aa r X i v : . [ m a t h . N T ] D ec GENERIC LEVEL p EISENSTEIN CONGRUENCES FOR GSp . DAN FRETWELL
Abstract
We investigate level p Eisenstein congruences for GSp , generalisations of level 1congruences predicted by Harder. By studying the associated Galois and automor-phic representations we see conditions that guarantee the existence of a paramodu-lar form satisfying the congruence. This provides theoretical justification for com-putational evidence found in the author’s previous paper.1. Introduction
Consider the discriminant function ∆( z ) = q Q ∞ n =1 (1 − q n ) (where q = e πiz ).A famous observation, due to Ramanujan, is that the Fourier coefficients τ ( n ) of∆ satisfy the congruence: τ ( n ) ≡ σ ( n ) mod 691 . A natural way to view this is as a congruence between the Hecke eigenvalues of theunique normalised weight 12 cusp form ∆ and the weight 12 Eisenstein series E .The modulus 691 appears since it divides the numerator of ζ (12) π , a quantity whichappears in the constant term of E .Since the work of Ramanujan there have been many generalizations of his con-gruences. Indeed by looking for big enough primes dividing the numerator of ζ ( k ) π k ,i.e. B k k , one can provide similar congruences at level 1 between cusp forms andEisenstein series of weight k [10]. In fact one can also give “local origin” congru-ences between higher level cusp forms and level 1 Eisenstein series by extending thedivisibility criterion to include Euler factors of ζ ( k ) rather than the global valuesof ζ ( s ) (see [12] for results and examples).One can study Eisenstein congruences for genus 2 Siegel modular forms. Thereare many ways to generalise. In this paper we consider a particular conjecturalcongruence for paramodular forms of level p , an extension of a congruence predictedat level 1 by Harder [16].Given k ′ ≥ N ≥ S k ′ (Γ ( N )) denote the space of weight k ′ ellipticcusp forms for Γ ( N ) and let S new k ′ (Γ ( N )) denote the subspace of new forms.For j, k ≥ V j,k denote the representation Symm j ( C ) ⊗ det k of GL ( C ).Then S j,k ( K ( N )) will denote the space of genus 2, V j,k -valued Siegel cusp forms Dan Fretwell, Heilbronn Institute for Mathematical Research, School of Mathematics, Univer-sity of Bristol, U.K. Email: [email protected] for the paramodular group: K ( N ) = ∗ N ∗ ∗ ∗∗ ∗ ∗ N − ∗∗ N ∗ ∗ ∗ N ∗ N ∗ N ∗ ∗ ∩ Sp ( Q ) , where the stars represent integers.For a normalised eigenform f ∈ S k (Γ ( N )) let Λ( f, s ) denote its completed L-function. For each critical value 1 ≤ m ≤ k − m such that Λ alg ( f, m ) = Λ( f,m )Ω m ∈ Q , well defined upto multiplication by O × Q f . Infact Ω m only depends on the parity of m .With this choice of period it makes sense to talk about divisibility of criticalvalues of the L-function. Harder’s original congruence suggests that for N = 1, largeenough primes dividing the numerators of these L -values should give Eisensteincongruences. In this paper we will be interested in the following level p version. Conjecture 1.1. (Level p paramodular Harder’s conjecture)Let j > and k ≥ and let f ∈ S new j +2 k − (Γ ( p )) be a normalized Hecke eigenformaway from p with eigenvalues a q ∈ O f . Suppose that ord λ (Λ alg ( f, j + k )) > forsome prime λ of Q f lying above a rational prime l > j + 2 k − (with l = p ).Then there exists a Hecke eigenform F ∈ S new j,k ( K ( p )) away from p with eigen-values b q ∈ O F satisfying b q ≡ q k − + a q + q j + k − mod Λ for all primes q = p (where Λ is some prime lying above λ in the compositum Q f Q F ). The j = 0 version of the above conjecture gives congruences between newformsfor K ( p ) and Saito-Kurokawa lifts of forms for Γ ( p ). Such congruences have beenstudied in detail, for example in [5].It should be noted that Harder’s level 1 congruence remains unproved and ev-idence is rare. The same can be said about higher level generalisations. In theauthor’s previous paper computational evidence was given for Conjecture 1.1 atlevels p = 2 , , , ( p ). The aim of this paper is to show that, given the existence of a“generic” level p congruence of the above type, it is likely that the genus 2 formcan be taken to be paramodular.More specifically we consider an automorphic representation π F = ⊗ π F,q at-tached to a Siegel modular form of genus 2, weight ( j, k ). Assuming π F,q is unrami-fied for all q = p and that F satisfies the congruence in Conjecture 1.1 the followingresult gives the required limitations on π F,p . Theorem 1.2.
