SAITO-KUROKAWA LIFTS AND APPLICATIONS TO ARITHMETIC

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1 SAITO-KUROKAWA LIFTS AND APPLICATIONS TO ARITHMETIC JIM L. BROWN Abstract. In this short survey paper we present an outline or using the Saito-Kurokawa correspondence to provide evidence or the Bloch-Kato conjecture or modular orms. Speciic results will be stated, but the aim is to provide the ramework or such results with an aim towards uture research. 1. The Bloch-Kato conjecture or modular orms In this section we will review the Bloch-Kato conjecture or modular orms. The conjecture is presented in the general ramework o using L-unctions and automorphic data to obtain arithmetic inormation. As such, we begin by reviewing Dirichlet s class number ormula and the Birch Swinnerton-Dyer conjecture in this ramework. Let K be a inite abelian Galois extension o Q. O particular arithmetic interest is the size o the class group o K, denoted by h K. In the spirit o the general ramework we are trying to establish, we associate automorphic objects to K in the orm o the group o characters X o K. Dirichlet s class number ormula gives a precise relationship between the special values o the L-unctions o these characters and the size o the class group o K. In particular, we have χ X (2π) L(1, χ) = 2r1 r2 R K h K ω K DK where r 1 is the number o real embeddings o K, r 2 the number o pairs o complex embeddings, R K the regulator o K, ω K the number o roots o unity in K, and D K the discriminant o K. Note that or the trivial character in X, we write L(1, 1) to denote the residue at s = 1 o the L-unction, in this case just the Riemann zeta unction. One sees that by studying the L-unctions o the automorphic data associated to K one can translate this back to the arithmetic data about K that is o interest to us. Let E be an elliptic curve over Q. The Mordell-Weil theorem gives that the rational points on E are given by E(Q) = E(Q) tors Z r where E(Q) tors is one o 15 possible inite groups and r is the rank o the elliptic curve. Arithmetically, the rank o the elliptic curve is o great interest, as is the 2000 Mathematics Subject Classiication. Primary 11F33, 11F67; Secondary 11F46, 11F80. Key words and phrases. modular orms, Saito-Kurokawa lits, Bloch-Kato conjecture, Selmer groups. 1

2 2 JIM L. BROWN Shaarevich-Tate group X(E/Q) which is deined as ollows: ( X(E/Q) = ker H 1 (Q, E) ) H 1 (Q p, E) p where we use the standard convention that H 1 (K, E) = H 1 cont (Gal(K/K), E(K)). The work o Taylor and Wiles ([TW], [W]) and subsequent work o Breuil, Conrad, Diamond and Taylor ([BCDT]) shows that associated to E one has a weight 2 neworm E so that L(s, E) = L(s, E ). The Birch and Swinnerton-Dyer conjecture can then be stated as lim s 1 L(s, E ) (s 1) r Ω E = ( 2 r ) R E Tam(E) (#E(Q) tors ) 2 #X(E/Q) where R E is the regulator o E, Ω E is a period associated to E, and Tam(E) is the Tamagawa actor. In light o the above ramework, one can consider the Bloch-Kato conjecture as a vast generalization o Dirichlet s class number ormula and the Birch and Swinnerton-Dyer conjecture to the ramework o motives. The reader interested in the general conjecture is directed to the original paper o Bloch and Kato ([BK]). We begin by recalling the deinition o the Selmer group. Let E be a inite extension o Q p, O the ring o integers o E, and a uniormizer. Let V be a p-adic Galois representation deined over E and let T V be a Gal(Q/Q)-stable O-lattice. Set W = V/T. Deine spaces H 1 (Q l, V ) by { H 1 (Q Hur (Q l, V ) = l, V ) l p ker(h 1 (Q p, V ) H 1 (Q p, V B crys )) l = p where H 1 ur(q l, M) = ker(h 1 (Q l, M) H 1 (I l, M)) or any Gal(Q l /Q l )-module M with I l the inertia group and B crys Fontaine s ring o p-adic periods. The Bloch-Kato groups H 1 (Q l, W) are deined by The Selmer group o W is given by ( H 1 (Q l, W) = im(h 1 (Q l, V ) H 1 (Q l, W)). H 1 (Q, W) = ker H 1 (Q, W) l ) H 1 (Q l, W) H 1 (Q, l, W) i.e., it consists o cocycles in H 1 (Q, W) that when restricted to the decomposition group at l lie in H 1 (Q l, W) or each prime l. When working with modular orms o level greater then 1, it is oten necessary to work with a larger Selmer group where we do not require any conditions at the primes dividing the level. Suppose is o level is N and set Σ = {l N}. The relevant Selmer group is then H 1,Σ (Q, W) = ker ( H 1 (Q, W) l/ Σ H 1 (Q l, W) H 1 (Q l, W) We are now in a position to state the Bloch-Kato conjecture or modular orms. For each prime p, let V p be the p-adic Galois representation arising rom a weight 2k 2 neworm. Let T p be a Gal(Q/Q)-stable lattice and W p = V p /T p. We denote ).

