Genus 2 paramodular Eisenstein congruences

We investigate certain Eisenstein congruences, as predicted by Harder, for level p paramodular forms of genus 2. We use algebraic modular forms to generate new evidence for the conjecture. In doing this we see explicit computational algorithms that generate Hecke eigenvalues for such forms.


Introduction
Congruences between modular forms have been found and studied for many years. Perhaps the first interesting example is found in the work of Ramanujan. He studied in great detail the Fourier coefficients τ (n) of the discriminant function ∆(z) = q ∞ n=1 (1 − q n ) 24 (where q = e 2πiz ). The significance of ∆ is that it is the unique normalized cusp form of weight 12.
Here σ 11 (n) = d|n d 11 is a power divisor sum. Naturally one wishes to explain the appearance of the modulus 691. The true incarnation of this is via the fact that the prime 691 divides the numerator of the "rational part" of ζ (12), i.e ζ (12) π 12 ∈ Q (a quantity that appears in the Fourier coefficients of the Eisenstein series E 12 ).
Since the work of Ramanujan there have been many generalizations of his congruences. Indeed by looking for big enough primes dividing numerators of normalized zeta values one can provide similar congruences at level 1 between cusp forms and Eisenstein series for other weights. In fact one can even give "local origin" congruences between level p cusp forms and level 1 Eisenstein series by extending the divisibility criterion to include single Euler factors of ζ(s) rather than the global values of ζ(s) (see [7] for results and examples).
There are also Eisenstein congruences predicted for Hecke eigenvalues of genus 2 Siegel cusp forms. One particular type was conjectured to exist by Harder [14]. There is only a small amount of evidence for this conjecture, the literature only contains examples at levels 1 and 2 (using methods specific to these levels). The conjecture is also far from being proved. Only one specific level 1 example of the congruence has been proved (p.386 of [5]).
In this paper we will see new evidence for a level p version of Harder's conjecture for various small primes (including p = 2 but not exclusively). The Siegel forms will be of paramodular type and the elliptic forms will be of Γ 0 (p) type. In doing this we will make use of Jacquet-Langlands style conjectures due to Ibukiyama. Explicitly it is expected that there is a Hecke equivariant isomorphism between the spaces S new j,k (K(p)) and certain spaces of algebraic modular forms. Bearing this in mind we give the reader a brief overview of the general theory of such forms.
3.1. The spaces A(G, K f , V ) of algebraic forms. Let G/Q be a connected reductive group with the added condition that the Lie group G(R) is connected and compact modulo center. Fix an open compact subgroup K f ⊂ G(A f ). Also let V be (the space of) a finite dimensional algebraic representation of G, defined over a number field F . Definition 3.1. The F -vector space of algebraic modular forms of level K f , weight V for G is: Fix a set of representatives T = {z 1 , z 2 , ..., z h } ∈ G(A f ) for G(Q)\G(A f )/K f . There is a natrual embedding: .., f (z h )).

Theorem 3.2. The map φ induces an isomorphism:
where Γ m = G(Q) ∩ z m K f z −1 m for each m. A pleasing feature of the theory is that the groups Γ m are often finite. Gross gives many equivalent conditions for when this happens [13]. One such condition is the following. 3.2. Hecke Operators. Let u ∈ G(A f ) and fix a decomposition K f uK f = r i=1 u i K f . It is well known that finitely many representatives occur. Then T u acts on f ∈ A(G, It is easy to see that this is independent of the choice of representatives u i since they are determined up to right multiplication by K f . We wish to find the Hecke representatives u i explicitly and efficiently. To this end a useful observation can be made when the class number is one. Finally we note that for G satisfying Proposition 3.4 there is a natural inner product on the space A(G, K, V ). This is given in Gross' paper [13] but we shall give the rough details here. Lemma 3.6. Let G satisfy the property of Proposition 3.4 and V be a finite dimensional algebraic representation of G, defined over Q. Then there exists a character µ : G → G m and a positive definite symmetric bilinear form , Taking adelic points we have a character µ ′ : Proposition 3.7. Let G satisfy the property of Proposition 3.4. Then A(G, K, V ) has a natural inner product given by:

