Abstract
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 \(\tau (n)\) of the discriminant function \(\Delta (z) = q\prod _{n=1}^{\infty }(1q^n)^{24}\) (where \(q = e^{2\pi i z}\)). The significance of \(\Delta \) is that it is the unique normalized cusp form of weight 12.
Amongst Ramanujan’s mysterious observations was a pretty congruence:
Here \(\sigma _{11}(n) = \sum _{d\mid n}d^{11}\) 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 \(\zeta (12)\), i.e., \(\frac{\zeta (12)}{\pi ^{12}}\in \mathbb {Q}\) (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 \(\zeta (s)\) rather than the global values of \(\zeta (s)\) (see [8] 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 [15]. There is only a small amount of evidence for this conjecture, and 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 [5, p. 386].
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 \(\Gamma _0(p)\) type. In doing this, we will make use of Jacquet–Langlands style conjectures due to Ibukiyama.
Harder’s conjecture
Given \(k\ge 0\) and \(N\ge 1\), let \(S_{k}(\Gamma _0(N))\) denote the space of elliptic cusp forms for \(\Gamma _0(N)\). Also for \(j\ge 0\) let \(S_{j,k}(K(N))\) denote the space of genus 2, vectorvalued Siegel cusp forms for the paramodular group of level N, taking values in the representation space \(\text {Symm}^j(V) \otimes \text {det}^k(V)\) of \(\text {GL}_2(\mathbb {C})\) (where V is the standard representation).
Given \(f\in S_k(\Gamma _0(N))\), we let \(\Lambda _{\text {alg}}(f,j+k) = \frac{\Lambda (f,j+k)}{\Omega }\), where \(\Lambda (f,s)\) is the completed Lfunction attached to f and \(\Omega \) is a Deligne period attached to f. The choice of \(\Omega \) is unique up to scaling by \(\mathbb {Q}_f^{\times }\) but Harder shows how to construct a more canonical choice of \(\Omega \) that is determined up to scaling by \(\mathcal {O}_{\mathbb {Q}_f}^{\times }\) [16].
In this paper, we consider the following paramodular version of Harder’s conjecture (when \(N=1\) this is the original conjecture found in [15]).
Conjecture 2.1
Let \(j>0\), \(k\ge 3\) and let \(f\in S_{j+2k2}^{\text {new}}(\Gamma _0(N))\) be a normalized Hecke eigenform with eigenvalues \(a_n\). Suppose that \(\text {ord}_\lambda (\Lambda _{alg}(f,j+k)) > 0\) for some prime \(\lambda \) of \(\mathbb {Q}_f\) lying above a rational prime \(l > j+2k2\) (with \(l\not \mid N\)).
Then there exists a Hecke eigenform \(F\in S_{j,k}^{\text {new}}(K(N))\) with eigenvalues \(b_n\in \mathbb {Q}_F\) such that
for all primes \(q\not \mid N\) (where \(\Lambda \) is some prime lying above \(\lambda \) in the compositum \(\mathbb {Q}_f\mathbb {Q}_F\)).
It should be noted that Harder’s conjecture has still not been proved for level 1 forms. However, the specific example with \(j=4, k=10\), and \(l=41\) mentioned in Harder’s paper has recently been proved in a paper by Chenevier and Lannes [5]. The proof uses the Niemeier classification of 24dimensional lattices and is specific to this particular case.
Following the release of the level 1 conjecture, Faber and Van der Geer were able to do computations when \(\text {dim}(S_{j,k}(\text {Sp}_4(\mathbb {Z})))=1\). They have now exhausted such spaces and in each case have verified the congruence for a significant number of Hecke eigenvalues. Ghitza, Ryan, and Sulon give extra evidence for the case \(j=2\) [12]. More recently, Cléry, Faber, and Van der Geer gave more examples for the cases \(j=4,6,8,10,12\), and 14 [6].