3 SAITO-KUROKAWA LIFTS AND APPLICATIONS TO ARITHMETIC 3 the j th Tate-twist o V p by V p (j) and similarly or W p. Let π be the natural map H 1 (Q, V p (j)) H 1 (Q, W p (j)). The Shaarevich-Tate group is deined to be X(j) = l H 1 (Q, W l(j))/π H 1 (Q, V l(j)). Deine the set o global points by Γ Q (j) = l H 0 (Q, W l (j)). One should think o these as the analogue o the rational torsion points on an elliptic curve. Conjecture 1.1. (Bloch-Kato) With the notation as above, one has L alg (k, ) = ( l c l(k)) #X(1 k) #Γ Q (k)#γ Q (k 2) where c p (j) are Tamagawa actors and we deine and the period Ω ± L alg (k, ) := L(k, ) π k Ω ± is chosen based on the parity o k. Remark It is known that away rom the central critical value the Selmer group is inite ([K], Theorem 14.2). Thereore we can identiy the p-part o the Selmer group with the p-part o the Shaarevich-Tate group. 2. I T l / T l is irreducible, then H 0 (Q, W l (j)) = The Tamagawa actors are integers. See ([BK], Section 5) or deinitions and discussion. The results we aim to prove are o the orm that i L alg (k, ) and does not divide some other product o special values o normalized L-unctions, then one has (Q, W p (1 k)). p #H 1,Σ 2. Ribet s proo o the converse o Herbrand s Theorem In 1976 Ribet published a short paper using modular orms and Galois representations to prove the converse o Herbrand s theorem ([R]). This work was subsequently generalized by Wiles in his proo o the main conjecture o Iwasawa theory or totally real ields. We briely outline this argument here as it provides a ramework or recent work on the Bloch-Kato conjecture; or example see [Kl] or [SU] or applications other then those described in this paper. Theorem 2.1. Let p be a ixed odd prime and χ : (Z/NZ) Q p an even primitive Dirichlet character o order prime to p, χ ω 2. Suppose χω Ip 1 as a character into F p where I p denotes the inertia group at p. Set ψ = χω and F = Q ψ. I L( 1, χ) is not a unit in Q p, then A ψ 1 F = A F Zp[ ] F p (ψ 1 ) 0 where = Gal(F/Q) and we write F p (ψ 1 ) to denote F p with an action o by ψ 1. To prove the theorem, one constructs a nontrivial unramiied abelian p-extension o F on which acts by ψ 1. This is accomplished by constructing an appropriate Galois representation.