3.3.
Trace of Hecke operators. The underlying representation V of G is typically big in dimension and so the action of Hecke operators is, although explicit, quite tough to compute. Fortunately, there is a simple trace formula for Hecke operators on spaces of algebraic modular forms. The details of the formula can be found in [8] but we give brief details here.
Note that G(A f ) acts on the set Z on the left by setting Let χ V denote the character of the representation of G(Q) on V . Then the trace formula is as follows.
More generally: Letting u = id we recover the following.
When h = 1 the situation becomes much simpler. In this case we may choose z 1 = id and γ 1,i = u i ∈ G(Q) for each i (this is possible by Corollary 3.5).
The trace formula was introduced to test a U(2, 2) analogue of Harder's conjecture. In this paper we will use it to test the level p paramodular version of Harder's conjecture given by Conjecture 2.1.

Eichler and Ibukiyama correspondences
4.1. Eichler's correspondence. From now on D will denote a quaternion algebra over Q ramified at {p, ∞} (for a fixed prime p) and O will be a fixed maximal order. Since D is definite, we have that D × ∞ = D × ⊗ R ∼ = H × is compact modulo center (and is also connected). Thus we may consider algebraic modular forms for the group G = D × .
Also note that in this case each Γ m will be finite since D × (Z) = O × is finite.
Let D q := D ⊗ Q q be the local component at prime q (no restriction on q) and let D A f be the restricted direct product of the D q 's with respect to the local maximal . Thus locally away from the ramified prime, D × behaves like GL 2 .
In fact more is true. It is the case that the reductive groups D × and GL 2 are inner forms of each other. So by the principle of Langlands functoriality we expect a transfer of automorphic forms between D × and GL 2 . Eichler gives an explicit description of this transfer.
Let V n = Symm n (C 2 ) (for n ≥ 0). Then V n gives a well defined representation of SU(2)/{±I} if and only if n is even. Thus we get a well defined action on V n by D × via: This is an open compact subgroup of D × A f . For k = 2 the above holds if on the right we quotient out by the space of constant functions.
It remains to describe how the Hecke operators transfer over the isomorphism. Fix a prime q = p. Choose u ∈ D × A f such that ψ(u q ) = diag(1, p) and is the identity at all other places. The corresponding Hecke operator T u,q corresponds to the classical T q operator under Eichler's correspondence.