For the level p conjecture, a substantial amount of evidence has been provided by Bergström et al for level 2 forms [1]. Their methods are specific to this level. A small amount of evidence is known beyond level 2. In particular, a congruence has been found with \((j,k,p,l) = (0,3,61,43)\) by Mellit [16, p. 99].
In this paper, we use the theory of algebraic modular forms to provide evidence for the conjecture at levels \(p=2,3,5,7\). The methods discussed can be extended to work for other levels.
Algebraic modular forms
In general, it is quite tough to compute Hecke eigensystems for paramodular forms. Fortunately, for a restricted set of levels, there is a (conjectural) Jacquet–Langlands style correspondence for \(\text {GSp}_4\) due to Ihara and Ibukiyama [19].
Explicitly, it is expected that there is a Hecke equivariant isomorphism between the spaces \(S_{j,k}^{\text {new}}(K(p))\) 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. For more details, see the introductory article of Loeffler in [26].
The spaces \(\mathcal {A}(G,K_f,V)\) of algebraic forms
Let \(G/\mathbb {Q}\) be a connected reductive group with the added condition that the Lie group \(G(\mathbb {R})\) is connected and compact modulo center. Fix an open compact subgroup \(K_f \subset G(\mathbb {A}_f)\). Also let V be (the space of) a finite dimensional algebraic representation of G, defined over a number field F.
Definition 3.1
The Fvector space of algebraic modular forms of level \(K_f\) and weight V for G is
Fix a set of representatives \(T = \{z_1,z_2,\ldots ,z_h\}\in G(\mathbb {A}_f)\) for \(G(\mathbb {Q})\backslash G(\mathbb {A}_f)/ K_f\). There is a natural embedding:
Theorem 3.2
The map \(\phi \) induces an isomorphism:
where \(\Gamma _m = G(\mathbb {Q})\cap z_m K_f z_m^{1}\) for each m.
Corollary 3.3
The spaces \(\mathcal {A}(G,K_f, V)\) are finite dimensional.
A pleasing feature of the theory is that the groups \(\Gamma _m\) are often finite. Gross gives many equivalent conditions for when this happens [14]. One such condition is the following.
Proposition 3.4
The groups \(\Gamma _m\) are finite if and only if \(G(\mathbb {Z})\) is finite.
Hecke operators
Let \(u\in G(\mathbb {A}_f)\) and fix a decomposition \(K_f u K_f = \coprod _{i=1}^r u_i K_f\). It is well known that finitely many representatives occur. Then \(T_u\) acts on \(f\in \mathcal {A}(G,K_f,V)\) via
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.
Proposition 3.5
If \(h=1\), then we may choose Hecke representatives that lie in \(G(\mathbb {Q})\).
Finally, we note that for G satisfying Proposition 3.4 there is a natural inner product on the space \(\mathcal {A}(G,K,V)\). This is given in Gross’ paper [14] 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 \(\mathbb {Q}\). Then there exists a character \(\mu : G \rightarrow \mathbb {G}_m\) and a positive definite symmetric bilinear form \(\langle ,\rangle : V\times V\rightarrow \mathbb {Q}\) such that
for all \(\gamma \in G(\mathbb {Q})\).
Taking adelic points, we have a character \(\mu ':G(\mathbb {A}) \rightarrow \mathbb {A}^{\times }\). Let \(\mu _{\mathbb {A}} = f\circ \mu '\), where \(f: \mathbb {A}^{\times } \rightarrow \mathbb {Q}^{\times }\) is the natural projection map coming from the decomposition \(\mathbb {A}^{\times } = \mathbb {Q}^{\times }\mathbb {R}^+\hat{\mathbb {Z}}^{\times }\).
Proposition 3.7
Let G satisfy the property of Proposition 3.4. Then \(\mathcal {A}(G,K,V)\) has a natural inner product given by