Let e (Λ) , f (Λ) be the ramification index and the inertia degree ofthe extension K Λ / Q l . Let k ′ = j + 2 k − . (1) If l ≥ max { f (Λ)+2 , e (Λ)+2 } then π F,p is induced from the Borel subgroupof GSp ( Q p ) . ENERIC LEVEL p EISENSTEIN CONGRUENCES FOR GSp . 3 (2) If further we have p j +2 t − for t = 0 , , , then π F,p is of typeI or II (see the Appendix for the classification). (3)
If further ord Λ (cid:18) B k ′ ( p k ′ − k ′ (cid:19) = 0 then either π F,p is of type II a or thereexists g ∈ S k ′ ( SL ( Z )) satisfying the congruence. Being of type II a guarantees the existence of a new K ( p ) fixed vector, hence thatwe may find F ∈ S new j,k ( K ( p )) satisfying the congruence.If ord Λ (cid:18) B k ′ ( p k ′ − k ′ (cid:19) > f will satisfy a simpler ramanujancongruence [12]. Thus we see that the condition in part three of the above iscontrolling the existence of a “simpler” congruence. The conditions in parts oneand two are not very restrictive since l is large in general relative to p, e (Λ) and f (Λ). 2. Proving Theorem 1.2
We wish to justify the use of paramodular forms in the statement of Conjecture1.1. As discussed in the introduction we will do this by proving Theorem 1.2. Inorder to do this we will fix the following notation: • π F = ⊗ q ≤∞ π F,q is an automorphic representation of GSp attached to someSiegel modular form of weight (j,k). We will assume that π F is unramifiedaway from p . The form F is assumed to be a Hecke eigenform away from p with eigenvalues b q ∈ O F . • f ∈ S new j +2 k − (Γ ( p )) is a normalized Hecke eigenform away from p , wtheigenvalues a q ∈ O f . We write k ′ = j + 2 k −
2. Attached to f is anautomorphic representation π f = ⊗ q ≤∞ π f,q of GL . • K = Q f Q F is the compositum of coefficient fields of f and F . • Λ is a prime of K lying above a rational prime l = p satisfying l > j + 2 k − >
4. Associated to Λ is a completion K Λ , valuation ring O Λ and residuefield F Λ . • ρ f is the 2-dimensional Λ-adic Galois representation associated to f , re-alised over O Λ . The mod Λ semisimple reduction of this is ρ F . Also foreach prime q we have the restriction ρ f,q to Gal( Q q / Q q ). • ρ F is the 4-dimensional Λ-adic Galois representation associated to F , alsorealised over O Λ . See [30] for details. Again we have a mod Λ semisimplereduction ρ F and restrictions ρ F,q to Gal( Q q / Q q ).From now on we assume that the pair ( f, F ) satisfy Conjecture 1.1, with Λ beingthe modulus of the congruence. We will prove the parts of Theorem 1.2 in reverseorder.We will need the following well known results about Galois representations at-tached to elliptic modular forms. Theorem 2.1. (Deligne) Let f ∈ S k (Γ ( N )) be a Hecke eigenform for all Heckeoperators q ∤ N with eigenvalues a q . Let l be a prime satisfying ≤ k ≤ l + 1 and a l l . Then ρ f,l is reducible and ρ f,l ∼ χ k − l λ a − l ⋆ λ a l ! , DAN FRETWELL where λ a is the unramified character Gal ( Q l / Q l ) −→ Z × l such that λ a ( φ l ) = a (here φ l is a frobenius element in Gal ( Q l / Q l ) ). Theorem 2.2. (Fontaine) Suppose that f and l are as above but that a l ≡ l .Then ρ f,l is irreducible. Naturally one asks about the structure of ρ f,p for p | N . The following theoremcan be found on p.309 of [18]. Theorem 2.3. (Langlands-Carayol) Suppose p is a prime such that ord p ( N ) = 1 .Then: ρ f,p ∼ (cid:18) χ l λ a ⋆ λ a (cid:19) for some fixed a . Proving Theorem 1.2 (3) . Assume that π F,p is of type I or II. Our aim isto show that if ord Λ (cid:18) B k ′ ( p k ′ − k ′ (cid:19) = 0 then either π F,p is of type II a (so that π F,p has new K ( p )-fixed vectors) or that there exists g ∈ S k ′ (SL ( Z )) replacing f in thecongruence.To do this we first we translate the congruence into a result about Galois repre-sentations. Lemma 2.4. ρ F ∼ ρ f ⊕ χ k − l ⊕ χ j + k − l , where χ l is the l -adic cyclotomic character.Proof. By assumption we have for each q = p : b q ≡ a q + q k − + q j + k − mod Λ . In terms of mod Λ representations this gives tr( ρ F ( φ q )) = tr(( ρ f ⊕ χ k − l ⊕ χ j + k − l )( φ q ))for all q = p, l .The Cebotarev density theorem givestr( ρ F ) = tr( ρ f ⊕ χ k − l ⊕ χ j + k − l ) . Then since l > (cid:3)
It will be handy to know when ¯ ρ f is irreducible. The Bernoulli criterion forcesthis. Lemma 2.5. If ρ f is reducible then ord Λ (cid:18) B k ′ ( p k ′ − k ′ (cid:19) > .Proof. Suppose ρ f is reducible. Then after a suitable choice of basis: ρ f = (cid:18) α ⋆ β (cid:19) , where α, β are two characters Gal( Q / Q ) → F × Λ . Notice that the image of thesecharacters is abelian.Now it is known that ρ f is unramified at all primes q ∤ pl and so α and β mustbe unramified at the same primes. This forces α = χ ml ǫ and β = χ nl ǫ where ǫ , ǫ are unramified outside p . ENERIC LEVEL p EISENSTEIN CONGRUENCES FOR GSp . 5 To see this let χ : Gal( Q / Q ) → F × Λ be a character unramified at all q ∤ pl .Note that by global class field theory α and β factor through Gal( Q ( µ p ∞ , µ l ∞ ) / Q )(where µ p ∞ denotes the set of p th power roots of unity, similarly for l ). The field Q ( µ p ∞ , µ l ∞ ) is the maximal abelian extension of Q unramified outside pl .We find that Gal( Q ( µ p ∞ , µ l ∞ ) / Q ) ∼ = Gal( Q ( µ p ∞ ) / Q ) × Gal( Q ( µ l ∞ ) / Q ) since p and l are coprime. Hence χ = δǫ where δ is unramified outside of l and ǫ isunramified outside of p .To prove the claim that δ is a power of χ l note that Gal( Q ( µ l ∞ ) / Q ) ∼ = Z × l ∼ =( Z / ( l − Z ) × Z l (using the fact that l > δ has to be trivial on l t Z l for some t ≥ δ induces a representa-tion of ( Z / ( l − Z ) × ( Z /l t Z ). But since l is coprime to | F × Λ | = N (Λ) − Z / ( l − Z ) ∼ = ( Z /l Z ) × are exactly the powers of χ l . Thus χ = χ sl ǫ for some integer s .Continuing we now see that since det( ρ f ( φ q )) ≡ q k ′ − mod Λ for all q ∤ pl itmust be that ǫ = ǫ − .Thus: ρ f = (cid:18) χ ml ǫ ⋆ χ nl ǫ − (cid:19) . A comparison of Artin conductors (p.39 of [31]) shows that ǫ is trivial. Indeed theArtin conductor of ρ f is known to be p whereas if ǫ is non-trivial then the Artinconductor would be at least p > p .Now recall 4 < k ′ < l . Also it must be the case that a l ρ f,l is irreducible by Theorem 2.2, contradicting the reducibility of ρ f ).Thus by Theorem 2.1 we see that ρ f,l must possess an unramified compositionfactor, hence one of χ ml , χ nl must be unramified at l . Since all non-trivial powers of χ l are ramified at l this means one of the composition factors is trivial. It is thenclear that the other composition factor must be χ k ′ − l .Hence: ρ f = (cid:18) ⋆ χ k ′ − l (cid:19) or (cid:18) χ k ′ − l ⋆ (cid:19) . In either case comparing traces of Frobenius at q = p, l gives the Ramanujancongruence: a q ≡ q k ′ − mod Λ . By Proposition 4 . Λ (cid:18) B k ′ ( p k ′ − k ′ (cid:19) > (cid:3) Proposition 2.6.