4 4 JIM L. BROWN Recall that there exists an Eisenstein series E 1,χ (z) M 2 (N, χ) with constant term L( 1,χ) 2 and l th Fourier coeicient given by c E (l) = 1 + χ(l)l. The irst step in Ribet s argument is to produce a cuspidal eigenorm so that the eigenvalues o are congruent to those o E 1,χ (z) modulo a prime that divides L( 1, χ). Note that modulo the Eisenstein series is a semi-cusp orm, i.e., it is a modular orm with a constant term o 0 in the Fourier expansion about the cusp ininity. One can show that there is a modular orm g M 2 (M, χ) with constant term 1 or some M with N M by studying the geometry o the modular curve X 1 (p). In act, one can choose M so that ord p (M) 1 and i l M with l Np, then χ(frob l ) l 2 modulo any prime above p. One then sets h = E 1,χ (z) L( 1,χ) 2 g. The modular orm h is congruent to E 1,χ (z) modulo and is in act a semi-cusp orm. Note that h is not necessarily an eigenorm, it is only an eigenorm modulo. One then applies the Deligne-Serre liting lemma to obtain a semi-cusp orm that is an eigenorm and congruent to E 1,χ (z) modulo. A short argument then yields an ordinary cusp orm that is an eigenorm and congruent to E 1,χ (z) modulo. It may be o interest to note that this can be phrased in terms o the Eisenstein ideal i one wishes. I I = T l 1 χ(l)l T (M) O is the Eisenstein ideal, then equivalently one has the ollowing isomorphism T (M) O /I O/(L( 1, χ)). One would be interested in the statement in this language i one wanted to ormulate the results on the Bloch-Kato conjecture in terms o the CAP-ideal (see [Kl]). We will not pursue such a ormulation here so only mention the Eisenstein ideal in passing. The reason we are interested in such a congruence is because o what it tells us about the residual Galois representation o. Let ρ : Gal(Q/Q) GL 2 (F) be the residual representation obtained upon composing ρ with the natural map O F = O/. Observe that the congruence to E 1,χ (z) modulo gives trace(ρ (Frob l )) = a (l) 1 + χ(l)l(mod ), i.e., ρ ss = 1 ψ where ψ = χω. Using the act that is ordinary at p so that we have ρ Dp ( ) χ1 0 χ 2 where χ 2 is unramiied at p and satisies χ 2 (Frob p ) is the unit root o x 2 a (p)x+ χ(p)p k 1, one shows that in act ρ ( ) 1 0 ψ and is non-split. One knows that ρ is unramiied away rom pm because has level M. Some relatively straight-orward calculations with tame inertia allow one to conclude that ρ is unramiied everywhere. Thus, i one deines h : Gal(Q/F) F by h(σ) = (σ) then one can show that the splitting ield Q h o h is the extension we seek. In the ollowing sections we will see how this type o argument can be adapted to our situation.

5 SAITO-KUROKAWA LIFTS AND APPLICATIONS TO ARITHMETIC 5 3. Saito-Kurokawa Lits In the previous section we outlined Ribet s proo o the converse o Herbrand s theorem. The main point o the argument was to produce a cuspidal eigenorm congruent to an Eisenstein series. The reason or this was that the particularly simple orm o the Galois representation o the Eisenstein series could be used to deduce inormation about the residual Galois representation o. In order to apply Ribet s argument to our situation we need to ind a suitable substitute or the Eisenstein series used in Ribet s argument. The substitute we seek is a Saito- Kurokawa lit. One begins with a cusp orm on SL 2 (Z) and associates a cusp orm F on Sp 4 (Z). In the language o automorphic orms, F is a CAP orm (cuspidal associated to parabolic). What this means is that F is a cusp orm that has the same Hecke eigenvalues almost everywhere as the Eisenstein series obtained by inducing the cuspidal automorphic representation π o GL 2 viewed on the Siegel parabolic to an automorphic representation on GSp 4. For the details o this construction in the language o automorphic representations one should consult ([PS]). We will be interested in the classical description o the Saito-Kurokawa correspondence as described in ([EZ], [MRV], [MR], [Z]). As this correspondence is well-known and the reerences listed provide all the details o the construction, we content ourselves here with stating the acts we will need. Let S2k 2 new (Γ 0(N)) be a neworm with Fourier coeicients in some ring O. The Saito-Kurokawa lit is constructed as a series o lits: irst liting to a halinteger weight orm in Kohnen s +-space, then liting this orm to a Jacobi orm, inally liting the Jacobi orm to a Siegel cuspidal eigenorm we denote by F. The orm F lies in the Maass space, denoted