(1 α 1 i,p p s )} 1 f(z) = m=1 a(m)e(mz) : a primitive form in S k (Γ 1 ) α p C s.t. α p + α 1 p = p k/2+1/2 a(p) χ : Dirichlet character L(s, f, χ) := 1 p (1 χ(p) α p p s+k/2 1/2 )(1 χ(p) α 1 p p s+k/2 1/2 ) L(s, f,ad, χ) := 1 p (1 χ(p)p s )(1 χ(p)α 2 p p s )(1 χ(p)α 2 p p s ) Rem. In case n = 1, L(s, f,st) = L(s, f,ad)

k 2Z >0, n 2Z >0 S + k n/2+1/2 (Γ 0(4)) := {g(z) = m=1 c(m)e(mz) c(m) = 0 unless ( 1) n/2 m 0,1 mod 4} Rem. There exists an isomorphism S + k n/2+1/2 (Γ 0(4)) = S 2k n (Γ 1 ) (Shimura correspondence) g(z) = f(z) = e=1 m=1 c g (e)e(ez) : a Hecke eigenform in S + k n/2+1/2 (Γ 0(4)) a f (m)e(mz) : the primitive form in S 2k n (Γ 1 ) corresponding to g via the Shimura correspondence Rem. The primitive form f is uniquely determined by g.

T L n >0 ( 1) n/2 det(2t) = 2 T T with T a fundamental discr. and T Z >0 b p (T, s) := e(tr(tr))µ p (R) s R Sym n (Q p )/Sym n (Z p ) µ p (R) := [RZ n p + Z n p : Z n p] Rem. (Y. Kitaoka) There exists a polynomial F p (T, X) in X s.t. b p (T, s) = F p (T, p s )(1 p s ) n/2 i=1 (1 p 2i 2s )(1 F p (T, X) := X ord p( T ) F p (T, p (n+1)/2 X) α p C such that α p + α 1 p = p k+n/2+1/2 a f (p) c In (g) (T) := c g( T ) k n/2 1/2 F p (T, α 1 p ), I n (g)(z) := T L n >0 T c In (g) (T)e(tr(TZ)) p ( (Z H n), T p ) p n/2 s ) 1

Thm. 2.1. I n (g)(z) is a Hecke eigenform in S k (Γ n ) whose standard L- function is ζ(s) n i=1 L(s + k i, f) (cf. T. Ikeda, Ann. of Math. 154(2001), 641-681). Rem. The exsitence of such a cuspidal Hecke eigenform was conjectured by W. Duke and O. Imamoglu. We call I n (g) the D-I-I lift of g (or of f) Rem. I 2 (g) is the Saito-Kurokawa lift of g (or of f) Rem. c In (g) (A) = c g(det(2a)) if ( 1) n/2 det(2a) is a fundamental discriminant.

Γ C (s) := 2(2π) s Γ(s) ξ(s) := Γ C (s)ζ(s) Λ(s, f, χ) := τ(χ) 1 Γ C (s)l(s, f, χ) τ(χ) the Gauss sum Λ(s, f,ad) := Γ C (s)γ C (s + 2k n 1)L(s, f,ad). S 2k n (Γ 1 ) S + k n/2+1/2 (Γ 0(4)) S k (Γ n ) f g I n (g) Thm. 2.2. (K-Kawamura) We have I n (g), I n (g) g, g = 2 α(n, k) Λ(k, f) ξ(n) where α(n, k) = (n 3)(k n/2) n + 1. n/2 1 i=1 Λ(2i + 1, f, Ad) ξ(2i), Rem. This was proved by Kohnen and Skoruppa [Invent. Math. 95 (1989), 541-558] in the case n = 2, and was conjectured by Ikeda [Duke Math. J. 131(2006), 469-497] in general case.

L(m, f, Ad) := Λ(m, f, Ad)/ f, f D : a fundamental discriminant such that ( 1) n/2 D > 0 c g ( D ) 2 g, g = 2a n,k( 1) b n,k D k n/2 Λ(k n/2, f,( D )) ( 1) n/2, f, f with some integers a n,k and b n,k. (cf. W. Kohnen and D. Zagier, Invent. Math. 64(1981), 175-198.) Thm 2.3. Assume that k n + 2. Then we have c g ( D ) 2 f, f n/2 I n (g), I n (g) = 2 a n,k( 1) b n,k D k n/2 Λ(k n/2, f,( D )) ξ(n)( 1) n/2 Λ(k, f). n/2 1 i=1 L(2i + 1, f, Ad) ξ(2i)