4.2.
Ibukiyama's correspondence. Ibukiyama's correspondence is a (conjectural) generalisation of Eichler's correspondence to Siegel modular forms. The details can be found in [18] but we explain the main ideas.
Given the setup in the previous subsection, consider the unitary similitude group: Hereḡ means componentwise application of the standard involution of D. This group is the similitude group of the standard Hermitian form on D n . Theorem 4.2. For any field K there exists a similitude-preserving isomorphism GU 2 (M 2 (K)) ∼ = GSp 4 (K).
One consequence of this is that the group GU 2 (D) behaves like GSp 4 locally away from the ramified prime. It is indeed true that these groups are also inner forms of each other.
A simple argument also shows that GU 2 (H)/Z(GU 2 (H)) ∼ = USp(4)/{±I}. Thus GU 2 (D ∞ ) is compact modulo center and connected. Thus we may consider algebaric modular forms for this group. Once again we are guaranteed that the Γ m groups are finite by the following.
Proof. Solving the equations gives: One consequence of Theorem 4.2 is that GU 2 (D q ) ∼ = GSp 4 (Q q )for all q = p.
Proposition 4.4. For any q = p there exists a similitude-preserving isomorphism ψ : GU 2 (D q ) → GSp 4 (Q q ) that preserves integrality: Proof. Choose an isomorphism of quaternion algebras D q ∼ = M 2 (Q q ) that preserves the norm, trace and integrality. This induces an isomorphism with the required properties: Let V j,k−3 be the irreducible representation of USp(4) with Young diagram parameters (j + k − 3, k − 3). This gives a well defined representation of USp(4)/{±I} if and only if j is even. Thus GU 2 (D) acts on this via: The groups GU(D) and GSp 4 are inner forms. Thus (as with Eichler) one expects a transfer of automorphic forms. The following is found in Ibukiyama's paper [18].
If (j, k) = (0, 3) then we also get an isomorphism after taking the quotient by the constant functions on the right.
Since our eventual goal is to study Harder's conjecture for paramodular forms we will neglect the first of these isomorphisms. However, it will turn out that the open compact subgroup U 1 will prove useful in later calculations.
A result of Shimura tells us the possibilities for L q (see [26]).
When D is ramified at {p, ∞} it is clear from this result that there are only two possibilities for L, up to local equivalence. Definition 4.7. Let D be ramified at p, ∞ for some prime p: • If L p is locally equivalent to O 2 p for all q then we say that L lies in the principal genus.
• If L p is locally inequivalent to O 2 p then we say that L lies in the non-principal genus.
Given L, results of Ibukiyama [21] allow us to write L = O 2 g for some g ∈ GL 2 (D) and determine the genus of L based on g.
• L lies in the principal genus if and only if gg T = mx for some positive m ∈ Q and some x ∈ GL n (O) such that x = x T and such that x is positive definite, i.e. yxy T > 0 for all y ∈ D n with y = 0.
• L lies in the non-principal genus if and only if gg T = m ps r r pt where m ∈ Q is positive, s, t ∈ N, r ∈ O lies in the two sided ideal of O above p and is such that p 2 st − N (r) = p (so that the matrix on the right has determinant p).
The lattice O 2 is clearly in the principal genus and corresponds to the choice g = I. Alternatively fix a choice of g such that O 2 g is in the non-principal genus. Let U 1 , U 2 respectively denote the corresponding open compact subgroups of GU 2 (D A f ) (as described above).

Hecke operators.
The transfer of Hecke operators in Ibukiyama's correspondence is similar to the Eichler correspondence but has subtle differences. Fix a prime q = p and let M q = diag(1, 1, q, q) ∈ GSp 4 (Q q ). Fixing an isomorphism as in Proposition 4.4 we may Under Ibukiyama's correspondence it is predicted that T u,q corresponds to the classical T q operator acting on S new j,k (K(p))).

The new subspace. Our final task in defining Ibukiyama's correspondence is to explain what is meant by the new subspace
. We will not go into too much detail but will refer the reader to Ibukiyama's papers [20], [22].
We start with the decomposition: He then associates an explicit theta series θ F to F . This is an elliptic modular form for SL 2 (Z) of weight j + 2k − 2 (if j + 2k − 6 = 0 then it is a cusp form). It is known that θ F is an eigenform for all Hecke operators if and only if θ F = 0. It should be noted that by Eichler's correspondence F 1 can be viewed as an elliptic modular form for Γ 0 (p) of weight j + 2. Further it will be a new cusp form precisely when j > 0. Thus computationally it is not difficult to find the new and old subspaces.

Finding evidence for Harder's conjecture
Now that we have linked spaces of Siegel modular forms S new j,k (K(p)) with spaces of algebraic modular forms , we can begin to generate evidence for Harder's conjecture.