by S,new k (Γ 4 0 (N)). The precise result is as ollows. Theorem 3.1. ([MRV], Theorem 8) The space S,new k (Γ 4 0 (N)) is isomorphic to the space S2k 2 new (Γ 0(N)) or N odd and square-ree. Given a neworm S2k 2 new (Γ 0(N)), the corresponding F S,new k (Γ 4 0 (N)) has modiied Spinor L-unction satisying (1) L spin(s, F) = ζ(s k + 1)ζ(s k + 2)L(s, ) where the modiied Spinor L-unction is deined by L spin(s, F) = p N[(1 p k 1 s )(1 p k 2 s )] 1 L spin (s, F). We will be interested in the arithmetic properties o this correspondence, so we will need the ollowing result as well. Corollary 3.2. ([B5], Corollary 2.7) I has Fourier coeicients in O, there is a normalization o the Saito-Kurokawa lit so that F has Fourier coeicients in O. In particular, i O is a discrete valuation ring, F has a Fourier coeicient in O. There are two other main results we will need about Saito-Kurokawa lits. The irst is the calculation o the inner product F, F in terms o,. We will need this result as one o the main steps in our argument is to produce a congruence between a Saito-Kurokawa lit and a Siegel eigenorm that is not a CAP orm. This is the analogous step to Ribet s construction o a cuspidal eigenorm congruent to to the Eisenstein series E 1,χ (z). In the case o level 1 this inner product calculation is a well-known result, see or example [F] or [KS]. For the case o square-ree level N > 1 the result is as ollows.

6 6 JIM L. BROWN Theorem 3.3. ([B2], Theorem 1.1) Let N = p 1... p n with the p i odd distinct primes, S2k 2 new (Γ 0(N)) a neworm, and F S,new k (Γ 4 0(N)) the Siegel modular orm associated to via the Saito-Kurokawa correspondence. Let D be a undamental discriminant with ( 1) k 1 D > 0, gcd(n, D) = 1, χ D the associated quadratic character, and c g ( D ) 0 where the c g are the Fourier coeicients o the hal-integral weight modular orm associated to via the Saito-Kurokawa correspondence. Then one has c g ( D ) 2 L(k, ) (2) F, F = B k,n, π D k 3/2 L(k 1,, χ D ) where B k,n = N k (k 1) n ( i=1 p 2m i 2 i (p 4 i + 1)) 2 n+3 3 [Sp 4 (Z) : Γ 4 0 (N)] [Γ 0(N) : Γ 0 (4N)]. One should note here that this calculation really only relies on the explicit nature o the Saito-Kurokawa correspondence and contains no deep results. It is possible to calculate such inner products in other similar situations. For example, the corresponding statement in terms o unitary groups and Hermitian modular orms can be ound in [Kl]. In the symplectic case, more generally one can calculate a similar relation or the Ikaeda lit F on Sp 2n (Z) where in this case the relation is between F, F and, n. This result is given in [CK] where the explicit ormula is not worked out, but the oundations or such a calculation are set. Finally, to complete the analogy with the Eisenstein series as used by Ribet we should have a particularly simple orm or the 4-dimensional Galois representation associated to F (see Theorem 5.1). This is in act the case and ollows rom the actorization o the Spinor L-unction o F. We have ρ F = ε k 2 ρ ε k 1 where ε is the p-adic cyclotomic character and ρ is the p-adic Galois representation associated to. Comparing this with the act that ρ E1,χ = 1 χω we see that the Saito-Kurokawa lit is a suitable replacement or the Eisenstein series in our situation. 4. Congruences In this section we will give a general outline or producing congruences between Saito-Kurokawa lits and cuspidal Siegel eigenorms that are not CAP orms. These congruences rely on being able to calculate certain inner products o pullbacks o Eisenstein series and Saito-Kurokawa lits. We will stick to giving a general outline o the argument. We outline the method in much greater generality then it is known to work or, thereby providing an outline or uture research. For a detailed treatment o such an argument in the cases it is known to work the reader can consult [B3], [B4], or [B5]. One begins by deining a Siegel Eisenstein series E on some symplectic space Sp 2m (Z) with the Eisenstein series normalized to have p-integral Fourier coeicients. The speciic Eisenstein series and symplectic space vary depending on the application. The important part is that one must be able to orm the pullback o this Eisenstein series to some product o the symplectic groups Sp 2 (Z) and Sp 4 (Z). We need at least one copy o Sp 4 (Z) to occur since we are interested in Saito- Kurokawa lits. I one wanted to work with Ikaeda lits, one would want copies o Sp 2 (Z) and Sp 2n (Z) instead.