Q(f) := Q(a f (1), a f (2),...) Fact. (1) L(m, f, Ad) Q(f) if 0 < m 2k n 1, m 1 mod 2. (2) Λ(k n/2, f,(d )) ( 1) n/2 Λ(k, f) Q(f). Cor. Assume thta k > n. Then we have for any m (cf. Furusawa, Choie-Kohnen). f, f n/2 I n (g), I n (g) Q(f) if c g(m) Q(f)

3. Congruence of Siegel modular forms L n := { g GSp(n,Q) + M 2n (Z) b g Γ n gγ n (a g Z)} (the integral Hecke ring) F S k (Γ n ) : a Hecke eigenform Q(F) := Q(λ F (T) T L n ) (the Hecke field of F ) Fact. (S. Mizumoto) If k n + 1, λ F (T) Q(F) for any T L n. M : a Hecke stable subspace of S k (Γ n ) such that M (CF), where (CF) is the orthogonal complement of CF in S k (Γ n ) with respect to the Petersson product. Def. Let K be an algebraic number field containing the Hecke field Q(H) for any Hecke eigenform H S k (Γ n ). A prime ideal in K is called a congruence prime of F with respect to M if there exists a Hecke eigenform G M such that for any T L n. λ G (T) λ F (T) mod

R : a subring of C S k (Γ n, R) : = {F(Z) = F(Z) = B L n >0 B L n >0 c F (B)e(tr(BZ)) S k (Γ n ) c F (B) R for any B L n >0} c F (B)e(tr(BZ)) S k (Γ n ) : Hecke eigenform ρ n = Λ(m, F,St) := { 3 n 5 and n 1 mod 4 1 otherwise L(m, F,St) F, F π n(n+1)/2+nk+(n+1)m. Fact. Let F be a Hecke eigenform in S k (Γ n,q(f)). Then for ρ n m k n, m n mod 2, Λ(m, F,St) belongs to Q(F) (S. Boecherer, S. Mizumoto). Assume that (*) cf S k (Γ n,q(f)) with some c C. Rem. (1) The condition (*) holds for D-I-I lift. (2) If the multiplicity one condition holds for S k (Γ n ), then the condition (*) holds for any Hecke eigenform in S k (Γ n ).

B L n >0 I F,B (m) := c F (B)c F (B)Λ(F, m,st) Fact. Assume (*). Then for any B L n >0 and ρ n m k n, m n mod 2, I F,B (m) belongs to Q(F). Thm. 3.1 (K, Math. Z. 259(2008),97-111) Let F be a Hecke eigenform in S k (Γ n ) satisfying the assumption (*). Let be a prime ideal of K not dividing (2k 1)!. Assume that divides the denominator of I F,B (m) for some B L n >0 and ρ n m k n 2, m n mod 2, Then is a congruence prime of F with respect to (CF).

4. Congruence between D-I-I lifts and non-d-i-i lifts g : a Hecke eigenform in S + k n/2+1/2 (Γ 0(4)) f : the primitive form in S 2k n (Γ 1 ) corresponding to g K : a sufficiently large algebraic number field as above A = K, : a prime ideal in K Ω (±) f = Ω(f, ±; A ( ) ) : Canonical periods of f arising from the Eichler-Shimura isomorphism (cf. H. Hida, Modular forms and Galois cohomology, Cambridge Univ. Press, 2000.) f, f Rem. A ( ) (the localization of A at ) Ω (+) f Ω ( ) f 0 < m 2k n 1 χ : a Dirichlet character j := χ( 1)( 1) m 1 L(m, f, χ) := Γ(m)L(m, f, χ) τ(χ)(2π 1) m Ω (j) f Rem. L(m, f, χ) belongs to the field K(χ) generated over K by all the values of χ (Shimura). S k (Γ n ) : the subspace of S k (Γ n ) spanned by all the D-I-I lifts I n (h) of Hecke eigenforms h S + k n/2+1/2 (Γ 0(4)) (generalized Maass subspace)

Thm. 4.1. (K) Let k 2n + 4. Assume that (1) divides L(k, f) (2) does not divide ξ(2m) n i=1 n/2 1 i=1 L(2i + 1, f,ad). L(2m + k i, f)l(k n/2, f,( D ))D(2k 1)! for some integer n/2+1 m k n/2 1, and for some fundamental discriminant D such that ( 1) n/2 D > 0. (3) does not divide where C p,k,n = 1 or n = 2 or not. C p,k,n f, f Ω(f,+, A )Ω(f,, A ), p (2k n)/12 (1 + p + + p n 1 ) according as Then is a congruence prime of I n (g) with respect to (S k (Γ n ) ).