5.1.
Brief plan of the strategy. In this paper we will deal with cases where h = 1 and dim(A new j,k−3 (D)) = 1.

Strategy
(1) Find all primes p such that h = 1.
(4) For each pair (j, k) look in the space of elliptic forms S new j+2k−2 (Γ 0 (p)) for normalized eigenforms f which have a "large prime" dividing Λ alg (f, j+k) ∈ Q f . (5) Find the Hecke representatives for the T u,q operator at a chosen prime q. (6) Use the trace formula to find tr(T u,q ) for T q acting on A j,k−3 (D). The above strategy can be modified to work for the case dim(A new j,k−3 (D)) = d > 1 but one must compute tr(T t u,q ) for 1 ≤ t ≤ d.
Theorem 5.1. The group Γ (2) consists of the following set of matrices: Proof. We know that: A simple calculation shows that any such matrix has similitude 1.
Recall also the open compact subgroup This is the stabilizer of a left O-lattice lying in the principal genus.
In this case the analogue of the group Γ (2) is the group Γ (1) = GU 2 (D) ∩ U 1 . We can employ identical arguments to the above to show the following: We already have an explicit description of Γ (1) (see Lemma 4.3). Computationally it is not straight forward to find the elements of Γ (2) due to the non-integrality of the entries of such matrices.
For θ ∈ Q × consider the sets Then in particular Y 1 = Γ (2) . Later the sets Y q for prime q = p will appear when finding Hecke representatives. Proposition 5.3. For each θ ∈ Q × conjugation by g gives a bijection: To calculate the sets W θ we diagonalize A. Choose a matrix P ∈ GL 2 (D) such that P AP T = B where B ∈ M 2 (D) is a diagonal matrix.
Proposition 5.4. For each θ ∈ Q × conjugation by P gives a bijection If we make an appropriate choice of g and P then we can diagonalize A in such a way as to preserve one integral entry in P νP −1 . Further P −1 λ,µ = P λ,µ .
To prove the second claim we note that r 2 = −p(p − 1) by the Cayley-Hamilton theorem (since tr(r) = 0 and N (r) = p(p − 1)). Then The final claim follows from the fact that P λ,µ P λ,µ = I (which again uses the fact that tr(r) = 0). It is in fact always possible to find some maximal order O of D where such λ, µ exist. For proof of this I refer to an online discussion with John Voight [29], of which the author is grateful. We fix such a choice from now on. Proof. Let ν = α β γ δ with α, β, γ, δ ∈ O. Then a simple calculation shows that P λ,µ νP λ,µ = α + rγ p ( αr p + β) + r p ( γr p + δ) γ Clearly these equations can have no solutions for θ < 0 and so we only consider θ ≥ 0.
A quick calculation shows that N (x) = N (w) and N (z) = p 2 N (y) (a fact we will use soon). The following algorithm allows us to comute W θ for θ ∈ N. Denote by X i the subset of O consisting of norm i elements.
Of course once the elements of W θ have been found it is straight forward to generate the elements of Y θ by inverting the bijection Φ θ in Proposition 5.3.
It should be noted that if we run this algorithm for p = 2 with the following choices then we get exactly the same elements for Y 1 = Γ (2) as Ibukiyama does on p.592 of [18].

5.3.
Finding h. We can use mass formulae to get information on class numbers h 1 and h 2 for U 1 and U 2 .
Define the mass of open compact U ⊂ GU 2 (D A f ) as follows: Ibukiyama provides the following formulae for M (U 1 ) and M (U 2 ) in [18].
Theorem 5.8. If D is ramified at p and ∞ then: This formula is analogous to the Eichler mass formula and is also a special case of the mass formula of Gan, Hanke and Yu [9]. Proof. A quick calculation shows that the only primes to satisfy 5760 (p−1)(p 2 +1) ∈ N are p = 2, 3. Recall |Γ (1) | = 2|O × | 2 . For p = 2, 3 we have |O × | = 24, 12 respectively and one checks that both values satisfy the equation.
Ibukiyama and Hashimoto have produced formulae in [16] and [17] that give the values of h 1 and h 2 for any ramified prime. Their formulae agree with this result.

5.4.
Finding the Hecke representatives. Now that we have found an algorithm to generate the elements of Γ (2) we consider the same question for the Hecke representatives for the T u,q operator on A j,k−3 (D) (where q = p is a fixed prime).
Proposition 5.11. Let D be a quaternion algebra over Q ramified at p, ∞ for some p ∈ {2, 3, 5, 7, 11}. Suppose u ∈ GU 2 (D A f ) is chosen as in Definition 4.9. Then Proof. Consider an arbitrary decomposition: By Proposition 3.5 we may take x i ∈ GU 2 (D) for each i. For the rest of the proof we embed GU 2 (D) ֒→ GU 2 (D A f ) diagonally.
Note that for any prime l = q we have To study the behaviour locally at q we fix a choice of h q ∈ GU 2 (D q ) such that Conjugation by h q gives a bijection between U 2,q u q U 2,q and G(h q u q h −1 q )G, where G = GU 2 (D q ) ∩ GL 2 (O q ). If we fix an isomorphism as in Proposition 4.4 then 1, q, q)).