Suppose π F,p is of type I or II and that ord Λ (cid:18) B k ′ ( p k ′ − k ′ (cid:19) = 0 .Then either π F,p is of type II a or there exists a level one normalized newform g ∈ S k ′ ( SL ( Z )) that satisfies Harder’s congruence with F .Proof. We know that ρ f is irreducible by the previous result. However by Theorem2.3 we have, under a suitable choice of basis: ρ f,p = (cid:18) λ a ⋆ χ l λ a (cid:19) or (cid:18) χ l λ a ⋆ λ a (cid:19) . DAN FRETWELL
In either case the restriction of ρ f,p to the inertia subgroup I p of Gal( Q p / Q p ) isas follows: ρ f,p (cid:12)(cid:12) I p = (cid:18) ⋆ ′ (cid:19) . We have two cases. First it could be the case that ⋆ ′ ≡ . g ∈ S k ′ (SL ( Z )) such that ρ g ∼ ρ f . We wouldthen observe a level one version of Harder’s congruence as required.Now suppose that ⋆ ′ ρ f,p is ramified. If π F,p is of type Ior II b then π F,p is unramified. By the Local Langlands Correspondence for GSp (proved in [15]) we see that ρ F,p is unramified so that ρ F,p is unramified, giving acontradiction. The only other possibility for π F,p is to be of type II a as required. (cid:3) To summarise our progress, given that π F,p is of type I or II then either: • f itself satisfies a simpler Ramanujan congruence, detected by a simpledivisibility criterion, • a replacement level 1 elliptic form satisfies Harder’s congruence with F .The level at which F appears could be 1 or p since we did not place anyramification restrictions on π F,p in our assumptions, • or π F,p is of type II a , implying that a new paramodular form of level p exists satisfying the congruence.The first possibility is a rare occurrence and is easy to check for in practice. Wewill see later that the second possibility rarely occurs for F ∈ S new j,k ( K ( p )). Thecase where F ∈ S j,k (Sp ( Z )) is of course the original Harder conjecture at level 1.From this discussion one should believe that the third possibility is most likelyto occur if F is not a lift from level 1.2.2. Proving Theorem 1.2 (2) . Let us now assume that π F,p is induced from theBorel subgroup of GSp ( Q p ). Then π F,p must be of type I-VI. We will show thatif p j +2 k − t = 0 , , , π F,p is of type I or II.Let W ′ Q p = C ⋊ W Q p be the Weil-Deligne group of Q p . The multiplication onthis group is given by ( z, w )( z ′ , w ′ ) = ( z + ν ( w ) z ′ , ww ′ ), where ν : W Q p −→ C × isthe character corresponding to | · | p by local class field theory.By the Local Langlands Correspondence for GSp we may associate to eachirreducible admissible representation π of GSp ( Q p ) its L -parameter, a certain rep-resentation: ρ π : W ′ Q p −→ GSp ( C ) . One can view such a representation as a pair ( ρ , N ) where: ρ : W Q p −→ GSp ( C )is a continuous homomorphism and N ∈ M n ( C ) is a nilpotent matrix such that: ρ ( w ) N ρ ( w ) − = ν ( w ) N, for all w ∈ W Q p . Given ρ and N we recover the L -parameter via ρ π ( z, w ) = ρ ( w )exp( zN ). ENERIC LEVEL p EISENSTEIN CONGRUENCES FOR GSp . 7 Let π , π be irreducible admissible representations of GSp ( Q p ). Then π ∼ = π implies ρ π ∼ = ρ π under the Local Langlands Correspondence. However theconverse does not hold. A fixed L -parameter can arise from different isomorphismclasses, but only finitely many (those in the same “ L -packet”).Roberts and Schmidt discuss the L -parameters of non-supercuspidal representa-tions of GSp ( Q p ) in [24]. If π is parabolically induced from the Borel subgroup ofGSp ( Q p ) then it is non-supercuspidal and ρ π is simple to describe. In particularthe ρ part is semisimple given by four characters µ , µ , µ , µ of W Q p (which bylocal class field theory correspond to four characters of Q × p ). See the Appendix fora complete table of L -parameters for Borel induced representations.The four complex numbers [ µ ( p ) , µ ( p ) , µ ( p ) , µ ( p )] are the Satake parametersof π . Unramified representations are uniquely determined by their Satake parame-ters up to scaling (much in the same way as unramified local Galois representationsare determined by the image of Frobenius).Let us now return to our congruence between f and F . We have already seenthat the existence of this congruence for all q = p leads to a residual equivalence ofglobal Galois representations: ρ F ∼ ρ f ⊕ χ k − l ⊕ χ j + k − l . In particular we can compare these representations locally at p , the level of f .Since we have the local equality χ l | W Q p = ν − it follows that: ρ F,p | W Q p ∼ ρ f,p | W Q p ⊕ ν − k ⊕ ν − j − k . Given the existence of the congruence we see that the local representations ρ F,p | W Q p and ρ f,p | W Q p ⊕ ν − k ⊕ ν − j − k of W Q p have the same composition fac-tors mod Λ.Recall that to F we have attached a “global” Galois representation ρ F and a“global” automorphic representation π F . Similarly for f . By local-global