7 SAITO-KUROKAWA LIFTS AND APPLICATIONS TO ARITHMETIC 7 Suppose we have such a pullback. Write our pullback as E(Z 1, Z 2,...,Z r ) or Z i h ni where h ni is the Siegel upper hal-space on which Sp 2ni (Z) acts and we have 2n 1 +2n n r = 2m. For simplicity assume that n 1 = 2 so that the irst actor will correspond to Sp 4 (Z). For example, the two cases in which we will state concrete results are the case o the space Sp 8 (Z) pulled back to Sp 4 (Z) Sp 4 (Z) and the space Sp 10 (Z) pulled back to Sp 4 (Z) Sp 4 (Z) Sp 2 (Z). The important and diicult (at least rom the author s point o view) step is now to calculate the inner product o E(Z 1,...,Z r ) with particular orms o our choosing on Sp 2ni (Z). For the actors o Sp 4 (Z), we choose our particular orm to be F. For the actors o Sp 2 (Z), any neworm will do as we will ultimately allow this orm to vary. Write F (i) or our particular orm at the ith place. One hopes to obtain a relationship o the orm (3) E(Z 1,..., Z r ), F (1) (Z 1 ) F (r) (Z r ) = C L-ctns periods where C is an explicitly determined constant and the product over periods is a product containing terms o the orm F (i), F (i) or some o the i. For example, using the inner product given in [Shim95] or Sp 4 (Z) Sp 4 (Z) one obtains E(Z 1, Z 2 ), F (Z 1 ) F (Z 2 ) = C L(3 k, χ)l(1,, χ)l(2,, χ) F, F where χ is a character that is used in the deinition o E and is o weight 2k 2. One should note that Shimura s ormula is true or Sp 2n (Z) Sp 2n (Z) so a similar statement holds or Ikaeda lits. The product o L-unctions that occur will play an important role in our divisibility conditions needed to ensure our congruence exists. Choose an orthogonal basis F 1 = F, F 2,..., F s o weight k Siegel orms on Sp 4 (Z) and an orthogonal basis g 1,...,g t o weight k elliptic modular orms on Sp 2 (Z) with g 1 being a neworm. For each Z j, write H (j) i j Thus, i Z j corresponds to Sp 4 (Z) then H (j) i j then H (j) i j or the i j th basis element. = F ij and i Z j corresponds to Sp 2 (Z) = g ij. We can expand E(Z 1,..., Z r ) in terms o these bases: (4) E(Z 1,...,Z r ) = c i1,...,i r H (1) i 1 (Z 1 ) H (r) i r (Z r ). i 1,...,i r For example, in the case o Sp 4 (Z) Sp 4 (Z) Sp 2 (Z) we write E(Z 1, Z 2, Z 3 ) = i 1,i 2,i 3 c i1,i 2,i 3 F i1 (Z 1 )F i2 (Z 2 )g i3 (Z 3 ). In order to ensure the congruence we construct does not produce a congruence betwween F and another Saito-Kurokawa lit, it is necessary to kill all the other Saito-Kurokawa lits that occur in the above expansion. This is accomplished by constructing an appropriate Hecke operator and then applying it to the expansion in equation (4). We omit the details and assume hereater that this has been perormed. For the details o such a construction see [B3] or [B5]. The coeicient c 1,1,...,1 is the coeicient that controls the congruence we seek to establish. Suppose that we are able to write c 1,1,...1 = 1 A or a uniormizer o some inite extension O o Z p and A O. Multiplying equation (4) through by and using that E has p-integral Fourier coeicients and a little work we obtain a congruence modulo o the orm AH (1) 1 (Z 1) H (r) 1 (Z r) c i1,...,i r H (1) i 1 (Z 1 ) H (r) i r (Z r )(mod ). i 1>1