Since by definition
However: thus u q ∈ GU 2 (D q ) ∩ g −1 q M 2 (O q ) × g q and the same can be said about the x i .
Also since both the conjugation and our chosen isomorphism respect similitude we find that µ(u q ) = µ(M q ) = q and so µ(U 2,q u q U 2,q ) ⊆ qZ × q . In particular µ(x i ) ∈ qZ × q .
Globally we now see that However in our case the similitude is positive definite so that µ(x i ) = q.
Thus the x i can be taken to lie in Y q . It is clear that each such element lies in the double coset.
It remains to see which elements of Y q generate the same left coset. We have (2) . So equivalence of left cosets is upto right multiplication by Γ (2) .
We have a nice formula for the degree of T u,q , found in the work of Ihara [23].
Employing similar arguments to Proposition 5.11 we get the following: Proposition 5.13. Let D be a quaternion algebra over Q ramified at p, ∞ for some p ∈ {2, 3}. Suppose u ∈ GU 2 (D A f ) is chosen as in Definition 4.9. Then Since Γ (1) is given explicitly it is possible to write down explicit representatives in this case.
Corollary 5.14. Let n ∈ N. For each k ∈ N let X k = {α ∈ O | N (α) = k}, t k = |X k /O × | and x 1,k , x 2,k , ..., x t k ,k be a set of representatives for X k /O × . For such a choice of k define: The following matrices are representatives for The finite subset R ′ m ⊂ R m is to be constructed in the proof.
We wish to study equivalence of these matrices under right multiplication by Γ (1) .
Letting k = N (α) choose x, y ∈ O × so that αx = x i,k and δy = x j,k for some . Clearly the matrices of this form are inequivalent.
It is now clear that R k gives representatives for the particular subcase N (α) = k > n 2 .
The matrices x j,m may now have extra anti-diagonal equivalences. Suppose Then x, y are uniquely determined: Thus each such matrix with v, w ∈ X m can only be equivalent to at most one other matrix: where v ∼ x s,m and w ∼ x t,m under the action of right unit multiplication.
Let R ′ m be a set consisting of a choice of matrix from each of these equivalence pairs (as x i,m and x j,m run through representatives for X m /O × and v, w run through elements of X m satisfying x i,m w + vx j,m = 0). Then it is now clear that R ′ m is a set of representatives for this subcase.
In the subcase k = n − 1 it is often easier to use anti-diagonal equivalence (since X 1 /O × = {1}). In this case we can identify: When n = 2 exactly half of these will form a set of representatives. In fact it is simple to see that the equivalent pairs would be: Thus: Example 5.15. If we apply Corollary 5.14 to the choices: we find that Hecke representatives for U 1 with ramified prime p = 2 and q = 3 are given by: There are 40 representatives here as expected and they agree with the explicit representatives given by Ibukiyama on p.594 of [18].
So far we have not needed the open compact subgroup U 1 but it is actually of use to us in studying U 2 .
Lemma 5.16. Let u ∈ GU 2 (D A f ) be chosen to form the T u,q operator with respect to both U 1 and U 2 (for prime q = p). Then the Hecke representatives for T u,q with respect to U 1 and U 2 can be taken to be the same.
Proof. Recall that u has identity component away from q and u q / ∈ U 2,q so the there is only one local condition to check, that U 2,q = U 1,q .
We know that U 2,q = Stab GU2(Dq) (O 2 q g q ). However by construction we know that O 2 q g q is equivalent to O 2 q (since q = p). Thus there exists h q ∈ GU 2 (D q ) such It is then clear that: . Thus U 2,q = U 1,q and so we are done.
This result is useful since we have seen that it is generally easier to generate Hecke representatives for T u,q with respect to U 1 .
Corollary 5.17. Let the ramified prime of D be p ∈ {2, 3}. Then we may use the representatives from Corollary 5.14 as Hecke representatives for T u,q with respect to U 2 (for q = p).
Proof. Since p ∈ {2, 3} we know that both the class numbers of U 1 , U 2 are 1. Hence both admit rational Hecke representatives.
We also know that given Hecke representatives for T u,q with respect to U 1 we may use them for U 2 . Thus the rational representatives from Corollary 5.14 can be used for U 2 .
5.5. Implementing the trace formula. Now that we have algorithms that generate the data needed to use the trace formula we discuss some of the finer details in its implementation, namely how to find character values. Denote by χ j,k−3 the character of the representation V j,k−3 .
Given g = α β γ δ ∈ GU 2 (D) we may produce a matrix A ∈ GSp 4 (C) via the embedding: where a = i 2 and b = j 2 in D.
We know that the image of GU 2 (H) 1 ∩ GU 2 (D) under this embedding is a subgroup of USp (4) In order to find χ j,k−3 (B) we first find the eigenvalues of B. This is equivalent to conjugating into the maximal torus of diagonal matrices. Since B ∈ USp(4) these eigenvalues will come in two complex conjugate pairs z, z, w, w for z, w on the unit circle.
The Weyl character formula gives: For any of the cases z 2 = 1, w 2 = 1, zw = 1, z = w one must formally expand this concise formula into a polynomial expression (not an infinite sum since each factor on the denominator except zw divides the numerator). It is easy for a computer package to compute this expansion for a given j, k.