compati-bility results (see [27] for GSp and [29] for GL ) we know that ρ F,p | W Q p correspondsto π F,p and ρ f,p | W Q p corresponds to π f,p under the corresponding local Langlandscorrespondences.Tying all of this together, the existence of the congruence forces the L -parameterof π F,p to be congruent modulo Λ to that of π f,p ⊕ | · | − kp ⊕ | · | − j − kp (up to scalingby p k ′− = p j +2 k − in the first component). In particular the Satake parametersshould match mod Λ.Since f is a newform of level p it is known that π f,p ∼ = St or π f,p ∼ = ǫ St whereSt is the Steinberg representation of GL ( Q p ) and ǫ is the unique unramified non-trivial quadratic character of Q × p . In either case the Satake parameters are knownto be α p and α − p where α p = p or ǫ ( p ) p = − p . Applying the scaling gives[ α p , α − p ] = [ p j +2 k − , p j +2 k − ] or [ − p j +2 k − , − p − j +2 k − ].It is now clear that the Satake parameters of π f,p ⊕ | · | − kp ⊕ | · | − j − kp are[ a, b, c, d ] = h ± p j +2 k − , ± p j +2 k − , p k − , p j + k − i (where the sign is the same for a and b ). Note that these are all integral powers of p . DAN FRETWELL
Theorem 2.7.
Suppose π F,p is of type I-VI and p j +2 t − for t =0 , , , . Then π F,p cannot be of type III,IV,V or VI.Proof.
Suppose π F,p is of one of the types III,IV,V,VI. We show that if the cor-responding Satake parameters are congruent mod Λ then p j +2 t − ≡ t = 0 , , ,
3. Then the result follows.We work in reverse order. Here ǫ will stand for the trivial character. Wheneverthere is a choice of sign this will be fixed by a choice of upper or lower row. Type VI ρ is given by the four characters ν σ, ν σ, ν − σ, ν − σ. Since the central character of π F,p is trivial we have σ = ǫ , so that σ is trivial orquadratic.Thus in some order the Satake parameters are given by ± p , ± p , ± p − , ± p − . Scaling by p k ′− gives ± p j +2 k − , ± p j +2 k − , ± p j +2 k − , ± p j +2 k − . Notice that there are two equal pairs here. Thus for [ a, b, c, d ] to be congruent tothese four numbers mod Λ we would have to have that a is equivalent to one of b, c or d mod Λ.Setting a ≡ b mod Λ gives p ≡ a ≡ c mod Λ gives p j +22 ≡ ± a ≡ d mod Λ gives p j ≡ ± Type V ρ is given by the four characters ν σ, ν ξσ, ν − ξσ, ν − σ. Since the central character of π F,p is trivial we have σ = ǫ , so that σ is trivial orquadratic.Thus in some order the Satake parameters are given by ± p , ∓ p , ∓ p − , ± p − . Scaling by p k ′− gives ± p j +2 k − , ∓ p j +2 k − , ∓ p j +2 k − , ± p j +2 k − . Notice that there are two pairs of the form ( α, − α ). Thus for [ a, b, c, d ] to becongruent to these four numbers mod Λ we would have to have that a is equivalentto one of − b, − c or − d mod Λ.Setting a ≡ − b mod Λ gives p ≡ a ≡ − c mod Λ gives p j +22 ≡ ∓ a ≡ − d mod Λ gives p j ≡ ∓ Type IV ρ is given by the four characters ν σ, ν σ, ν − σ, ν − σ. ENERIC LEVEL p EISENSTEIN CONGRUENCES FOR GSp . 9 Since the central character of π F,p is trivial we have σ = ǫ , so that σ is trivial orquadratic.Thus in some order the Satake parameters are given by ± p , ± p , ± p − , ± p − . Scaling by p k ′− gives ± p j +2 k , ± p j +2 k − , ± p j +2 k − , ± p j +2 k − . If [ a, b, c, d ] are congruent to these numbers mod Λ then there are four possibilitiesfor c .Setting c ≡ ± p j +2 k mod Λ gives p j +42 ≡ ± c ≡ ± p j +2 k − mod Λ gives p j +22 ≡ ± c ≡ ± p j +2 k − mod Λ gives p j ≡ ± c ≡ ± p j +2 k − mod Λ gives p j − ≡ ± Type III ρ is given by the four characters ν χσ, ν − χσ, ν σ, ν − σ. Since the central character of π F,p is trivial we have χσ = ǫ , so that χσ = σ − .Thus in some order the Satake parameters are given by p β − , p − β − , p β, p − β, where β = σ ( p ). Scaling by p k ′− gives p j +2 k − β − , p j +2 k − β − , p j +2 k − β, p j +2 k − β. If [ a, b, c, d ] are congruent to these numbers mod Λ then there are four possibilitiesfor a (each giving the value of β mod Λ). However replacing β by β − gives thesame Satake parameters, so it suffices to set a congruent to just the last two Satakeparameters.Setting a ≡ p j +2 k − β mod Λ gives β ≡ ± ± p j +2 k − , ± p j +2 k − , ± p j +2 k − , ± p j +2 k − . However we have already dealt with these in Type VI.Setting a ≡ p j +2 k − β mod Λ gives β ≡ ± p mod Λ. This gives Satake parametersequivalent to ± p j +2 k , ± p j +2 k − , ± p j +2 k − , ± p j +2 k − . However we have already dealt with these in Type IV.Suppose now that none of the following holds: p j − ≡ p j ≡ p j +2 ≡ p j +4 ≡ . Then none of the conditions found above hold and so we must have that π F,p is oftype I or II, as required. (cid:3)
Note that if one compares the Satake parameters [ a, b, c, d ] to those from a rep-resentation of type I or II then no conditions arise. It is always possible for theseto be congruent mod Λ.2.3.