Outline of the proof Step 1. Recall the following two facts: (*) L(s, I n (g),st) = ζ(s) n i=1 L(s + k i, f) (**) c g( D ) 2 f, f n/2 I n (g), I n (g) c n,k D k n/2 L(k n/2, f,( D = )) ξ(n)l(k, f) n/2 1 i=1 L(2i + 1, f, Ad) ξ(2i) c n,k Step 2. A L n >0 s.t c I n (g) (A). Thus we have c g (D) Λ(2m, I n (g),st) c In (g) (A) 2 = ǫ k,n,m n i=1 L(2m + k i, f) D k/2 n/2 L(k n/2, f, χ D ) L(k, f) ξ(n) n/2 1 i=1 L(2i + 1, f, St) ξ(2i) Ω(f,+; )Ω(f, ; A ) ( ) n/2, ǫ k,n,m f, f

Step 3. Thus by (1),(2), and Thm. 3.1, there exists a Hecke eigenform G in CI n (g) such that λ G (T) λ In (g) (T) mod T L n. Step 4. To show that G S k (Γ n ), we use : *Assumption (3) *A charcterization of congruence primes of elliptic modular forms in terms of f,f Ω(f,+;A )Ω(f, ;A ) due to Hida and Ribet (cf. H. Hida, Invent Math. 63(1981), 225-261, Invent. Math. 64(1981), 221-262, K. Ribet, Invent. Math. 71(1983), 193-205.) *Strum s theorem for congruence of elliptic modular forms

Conjecture A. Let g, f and K be as above. Assume that k > n. Let be a prime ideal of K not dividing (2k 1)!. Then is a congruence prime of I n (g) with respect to (S k (Γ n ) ) if and only if divides L(k, f) n/2 1 i=1 L(2i + 1, f,ad). Rem. By using the congruence between the Saito-Kurokawa lifts and non-saito Kurokawa lifts combined with the Galois representations associated to automorphic representations of GSp(2), J. Brown constructed a torsion element of a certain Bloch-Kato Selmer group for an elliptic modular form. (cf. J. Brown, Composit. Math. )

f S 2k 2 (Γ 1 ) : a primitive form p 2k 3 : a prime number K : a sufficiently large number field as above : a prime ideal of K such that p K : the -adic completion of K, : a uniformizer of K (ρ f, V p ) : p-adic rep. of Gal( Q/Q) over K assoc. f T p : Gal( Q/Q)-stable lattice in V p, W p = V p /T p Hf 1(Q, W p(1 k)) : Bloch-Kato s Selmer group of W p (1 k) ( J. Brown, Compos. Math. 143 (2007), 290 322 ) Assume divides L(k, f) and satisfies the other conditions in Theorem 4.1. Then p divides the order of Hf 1(Q, W p(1 k)).

Outline of the proof Let g be a Hecke eigenform in S + k 1/2 (Γ 0(4)) corresponding to f. Theorem 4.1 = F (S k (Γ 2 ) ) such that I 2 (g) F mod (Piatetski-Shapiro) F (S k (Γ 2 ) ) = F is tempered (G. Laumon?) ρ F an absolutely irreducible nice rep. of Gal( Q/Q) assoc. F ρ I2 (g) = ρ F = ǫ k 2 0 0 0 ρ f 0 0 0 ǫ k 1 ǫ k 2 a 12 0 a 21 ρ f a 23 0 0 ǫ k 1 = ρ I2 (g) ρ F mod ǫ : p-adic cyclotomic character, a 12a 21 = 0 and a 23 0. Then ρ f a 23 0 ǫ k 1 gives rise to a non-trivial -torsin element of Hf 1(Q, W p(1 k)).

Consider the case n = 4. f : a primitive form in S 2k 4 (Γ 1 ) g : a Hecke eigenform in S + k 3/2 (Γ 0(4)) corresponding to f. Fact: ρ I4 (g) = ρ f (k 4) 0 0 0 0 0 ρ f (k 3) 0 0 0 0 0 ρ f (k 2) 0 0 0 0 0 ρ f (k 1) 0 0 0 0 0 ρ f,ad where ρ f (i) = ρ f ǫ i 0 0 ǫ k 2 ǫ i, ρ f,ad = ρ f,ad 0 0 ǫ 2k 7

Problem C. Give a torsion element of another Bloch-Kato Selmer group attached to ρ f ǫ i or to ρ f,ad by using Theorem 4.1. Obstacles to proceed further: (1) There is no Galois reresentation attached to a tempered automorophic representation of GSp(4) (?) (2) There is no classification of non-tempered automorphic representations of GSp(4).