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 [9] but we give brief details here.
Note that \(G(\mathbb {A}_f)\) acts on the set Z on the left by setting \(w\cdot z_i = z_j\) if and only if \(G(\mathbb {Q})(wz_i)K_f = G(\mathbb {Q})z_jK_f\). For each \(m=1,2,\ldots ,h\), we consider the set \(S_m = \{i\,\, u_i\cdot z_m = z_m\}\). Next for each \(i\in S_m\) choose elements \(k_{m,i}\in K_f\) and \(\gamma _{m,i}\in G(\mathbb {Q})\) such that \(\gamma _{m,i}^{1} u_i z_m k_{m,i} = z_m\).
Let \(\chi _V\) denote the character of the representation of \(G(\mathbb {Q})\) on V. Then the trace formula is as follows.
Theorem 3.8
(Dummigan)
More generally,
Letting \(u=\text {id}\), we recover the following:
Corollary 3.9
When \(h=1\), the situation becomes much simpler. In this case, we may choose \(z_1 = \text {id}\) and \(\gamma _{1,i} = u_i\in G(\mathbb {Q})\) for each i (this is possible by Corollary 3.5).
Corollary 3.10
If \(h=1\), then we have
where \(\Gamma = G(\mathbb {Q}) \cap K_f\).
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
Eichler’s correspondence
From now on, D will denote a quaternion algebra over \(\mathbb {Q}\) ramified at \(\{p,\infty \}\) (for a fixed prime p) and \(\mathcal {O}\) will be a fixed maximal order. Since D is definite, we have that \(D_{\infty }^{\times } = D^{\times }\otimes \mathbb {R} \cong \mathbb {H}^{\times }\) is compact modulo center (and is also connected). Thus, we may consider algebraic modular forms for the group \(G = D^{\times }\).
Also note that in this case each \(\Gamma _m\) will be finite since \(D^{\times }(\mathbb {Z}) = \mathcal {O}^{\times }\) is finite.
Let \(D_q:= D \otimes \mathbb {Q}_q\) be the local component at prime q (no restriction on q) and let \(D_{\mathbb {A}_f}\) be the restricted direct product of \(D_q\)’s with respect to the local maximal orders \(\mathcal {O}_q:= \mathcal {O} \otimes \mathbb {Z}_q\).
Note that if \(q\ne p\) then \(D_q^{\times } \cong (\text {M}_2(\mathbb {Q}_q))^{\times } = \text {GL}_2(\mathbb {Q}_q)\). Thus, locally away from the ramified prime, \(D^{\times }\) behaves like \(\text {GL}_2\).
In fact more is true. It is the case that the reductive groups \(D^{\times }\) and GL\(_2\) are inner forms of each other. So by the principle of Langlands functoriality we expect a transfer of automorphic forms between \(D^{\times }\) and GL\(_2\). Eichler gives an explicit description of this transfer.
Let \(V_n = \text {Symm}^n(\mathbb {C}^2)\) (for \(n\ge 0\)). Then \(V_n\) gives a welldefined representation of \(\text {SU}(2)/\{\pm I\}\) if and only if n is even. Thus, we get a welldefined action on \(V_n\) by \(D^{\times }\) through
Take \(U = \prod _{q}\mathcal {O}_q^{\times }\). This is an open compact subgroup of \(D_{\mathbb {A}_f}^{\times }\).
Theorem 4.1
(Eichler) Let \(k>2\). Then there is a Hecke equivariant isomorphism:
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 \(q\ne p\) and an isomorphism \(\psi : D_q^{\times } \cong \text {GL}_2(\mathbb {Q}_q)\). Choose \(u \in D_{\mathbb {A}_f}^{\times }\) such that \(\psi (u_q) = \text {diag}(1,q)\) and is the identity at all other places. The corresponding Hecke operator corresponds to the classical \(T_q\) operator under Eichler’s correspondence.
Ibukiyama’s correspondence
Ibukiyama’s correspondence is a (conjectural) generalization of Eichler’s correspondence to Siegel modular forms. The details can be found in [19] but we explain the main ideas.
Given the setup in the previous subsection, consider the unitary similitude group:
Here \(\bar{g}\) means componentwise application of the standard involution of D. This group is the similitude group of the standard Hermitian form on \(D^n\).
Lemma 4.2
For any field K, there exists a similitudepreserving isomorphism \(\mathrm {GU}_2(\mathrm {M}_2(K)) \cong \mathrm {GSp}_4(K)\).
Proof
Conjugation by the matrix \(M = \text {diag}(1,A,1)\in \text {M}_4(K)\), where \(A = \left( \begin{array}{cc} 0 &{} 1\\ 1 &{} 0\end{array}\right) \) gives such an isomorphism. \(\square \)
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 \(\text {GU}_2(\mathbb {H})/Z(\text {GU}_2(\mathbb {H})) \cong \text {USp}(4)/\{\pm I\}\). Thus GU\(_2(D_{\infty })\) is compact modulo center and connected. Thus, we may consider algebraic modular forms for this group. Once again, we are guaranteed that the \(\Gamma _m\) groups are finite by the following.
Lemma 4.3
\(\mathrm {GU}_2(\mathcal {O})= \{\gamma \in \mathrm {GU}_2(D)\cap \mathrm {M}_2(\mathcal {O})\,\,\mu (\gamma )\in \mathbb {Z}^{\times }\}\) is finite.
Proof
Solving the equations gives
\(\square \)
One consequence of Lemma 4.2 is that GU\(_2(D_q) \cong \text {GSp}_4(\mathbb {Q}_q)\)for all \(q\ne p\).
Lemma 4.4
For any \(q\ne p\), there exists a similitudepreserving isomorphism \(\psi : \mathrm {GU}_{2}(D_q) \rightarrow \mathrm {GSp}_4(\mathbb {Q}_q)\) that preserves integrality:
Proof
Choose an isomorphism of quaternion algebras \(D_q \cong \mathrm {M}_2(\mathbb {Q}_q)\) that preserves the norm, trace, and integrality. This induces an isomorphism with the required properties since
\(\square \)
Let \(V_{j,k3}\) be the irreducible representation of USp(4) with Young diagram parameters \((j+k3,k3)\). This gives a welldefined representation of USp\((4)/\{\pm I\}\) if and only if j is even. Thus, GU\(_2(D)\) acts on this through
The groups \(\text {GU}(D)\) and \(\text {GSp}_4\) are inner forms. Thus (as with Eichler), one expects a transfer of automorphic forms. The following is found in Ibukiyama’s paper [19].
Conjecture 4.5
(Ibukiyama) Let \(j\ge 0\) be an even integer and \(k\ge 3\). Suppose \((j,k) \ne (0,3)\). Then there is a Hecke equivariant isomorphism:
where \(U_1, U_2, V_{j,k3}\) are to be defined.
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.
The levels \(U_1\) and \(U_2\).
In Eichler’s correspondence, the “level 1” open compact subgroup \(U = \prod _q \mathcal {O}_q^{\times } \subset D_{\mathbb {A}_f}^{\times }\) can be viewed as Stab\(_{D_{\mathbb {A}_f}^{\times }}(\mathcal {O})\) under an action defined by right multiplication. Similarly, one can produce open compact subgroups Stab\(_{\text {GU}_2(D_{\mathbb {A}_f})}(L)\subseteq \text {GU}_2(D_{\mathbb {A}_f})\), where L is a left \(\mathcal {O}\)lattice of rank 2 in \(D^2\) (a free left \(\mathcal {O}\)module of rank 2).
A left \(\mathcal {O}\)lattice \(L\subseteq D^2\) gives rise to a left \(\mathcal {O}_q\)lattice \(L_q = L\otimes \mathbb {Z}_q \subseteq D_q^2\) for each prime q. A result of Shimura tells us the possibilities for \(L_q\) (see [28]).
Theorem 4.6
Let D be a quaternion algebra over \(\mathbb {Q}\). If D is split at q, then \(L_q\) is right \(\mathrm {GU}_2(D_q)\) equivalent to \(\mathcal {O}_q^2\). If D is ramified at q, then there are exactly two possibilities for \(L_q\), up to right \(\mathrm {GU}_2(D_q)\) equivalence (one being \(\mathcal {O}_q^2\)).
When D is ramified at \(\{p,\infty \}\), 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 \(p,\infty \) for some prime p:

If \(L_p\) is locally right equivalent to \(\mathcal {O}_p^2\) for all q, then we say that L lies in the principal genus.

If \(L_p\) is locally right inequivalent to \(\mathcal {O}_p^2\), then we say that L lies in the nonprincipal genus.
\(\square \)
Given L, results of Ibukiyama [23] allow us to write \(L = \mathcal {O}^2 g\) for some \(g\in \mathrm {GL}_2(D)\) and determine the genus of L based on g.
Theorem 4.8

(1)
L lies in the principal genus if and only if \(g\overline{g}^T = mx\) for some positive \(m\in \mathbb {Q}\) and some \(x\in \mathrm {GL}_n(\mathcal {O})\) such that \(x = \overline{x}^T\) and such that x is positive definite, i.e., \(yx\overline{y}^T > 0\) for all \(y\in D^n\) with \(y\ne 0\).

(2)
L lies in the nonprincipal genus if and only if \(g\overline{g}^T = m\left( \begin{array}{cc} ps &{} r\\ \overline{r} &{} pt\end{array}\right) \), where \(m\in \mathbb {Q}\) is positive, \(s,t\in \mathbb {N}, r\in \mathcal {O}\) lies in the twosided ideal of \(\mathcal {O}\) above p and is such that \(p^2st  N(r) = p\) (so that the matrix on the right has determinant p).
The lattice \(\mathcal {O}^2\) is clearly in the principal genus and corresponds to the choice \(g=I\). Alternatively, fix a choice of g such that \(\mathcal {O}^2 g\) is in the nonprincipal genus. Let \(U_1,U_2\) denote, respectively, the corresponding open compact subgroups of \(\text {GU}_2(D_{\mathbb {A}_f})\) (as described above).
Hecke operators
The transfer of Hecke operators in Ibukiyama’s correspondence is similar to the Eichler correspondence but there are subtle differences. Fix a prime \(q\ne p\) and an isomorphism \(\psi \) as in Lemma 4.4. Then we may find \(v_q\in \text {GU}_2(D_q)\) such that \(\psi (v_q) = \text {diag}(1,1,q,q)\).
Recall that D is split at q. By Theorem 4.6, there exists \(h_q\in \text {GU}_2(D_q)\) such that \(\mathcal {O}^2g_q = \mathcal {O}^2h_q\) (where \(g_q\) is the image of g under the standard embedding \(\text {GU}_2(D)\hookrightarrow \text {GU}_2(D_q)\)). Fix such a choice.
Let \(u\in \text {GU}_2(D_{\mathbb {A}_f})\) have \(u_q = h_q v_q h_q^{1}\) as the component at q and have identity component elsewhere.
Definition 4.9
Given u as above, the corresponding Hecke operator on \(\mathcal {A}^{\text {new}}(\text {GU}_2(D),U_2, V_{j,k3})\) will be called \(T_{u,q}\). \(\square \)
Under Ibukiyama’s correspondence, it is predicted that \(T_{u,q}\) corresponds to the classical \(T_q\) operator acting on \(S_{j,k}^{\text {new}}(K(p))\)).
The new subspace
Our final task in defining Ibukiyama’s correspondence is to explain what is meant by the new subspace \(\mathcal {A}^{\text {new}}(\text {GU}_2(D),U_2,V_{j,k3})\). We will not go into too much detail but will refer the reader to Ibukiyama’s papers [21, 22].
Let \(G = D^{\times }\times \text {GU}_2(D)\). Then we have an open compact subgroup \(U' = U\times U_2\) and finite dimensional representations \(W_{j,k3}:= V_{j}\otimes V_{j,k3}\) of \(G(\mathbb {A}_f)\).
We start with the decomposition:
Ibukiyama takes \(F\in \mathcal {A}(G,U',W_{j,k3})\). If F is an eigenform, then \(F = F_1\otimes F_2\) for eigenforms \(F_1,F_2\). He then associates an explicit theta series \(\theta _F\) to F. This is an elliptic modular form for \(\mathrm {SL}_2(\mathbb {Z})\) of weight \(j+2k2\) (if \(j+2k6\ne 0\), then it is a cusp form). It is known that \(\theta _F\) is an eigenform for all Hecke operators if and only if \(\theta _F \ne 0\).
Definition 4.10
The subspace of old forms \(A_{j,k3}^{\text {old}}(D) \subseteq A_{j,k3}(D)\) is generated by the eigenforms \(F_2\) such that there exists an eigenform \(F_1\) satisfying \(\theta _{F_1 \otimes F_2} \ne 0\).
The subspace of new forms \(A_{j,k3}^{\text {new}}(D)\) is the orthogonal complement of the old space with respect to the inner product in Proposition 3.7. \(\square \)
It should be noted that by Eichler’s correspondence \(F_1\) can be viewed as an elliptic modular form for \(\Gamma _0(p)\) 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 \(S_{j,k}^{\text {new}}(K(p))\) with spaces of algebraic modular forms \(A_{j,k3}^{\text {new}}(D) = \mathcal {A}^{\text {new}}(\text {GU}_2(D),U_2, V_{j,k3})\), we can begin to generate evidence for Harder’s conjecture.
Brief plan of the strategy
In this paper, we deal with cases where \(h=1\) and dim\((A_{j,k3}^{\text {new}}(D)) = 1\).
Strategy