Proving Theorem 1.2 (1) . We now move on to our final task, finding con-ditions that guarantee π F,p is induced from the Borel subgroup of GSp ( Q p ). Wewill show that if l ≥ max { f (Λ) + 2 , e (Λ) + 2 } then π F,p is induced from the Borelsubgroup of GSp .In this section Λ ′ will be an arbitrary prime of K = Q F Q f , lying above a rationalprime l ′ .Recall that π F,p corresponds via Local Langlands to a representation of theWeil-Deligne group W ′ p , which itself is parametrized by a continuous representation ρ : W Q p → GSp ( C ) and a nilpotent matrix N ∈ M ( C ) with certain properties(mentioned in the previous subsection). However if we fix a choice of embeddings Q ֒ → C and Q ֒ → Q l ′ then one can convert these representations into l ′ -adicrepresentations with open kernel (p.77 of [28]).It is also known that local Galois representations give rise to Weil-Deligne rep-resentations. Theorem 2.8. (Grothendieck-Deligne) Let p = l ′ and fix a continuous n -dimensional Λ ′ -adic representation: ρ : Gal ( Q p / Q p ) −→ GL n ( K Λ ′ ) . Then associated to ρ is a unique l ′ -adic representation of W ′ Q p , given by a pair ( ρ ′ , N ′ ) satisfying: • ρ ′ : W Q p −→ GL n ( K Λ ′ ) is continuous with respect to the discrete topologyon GL n ( K Λ ′ ) . In particular ρ ′ ( I p ) is finite. • ρ ′ ( φ p ) has characteristic polynomial defined over O Λ ′ with constant term a Λ ′ -adic unit. • N ′ ∈ M n ( K Λ ′ ) is nilpotent and satisfies ρ ′ ( σ ) N ′ ρ ′ ( σ ) − = ν ( σ ) N ′ , for all σ ∈ W Q p .Fixing a tamely ramified character t l ′ : I p → Z l ′ , the relationship between ρ and ρ ′ is: ρ ( φ np u ) = ρ ′ ( φ np u ) exp ( t l ′ ( u ) N ′ ) , for all n ∈ Z , u ∈ I p . Now consider the local Galois representation ρ F,p . By the above theorem ithas an associated Weil-Deligne representation, given by a pair ( ρ ′ , N ′ ). A Local-Global Compatibility conjecture of Sorensen (pages 3-4 of [27], proved in certaincases by Mok in Theorem 4 .
14 of [21]) shows that the Weil-Deligne representationsattached to π F,p and ρ F,p are isomorphic (up to Frobenius semi-simplification).In particular this implies that ρ ∼ = ρ ′ up to semi-simplification. We make thisidentification from now on and use ρ to denote the Frobenius semi-simplificationof both representations.A useful corollary of the above theorem is the following: ENERIC LEVEL p EISENSTEIN CONGRUENCES FOR GSp . 11 Corollary 2.9. (Grothendieck Monodromy Theorem) With the above setup thereexists a finite index subgroup J Λ ′ ⊆ I p such that ρ ( σ ) = exp ( t l ′ ( σ ) N ) for each σ ∈ J Λ ′ , i.e. each element of J Λ ′ acts unipotently. See the appendix of [26] for a proof of this.By the Grothendieck Monodromy Theorem there exists a (maximal) finite indexsubgroup J Λ ′ ⊆ I p acting by unipotent matrices, i.e. if σ ∈ J Λ ′ then: ρ F,p ( σ ) = exp( t l ′ ( σ ) N ) . Note then that as a consequence, for each σ ∈ J Λ ′ : ρ ( σ ) = ρ F,p ( σ )exp( − t l ′ ( σ ) N ) = I. Thus ρ factors through I p /J Λ ′ : ρ : I p −→ I p /J Λ ′ −→ GL ( O Λ ′ ) . Note that ρ ( I p /J Λ ′ ) is finite. It is conjectured that the size of this image isindependent of Λ ′ (see Conjecture 1 . π F,p is induced from the Borel subgroup of GSp . It sufficesto show that J Λ ′ = I p for some Λ ′ (this case is commonly known as “semi-stable”). Proposition 2.10. If Λ ′ satisfies J Λ ′ = I p then π F,p is induced from the Borelsubgroup of GSp ( Q p ) .Proof. It suffices to show that there is a basis of K ′ such that ρ ∼ = χ ⋆ ⋆ ⋆ χ ⋆ ⋆ χ ⋆ χ , for four unramified characters χ , χ , χ , χ of W Q p . Then since the image of ρ lies in GSp we must have that χ = χ − and χ = χ − . Then by Local Langlandsfor GSp it must be that π F,p is induced from the Borel subgroup.To this end we already know that I p acts unipotently and so it remains to studythe action of Frobenius φ p . Recall the condition ρ ( φ p ) N ρ ( φ p ) − = p − N . Wewill rewrite this as ρ ( φ p ) N = p − N ρ ( φ p ).By Theorem 2.8 the characteristic polynomial of ρ ( φ p ) has constant term in O × Λ ′ . Choosing an eigenvector v of ρ ( φ p ) with non-zero eigenvalue α ∈ O Λ ′ , noticethat ρ ( φ p )( N v ) = p − N ρ ( φ p ) = αp − ( N v ) . This shows that if
N v = 0 then N v is another eigenvector of ρ ( φ p ) with eigenvalue αp − = α .Consider the list v, N v, N v, N v . If all of these vectors are non-zero then wehave a basis of eigenvectors for ρ ( φ p ). Then ρ ( φ p ) is diagonal.If for some i ≤ N i v = 0 then we can quotient out by the subspacegenerated by v, N v, ..., N i − v and apply the same argument to the quotient, liftingbasis vectors to K ′ where necessary.Continuing in this fashion we then construct a basis of K ′ such that: ρ ( φ p ) = α ⋆ ⋆ ⋆ α ⋆ ⋆ α ⋆ α . It is then clear that ρ is of the required form with unramified characters definedby χ i ( φ p ) = α i for i = 1 , , , I p acts unipotently). (cid:3) If J Λ = I p then the Proposition shows that π F,p is induced from the Borel. It isour aim to find a condition guaranteeing this. First we study the possible sizes of ρ ( I p /J Λ ′ ). Lemma 2.11.