5.6.
Finding the trace contribution for the new subspace. Let tr(T u,q ) new and tr(T u,q ) old be the traces of the action of T u,q on A new j,k−3 (D) and A old j,k−3 (D) respectively. Then tr(T u,q ) new = tr(T u,q ) − tr(T u,q ) old .
Theorem 5.18. For q = p we have the following identity in C(t): .
Ibukiyama conjectures that there is a bijection between pairs of eigenforms (F 1 , θ F ) and eigenforms F 2 . With this in mind it is now possible to calculate the oldform trace contribution.
Then for q = p and j > 0: a q,hi .

Examples and Summary
The following table highlights the choices for D, O, λ, µ that were used.
Using these choices along with the algorithms and results mentioned previously one can calculate the groups Γ (1) , Γ (2) for each such p, hence generating tables of dimensions of the spaces A new j,k−3 (D). These tables are given in Appendix A.1.
From these tables one isolates 1-dimensional spaces. For each possibility the MAGMA command LRatio allows us to test for large primes dividing Λ alg on the elliptic side. The cases that remained were ones where we expect to find examples of Harder's congruence.
Tables of the congruences observed can be found in Appendix A.2. In particular for p = 2 one observes congruences provided in Bergström [1]. We finish with some new examples for p = 3. One easily checks that dim(S new 16 (Γ 0 (3))) = 2. This space is spanned by the two normalized eigenforms with q-expansions: Indeed MAGMA informs us that ord 109 (Λ alg (f 1 , 10)) = 1 and so we expect a congruence of the form: b q ≡ a q + q 9 + q 6 mod 109 for all q = 3, where b q are the Hecke eigenvalues of F and a q the Hecke eigenvalues of f 1 . As discussed earlier we will only work with the case q = 2 for simplicity.
MAGMA informs us that ord 67 (Λ alg (f 2 , 13)) = 1 and so we expect a congruence of the form: b q ≡ a q + q 12 + q 3 mod 67 for all q = 3.
Applying the trace formula this time gives tr(T u,2 ) = 300. However since dim(A 8,2 (D)) > dim(A new 8,2 (D)) there is an oldform contribution to this trace. In order to find it we need Hecke eigenvalues of normalized eigenforms for the spaces S 16 (SL 2 (Z)) and S new 10 (Γ 0 (3)).
The congruence is then simple to check: 12 ≡ −72 + 2 12 + 2 3 mod 67. Example 6.3. Our final example is a case where the Hecke eigenvalues of the elliptic modular form lie in a quadratic extension of Q.
One easily checks that dim(S new 14 (Γ 0 (3))) = 3. This space is spanned by the three normalized newforms with q-expansions: This is divisible by 47 and so the congruence holds for q = 2.