8 8 JIM L. BROWN For j > 1, we expand the Z j terms in their Fourier expansions and equate the appropriate Fourier coeicients to obtain a congruence F (Z 1 ) i>1 c i F i(z 1 )(mod ). Thus, it only remains to study the coeicient c 1,1,...,1. To do this we combine equation (3) with equation (4). Using these equations and the act that Z 1 corresponds to Sp 4 (Z), we will obtain a F, F in the denominator o c 1,1,...,1. Thus, we can apply Theorem 3.3 to replace F, F and obtain L(k, ) in the denominator o c 1,1,...,1. The rest o c 1,1,...,1 will consist o things that are p-units and then the rest o the L-unctions coming rom equation (3) and the L-unction L(k 1,, χ D ) rom Theorem 3.3. The periods occurring are used to normalize the L-unctions. Thus, we obtain statements o the orm that i L alg (k, ) and does not divide some product o L-unctions, then there is a congruence between F and a Siegel modular orm G that is not a Saito-Kurokawa lit. In order to make G into a cuspidal eigenorm some more work is required, but we omit the details. One should also note that we assume ρ is irreducible and so one cannot have F congruent to a CAP orm that is not a Saito-Kurokawa lit because o the shapes o the corresponding Galois representations. We now list two speciic results as an illustration o theorems proven in this manner. Theorem 4.1. ([B3], Theorem 6.5) Let k > 3 be an even integer and p a prime so that p > 2k 2. Let S 2k 2 (SL 2 (Z), O) be a neworm with real Fourier coeicients and F the Saito-Kurokawa lit o. Suppose that ρ is irreducible and is ordinary at p. I there exists an integer M > 1, a undamental discriminant D so that ( 1) k 1 D > 0, χ D ( 1) = 1, p MD[Sp 4 (Z) : Γ 4 0 (M)], and a Dirichlet character χ o conductor M so that with m 1 and m L alg (k, ) n L(3 k, χ)l alg (k 1,, χ D )L alg (1,, χ)l alg (2,, χ) with n < m, then there exists a cuspidal Siegel eigenorm G on Sp 4 (Z) o weight k that is not a CAP orm so that F G(mod ) where the congruence is a congruence o eigenvalues. Theorem 4.2. ([B4]) Let k 2Z, k > 6 and let p > 2k +3 be a regular prime. Let S 2k 2 (SL 2 (Z)) and h S k (Γ 0 (M)) be neworms with F the Saito-Kurokawa lit o. Assume urther that is ordinary at p. I and m L alg (k, ) n L alg (2k 4, )L alg (2k 3, F h) with n < m, then then there exists a cuspidal Siegel eigenorm G on Sp 4 (Z) o weight k that is not a CAP orm so that F G(mod ) where the congruence is a congruence o eigenvalues. Note that L alg (s, F h) is a convolution L-unction. Note the essential dierence in the two theorems is the L-unction we want our prime to miss. In the irst theorem we have the reedom to move a character χ around to try and cause to miss the L-values. The second theorem allows us to

9 SAITO-KUROKAWA LIFTS AND APPLICATIONS TO ARITHMETIC 9 move a modular orm around instead, providing more reedom and hopeully more instances where the L-values are missed by. It should also be mentioned that Theorem 4.2 will appear in [B4] only in the case o h having level 1. The general case is still work in progress and will appear in a uture paper. 5. Galois representations and Selmer groups This inal section deals with the problem o using an eigenvalue congruence o the orm F G(mod ) between a Saito-Kurokawa lit and a cuspidal Siegel eigenorm that is not a CAP orm to produce nontrivial p-torsion elements in an appropriate Selmer group. We begin with the ollowing result stating the existence o the desired Galois representations. Theorem 5.1. ([SU], Theorem 3.1.3) Let F S k (Γ 4 0(N)) be an eigenorm, K F the number ield generated by the Hecke eigenvalues o F, and p a prime o K F over p. There exists a inite extension E o the