Suppose G is a finite subgroup of GL n ( O Λ ′ ) and that l ′ > e (Λ ′ ) + 1 (where e (Λ ′ ) is the ramification index of K Λ ′ / Q l ′ ). Then the reduction map injects G into GL n ( F Λ ′ ) .Proof. We show that the reduction map GL n ( O Λ ′ ) → GL n ( F Λ ′ ) is torsion-free.Then the restriction of this map to G must have trivial kernel, so that G injectsinto GL n ( F Λ ′ ).To prove the claim we take A ∈ GL n ( O Λ ′ ) with A = I and A ≡ I mod Λ ′ . Wewish to prove that A m = I for each m . We already know this for m = 1.Suppose that A has finite order m >
1. Choose a prime q | m , so that m = qk forsome k ≥
1. Letting B = A k we see that B q = I , B = I and that B ≡ I mod Λ ′ .Thus we can assume without loss of generality that A has prime order q .To this end we write A = I + M with M = 0 and M having entries in Λ ′ .Choose an entry m u,v of M such that | m u,v | Λ ′ = δ is maximal among all entries of M . Then 0 < δ ≤ N (Λ ′ ) .Note that: A q = ( I + M ) q = I + qM + (cid:18) q (cid:19) M + ... + (cid:18) qq − (cid:19) M q − + M q . Case 1:
Suppose q = l ′ . Then the entries of (cid:0) qj (cid:1) M j for j ≥ ′ -adicabsolute value less than or equal to δ . However qM contains the entry qm u,v ofabsolute value δ > δ (since q = l ′ ). Hence A q − I must contain an entry of absolutevalue δ > A q − I = 0 as required. Case 2: q = l ′ . We need sharper inequalities for this case since qM has no entryof absolute value δ . It is now the case that qm u,v has absolute value δN (Λ ′ ) e (where e = e (Λ ′ )).For 2 ≤ j ≤ q − q divides (cid:0) qj (cid:1) so the matrices (cid:0) qj (cid:1) M j haveentries of maximal absolute value δ N (Λ ′ ) e < δN (Λ ′ ) e . Also the matrix M q has entriesof absolute value greater than or equal to δ q < δ e +1 ≤ δN (Λ ′ ) e (using here thecondition q = l ′ > e + 1).Thus we see that A q − I contains an entry of absolute value δN (Λ ′ ) e > A q − I = 0 as required. (cid:3) ENERIC LEVEL p EISENSTEIN CONGRUENCES FOR GSp . 13 We now know that m Λ ′ = | ρ ( I p /J Λ ′ ) | divides | GL ( F Λ ′ ) | whenever l ′ > e (Λ ′ )+1.By using the existence of the congruence we may find a further restriction on thesize of this image. Lemma 2.12. If l > e (Λ) + 1 and J Λ = I p then N (Λ) | m Λ ′ for all Λ ′ .Proof. Note that m Λ ′ = | ρ F,p ( I p /J Λ ′ ) | . As mentioned earlier it is conjectured that m Λ ′ has order independent of Λ ′ . Thus it suffices to show that N (Λ) | m Λ .Now G = ρ F,p ( I p /J Λ ) is a finite subgroup of GL ( O Λ ) and l > e (Λ) + 1. ByLemma 2.11 the reduction map injects G into GL ( F Λ ). Thus | G | = | ρ F,p ( I p /J Λ ′ ) | .However by the existence of the congruence we know that the mod Λ reduction ρ F,p has composition factors ρ f,p , χ k − l , χ j + k − l .Then since ρ f,p | I p = (cid:18) ⋆ (cid:19) and χ l is unramified at p we have: ρ F,p ( I p /J Λ ) ⊆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ . However J Λ = I p by assumption, so that ρ F,p ( I p /J Λ ) is non-trivial. This showsthat N (Λ) divides m Λ , as required. (cid:3) Corollary 2.13. If l ≥ max { f (Λ) + 2 , e (Λ) + 2 } then J Λ = I p (here f (Λ) is theresidue degree of K Λ / Q l ).Proof. Suppose J Λ = I p . Then we know that N (Λ) | m Λ ′ for all Λ ′ . But for eachΛ ′ satisfying l ′ > e (Λ ′ ) + 1 we know that m Λ ′ divides | GL ( F Λ ′ ) | and so (writing f = f (Λ)) l | l ′ f ( l ′ f − l ′ f − l ′ f − l ′ f − . It remains to prove that l ′ can be chosen to contradict this. To contradictthe divisibility condition it suffices to choose l ′ = l such that l ′ f l and l ′ f l .There are at most 3 f + 4 f = 7 f classes mod l that have order dividing 3 f or 4 f .However note that the classes of order dividing hcf(3 f, f ) = f are counted twiceand so there must be at most 7 f − f = 6 f classes of order dividing 3 f and 4 f . Butsince l ≥ f + 2 there must be a non-zero class a mod l that has order coprime to3 f and 4 f . By Dirichlet’s theorem there are infinitely many primes in this classmod l . It suffices to choose l ′ ≡ a mod l such that l ′ > e (Λ ′ ) + 1. (cid:3) Of course it is highly likely that l ≥ max { f (Λ) + 2 , e (Λ) + 2 } in practice since l is a “large” prime. 3. Congruences of local origin
One can use similar techniques to Subsection 3 . Recall that, for all primes p we have the Ramanujan congruence: τ ( p ) ≡ p mod 691 . This shows a congruence between Hecke eigenvalues of a level 1 cuspform of weight12 and the Hecke eigenvalues of the weight 12 Eisenstein series.The modulus 691 can be interpreted in many ways. Naively this prime justhappens to appear in the q -expansion of E . A better interpretation is that itdivides the numerator of B (the relevant quantity in the coefficients of E ).However the best interpretation is that it divides the numerator of ζ (12) π .Ramanujan’s congruence can be extended to give other Eisenstein congruencesfor even weights k ≥