5. Ikeda s conjecture on the period of the Ikeda lift for U(m, m) K : an imaginary quadratic extension of Q with the discr. D, O : the ring of integers in K Gal(K/Q) = σ χ := ( D ) U(m, m)(r) := {g GL m (R K) t g σ J m g = J m } for any Q-algebra R. Γ (m) := U(m, m)(q) GL 2m (O). det l : Γ (m) γ (det(γ)) l F, G S 2l (Γ (m),det l ) F, G = Γ (m) \ m F(Z)G(Z)det(Im(Z)) l 2m dxdy.

f(z) = N=1 a(n)e(nz) : a primitive form in S 2k+1 (Γ 0 (D), χ). p D, α p C s.t. α p + χ(p)α 1 p = p k a(p) p D, α p := p k a(p) L(s, f, χ i ) := {(1 α p p s+k χ(p) i )(1 α 1 p p s+k χ(p) i+1 )} 1 p D p D p D (1 α p p s+k ) 1 if i is even (1 α 1 p p s+k ) 1 if i is odd.

Hermitian Ikeda lift: (1) S 2k+1 (Γ 0 (D), χ) S 2k+2n (Γ (2n),det k n ) (2) S 2k (SL 2 (Z)) S 2k+2n (Γ (2n+1 ),det k n ) k : a non-negative integer, m = 2n or 2n + 1 S (m,k) := S 2k+1 (Γ 0 (D), χ) if m = 2n, S 2k (SL 2 (Z)) if m = 2n + 1.

For simplicity, from now on, assume that the class number of K is one. Theorem 5.1. Let m = 2n or 2n + 1. Let f be a primitive in S (m,k). Then there exsits a Hecke eigenform Lift (m) (f) in S 2k+2n (Γ (m),det k n ) whose standard L-function is m L(s + k + n i + 1/2, f)l(s + k + n i + 1/2, f, χ) i=1 except in the case: m = 2n with n odd and f comes from a Hecke character of K. (cf. T. Ikeda, Composit. Math. 144(2008) 1107-1154.)

f(z) = m=1 a(m)e(mz) : a primitive form in S 2k+1 (Γ 0 (D), χ) p D, α p C s.t. α p + χ(p)α 1 p p D, α p := p k a(p) = p k a(p) L(s, f,ad, χ i ) := {(1 α 2 pχ(p) i+1 p s )(1 α 2 p χ(p) i+1 p s )(1 χ(p) i p s )} 1 p D p D p D (1 p s ) 1 if i is even {(1 α 2 pp s )(1 α 2 p p s )} 1 if i is odd Λ(s, f,ad, χ i ) := Γ C (s)γ C (s + 2k n 1)L(s, f,ad, χ i ) L(i, f,ad, χ i+1 ) := Λ(i, f,ad, χ i+1 )/ f, f Λ(s, χ i ) := Γ C (s)l(s, χ i ) L(j, χ j Λ(2i + 1, χ)d ) = 2i+1/2 j = 2i + 1 Λ(2i) j = 2i

Theorem 5.2. (K) Let f be a primitive form in S (m,k). Then Lift (m) (f), Lift (m) (f) f, f = 2 α n,k m i=2 Λ(i, f, Ad, χ i 1 ) m i=2 Λ(i, χ i ) D 2nk+5n2 3n/2 1/2 q D (1 + q 1 ) if m = 2n D 2nk+5n2 +5n/2 if m = 2n + 1. Rem. This was conjectured by Ikeda (cf. Composit. Math. 144(2008) 1107-1154).

Theorem 5.3. Let the notation be as above. Then we have Lift (m) (f), Lift (m) (f) f, f m = 2 β n,k m i=2 L(i, f, Ad, χ i+1 )L(i, χ i ) D 2nk+4n2 n q D (1 + q 1 ) if m = 2n D 2nk+4n2 +n if m = 2n + 1, where β n,k is an integer depending on n and k. Corollary. In addition to the above notation and the assumption, assume that m 2k. Then Lift(m) (f), Lift (m) (f) f, f m is algebraic, and in particular it belongs to Q(f).

S 2k+2n (Γ (m),det k n ) :=< Lift (m) (f) f primitive form in S (k,m) > C. (generalized Maass space) Conjecture B. Let K and f be as above. Assume that 2k m. Let be a prime ideal of a certain sufficiently large number field not dividing (2k + 2n 1)!. Then is a congruence prime of Lift (m) (f) with respect to (S 2k+2n (Γ (m),det k n ) ) if and only if divides m L(i, f, Ad, χ i+1 ). i=2