completion K F,p o K F at p and a continuous semi-simple Galois representation ρ F,p : Gal(Q/Q) GL 4 (E) unramiied at all primes l pn and so that or all l pn, we have We also need the ollowing result. det(x I ρ F,p (Frob l )) = L (l) spin (X). Theorem 5.2. ([Falt], [U]) Let F be as in Theorem 5.1 with p a prime not dividing the level o F. The restriction o ρ F,p to the decomposition group D p is crystalline at p. In addition i p > 2k 2 then ρ F,p is short. Recall that in section 3 it was stated that or F a Saito-Kurokawa lit, the Galois representation associated to F is o the orm ρ F,p = εk 2 ρ,p ε k 1 where ε is the p th cyclotomic character. This act along with the eigenvalue congruence F G(mod ) allows one to deduce that ρ ss G,p = ρ F,p = ω k 2 ρ,p ω k 1 where we use ω to denote the reduction o ε modulo. The goal is to use this result on the semi-simpliication o ρ G,p to deduce the orm o ρ G,p. In particular, simple matrix calculations allow one to conclude that there is a Gal(Q/Q)-stable lattice so that one has ρ G,p = ωk ρ,p 4 ω k 1 where either 1 or 3 is zero. For the details o these calculations one can consult [B1]. In analogy with Ribet s result, we would like to show that 4 gives a non-zero class in the Selmer group H 1,Σ (Q, W,p (1 k)). We irst need the ollowing result.

10 10 JIM L. BROWN The proo o this proposition is a direct generalization o the corresponding result in [R]. Proposition 5.3. Let ρ G,p be such that it does not have a sub-quotient o dimension 1 and ρg,p ss ρ,p ω k 1. Then there exists a Gal(Q/Q)-stable O-lattice in V G having an O-basis such that the corresponding representation ρ = ρ G,p : Gal(Q/Q) GL 4 (O) has reduction o the orm ω k (5) ρ G,p = 3 ρ,p 4 ω k 1 and such that there is no matrix o the orm 1 n 1 (6) U = 1 n 2 1 n 3 GL 4(O) 1 such that ρ = UρU 1 has reduction o type (5) with 2 = 4 = 0. We split into two cases, 3 = 0 and 1 = 0. Suppose 3 = 0. We wish to show that the quotient extension ( ) ρ,p (7) 4 0 ω k 1 is not split. Suppose it is split. Then by Proposition 5.3 we know that the extension ( ) ω k 2 (8) 2 0 ω k 1 cannot be split as well. One can then use class ield theory and some arguments in tame inertia to show that this gives a non-trivial quotient o the ω 1 -isotypical piece o the p-part o the class group o Q(ζ p ). However, this is impossible by Herbrand s theorem. Thus we have that the quotient extension (7) is non-split. A similar argument deals with the case o 1 = 0. The properties o the Galois representation ρ G,p and the act that p > 2k 2 so that ρ G,p is crystalline and short at p give that 4 gives a non-trivial p-torsion element in H 1,Σ (Q, W,p (1 k)). Thus, we have the ollowing theorem. Theorem 5.4. Let S 2k 2 (Γ 0 (N)) be a neworm with F the Saito-Kurokawa lit o. I there is a non-cap cuspidal Siegel eigenorm G so that F G(mod ), then H 1,Σ (Q, W,p (1 k)) is non-trivial and p #H 1,Σ (Q, W,p (1 k)). The main point to notice here is that the input to these Galois representation arguments is an eigenvalue congruence F G(mod ). Once one knows such a congruence, the arguments given produce the non-trivial p-torsion element in the Selmer group regardless o how one achieved such a congruence. The hope is that by looking at more general inner product ormulas as in the outline given in section 4 one can produce congruences with conditions that are easier to satisy.

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