12. The following is proved in [10].
Theorem 3.1.
Suppose ord l (cid:16) ζ ( k ) π k (cid:17) > for some prime l . Then there exists anormalised eigenform f ∈ S k ( SL ( Z )) with eigenvalues a n such that: a p ≡ p k − mod Λ , for all primes p (here Λ | l in Q f ). One can ask whether such Eisenstein congruences arise for elliptic modular formsof higher level. Indeed they do. Consider the question of finding a normalizedeigenform f ∈ S k (Γ ( p )) satisfying for all q = p : a q ≡ q k − mod λ where λ is some prime of Q f . For technical reasons we must demand that k = 2and that λ does not lie above 2 or 3.Of course if ord λ (cid:16) ζ ( k ) π k (cid:17) > S k (Γ ( p ))). However newforms cansatisfy such congruences too. How do we account for these?It turns out that instead of looking for primes dividing (global) zeta values wecan instead look for primes dividing incomplete zeta values. Let: ζ { p } ( s ) = Y q = p (cid:18) − q s (cid:19) − = (cid:18) − p s (cid:19) ζ ( s ) = ( p k − p s ζ ( s ) . The following is proved in [12].
Theorem 3.2.
Let p be prime and k ≥ be even. Suppose l > satisfiesord l (cid:16) ζ { p } ( k ) π k (cid:17) > . Then there exists f ∈ S k (Γ ( p )) , a normalized Hecke eigen-form away from p with eigenvalues a q ∈ O f satisfying: a q ≡ q k − mod Λ , for all q = p and for some Λ | l in Q f . Notice ζ { p } ( k ) π k ∼ B k ( p k − k , a condition we saw in Theorem 1.2. The term “localorigin” is used to describe the new congruences arising from divisibility of a (local)Euler factor. As mentioned above, the local origin congruences generally come fromnewforms (since divisibility of the zeta value gives a congruence at level 1). ENERIC LEVEL p EISENSTEIN CONGRUENCES FOR GSp . 15 We can do this in more generality. Let Σ be a finite set of primes and set ζ Σ ( s ) = Y p/ ∈ Σ (cid:18) − p s (cid:19) − = Y p ∈ Σ (cid:18) − p s (cid:19) ζ ( s ) = Y p ∈ Σ ( p k − p s ζ ( s ) . Then one can predict similar congruences for higher level newforms coming fromdivisibility of the special values ζ Σ ( k ) π k .Naturally we may ask whether “local origin” analogues of Conjecture 1.1 exist.Indeed these are also predicted to occur and a plentiful supply of evidence has beenfound [3]. However, unlike the GL case, these congruences are still conjectural.Given a normalized Hecke eigenform f ∈ S k ′ (SL ( Z )) with eigenvalues a q ∈ O f and a fixed prime p we define an incomplete L-function of f : L { p } ( f, s ) = (1 − a p p − s + p k ′ − − s ) L ( f, s ) = ( p s − a p p s + p k ′ − ) p s L ( f, s ) . The following congruences are then predicted by Harder in [17].
Conjecture 3.3.
Let j > and k ≥ . Let f ∈ S j +2 k − ( SL ( Z )) be a normalizedHecke eigenform with eigenvalues a p ∈ O f . Suppose ord λ (cid:16) L { p } ( f,j + k )Ω j + k (cid:17) > forsome prime λ in Q f . Then there exists F ∈ S j,k (Γ ( p )) , a Hecke eigenform awayfrom p with eigenvalues b q ∈ O F satisfying b q ≡ q k − + a q + q j + k − mod Λ for all q = p and for some Λ | λ in Q f Q F . As in the GL case, congruences arising from divisibility of the Euler factor aredescribed as local origin congruences. It is also expected, in analogy, that localorigin congruences generally come from newforms (since divisibility of the L-valuewould give a congruence at level 1, by the original Harder conjecture).One may ask whether it is possible to find local origin congruences for paramod-ular newforms. We will see below that these are surprisingly very rare. To thisend suppose an eigenform F ∈ S new j,k ( K ( p )) away from p satisfies a local origincongruence with a normalized eigenform f ∈ S k ′ (SL ( Z )) and modulus Λ | λ in Q f Q F .Recall that, by discussions in Subsection 3 .
2, the existence of the congruenceforces π F,p to have Satake parameters congruent to α p , α − p , p j + k − , p k − mod λ .However now that f is of level 1 the values of α p , α − p are different.Fortunately we only need to know these values mod Λ and the divisibility of theEuler factor at p gives this. Indeed:Λ | ( p j + k ) − a p p j + k + p k ′ − ) = ( p j + k + α p p k ′− )( p j + k + α − p p k ′− )and so α p ≡ p ± ( j +32 ) mod Λ. Scaling by p j +2 k − gives Satake parameters [ p j + k , p k − ]for π f,p .So, assuming the existence of a local origin congruence the Satake parameters[ a, b, c, d ] of π F,p must be congruent to [ p j + k , p k − , p j + k − , p k − ] in some order. Theorem 3.4.
If a local origin congruence occurs for F ∈ S new j,k ( K ( p )) with mod-ulus Λ then p j +2 t ≡ for some t = 0 , , , . Proof.
Since F ∈ S new j,k ( K ( p )) we know that π F,p is of type II a , IV c , V b , V c or VI c (these are the only types with new K ( p ) fixed vectors).Comparing Satake parameters as in Theorem 2.7 gives the result. The detailsare omitted. (cid:3) ENERIC LEVEL p EISENSTEIN CONGRUENCES FOR GSp . 17 Appendix A. Borel induced representations of GSp The following table, extracted from p.297 of Roberts and Schmidt [24], liststhe classification of all non-supercuspidal irreducible admissible representations ofGSp ( Q p ) induced from the Borel subgroup. For simplicity only the induced rep-resentations are given, rather than their irreducible constituents.Contained in the table is information about dim( V K ) for certain interestingopen compact subgroups K of GSp ( Q p ) (i.e. GSp ( Z p ) and the local paramodulargroup K ( p )).Here ǫ is the trivial character and ξ is the unique unramified quadratic characterof Q p . Also χ , χ , σ are unramified characters.Type Constituent of Conditions dim( V GSp ( Z p ) ) dim( V K ( p ) )I χ × χ ⋊ σ χ , χ = | · | ± p , χ = | · | ± p χ ± a | · | p χ × | · | − p χ ⋊ σ χ = | · | ± p , χ = | · | ± p b a χ × | · | p ⋊ | · | − p σ χ = ǫ , | · | ± p b a | · | p × | · | p ⋊ | · | − p σ b c d a | · | p ξ × ξ ⋊ | · | − p σ ξ = ǫ , ξ = ǫ b c d a | · | p × ǫ ⋊ | · | − p σ b c d L -parameters. The matrices N will not be needed so have beenomitted. Type ρ Central characterI χ χ σ, χ σ, χ σ, σ χ χ σ II χ σ, ν χσ, ν − χσ, σ ( χσ ) III ν χσ, ν − χσ, ν σ, ν − σ χσ IV ν σ, ν σ, ν − σ, ν − σ σ V ν σ, ν ξσ, ν − ξσ, ν − σ σ VI ν σ, ν σ, ν − σ, ν − σ σ
28 DAN FRETWELL
References [1] M. Asgari, R. Schmidt,
Siegel modular forms and representations . Manuscripta math. 104,173 − Siegel Modular Forms of Genus and Level :Cohomological Computations and Conjectures . Int. Math. Res. Not. IMRN, 2008.[3] J. Bergstr¨om, N. Dummigan, Eisenstein congruences for split reductive groups . 2014. http://people.su.se/~jonab/eiscong1.pdf .[4] J. Bernstein, S. Gelbart at al,
An introduction to the Langlands Program . Birkhauser Basel,2004.[5] J. Brown
Congruence primes of Saito-Kurokawa lifts
Mathematical Research Letters,17(5) , − The - - of Modular Forms . Springer,Universitext, 2008.[7] J. W. S. Cassels, Local Fields . Cambridge University Press, London Mathematical SocietyStudent Texts (No. 3), 1986.[8] G. Chenevier, J. Lannes,
Formes automorphes et voisins de Kneser des rseaux de Niemeier . http://arxiv.org/pdf/1409.7616v2.pdf .[9] N. Childress, Class Field Theory . Springer, Universitext, 2008.[10] B. Datskovsky, P. Guerzhoy,
On Ramanujan Congruences for Modular Forms of Integral andHalf-Integral Weights
Proceedings of the American Mathematical Society, Volume 124, No.8, Pages 2283 − Values of L-Functions and Periods of Integrals . Proc. Symp. Pure Math. AMS33, 313 − Ramanujan style congruences of local origin . Journal of NumberTheory, Volume 143, Pages 248 − The weight in Serre’s conjectures on modular forms . Inventiones mathemati-cae, Volume 109, Issue 1, 563 − Level p paramodular congruences of Harder type . arXiv:1603.07088 [math.NT],2016.[15] W. T. Gan, S. Takeda, The Local Langlands Conjecture for GSp (4). Annals of Mathematics,Pages 1841 − A congruence between a Siegel and an Elliptic Modular Form . Featured in “The1-2-3 of Modular Forms”.[17] G. Harder,
Secondary Operations in the Cohomology of Harish-Chandra modules , 2013. .[18] H. Hida,
Geometric Modular Forms and Elliptic Curves . World Scientific publishing, 2000.[19] F. Jarvis,
Level lowering for modular mod l representations over totally real fields . Mathe-matische Annalen 12; 313(1) : 141 − The vanishing of Ramanujans function τ ( n ). Duke Math. J. 14: 429433, 1947.[21] C. P. Mok, Galois representations attached to automorphic forms on GL2 over CM fields .Compositio Math. 150, 523-567, 2014.[22] C. Poor, D. Yuen,
Paramodular cusp forms . Journal Math. Comp. 84, 1401 − On Modular Representations of Gal(Q/Q) arising from modular forms . Invent.Math. 100 , Local Newforms for GSp . Springer, Lecture notes in mathematics1918, 2007.[25] B. Roberts, R. Schmidt, On modular forms for the paramodular group . Automorphic Formsand Zeta Functions. Proceedings of the Conference in Memory of Tsuneo Arakawa. WorldScientific, 2006.[26] J. P. Serre, J. Tate,
Good reduction of Abelian varieties . Annals of Mathematics, Vol 88, 3,492 − Galois representations and Hilbert-Siegel modular forms . Doc. Math. 15,623 − Galois Representations . .[29] R. Taylor, T. Yoshida, Compatibility of local and global Langlands correspondences . J. Amer.Math. Soc., 20 − , − Four dimensional Galois representations . 2000.[31] G. Wiese