(1)
Find all primes p such that \(h=1\).

(2)
For each such p calculate \(\Gamma ^{(2)} = \text {GU}_2(D)\cap U_2\).

(3)
Using Corollary 3.9 find all j, k such that \(\text {dim}(A_{j,k}^{\text {new}}(D))=1\).

(4)
For each pair (j, k) look in the space of elliptic forms \(S_{j+2k2}^{\text {new}}(\Gamma _0(p))\) for normalized eigenforms f which have a “large prime” dividing \(\Lambda _{\text {alg}}(f,j+k)\in \mathbb {Q}_f\).

(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 \(A_{j,k3}(D)\).

(7)
Subtract off the trace contribution of \(T_{u,q}\) acting on \(A_{j,k3}^{\text {old}}(D)\) in order to get the trace of the action on \(A_{j,k3}^{\text {new}}(D)\). Since dim\((A_{j,k3}^{\text {new}}(D))=1\), this trace should be exactly the Hecke eigenvalue of a new paramodular eigenform by Ibukiyama’s conjecture.

(8)
Check that Harder’s congruence holds.
The above strategy can be modified to work for the case \(\text {dim}(A_{j,k3}^{\text {new}}(D)) = d > 1\) but one must compute \(\text {tr}(T_{u,q}^t)\) for \(1\le t \le d\).
Finding \(\Gamma ^{(2)}\)
For \(\theta \in \mathbb {Q}^{\times }\), consider the subset:
In particular let SU\(_2(D):= \text {GU}_2(D)_1\).
Lemma 5.1
The group \(\Gamma ^{(2)}\) consists of the following set of matrices:
where \(g\in \mathrm {GL}_2(D)\) satisfies the condition in Theorem 4.8.
Proof
We know that
A simple calculation shows that any such matrix has similitude 1. \(\square \)
Recall also the open compact subgroup \(U_1 = \text {Stab}_{\text {GU}_2(\mathbb {A}_f)}(\mathcal {O}^2) \subset \text {GU}_2(D_{\mathbb {A}_f})\). This is the stabilizer of a left \(\mathcal {O}\)lattice lying in the principal genus.
In this case, the analogue of the group \(\Gamma ^{(2)}\) is the group \(\Gamma ^{(1)} = \text {GU}_2(D) \cap U_1\). We can employ identical arguments to the above to show the following:
Lemma 5.2
We already have an explicit description of \(\Gamma ^{(1)}\) (see Lemma 4.3). Computationally, it is not straightforward to find the elements of \(\Gamma ^{(2)}\) due to the nonintegrality of the entries of such matrices.
For \(\theta \in \mathbb {Q}^{\times }\), consider the sets
and
where \(\mathrm {M}_2(\mathcal {O})^{\times } = \mathrm {GL}_2(D)\cap \mathrm {M}_2(\mathcal {O})\) and \(A = g\bar{g}^T\).
Then in particular \(Y_1 = \Gamma ^{(2)}\). Later, the sets \(Y_q\) for prime \(q\ne p\) will appear when finding Hecke representatives.
Proposition 5.3
For each \(\theta \in \mathbb {Q}^{\times }\), conjugation by g gives a bijection
To calculate the sets \(W_{\theta }\) we diagonalize A. Choose a matrix \(P\in \text {GL}_2(D)\) such that \(PA\overline{P}^{T} = B\), where \(B\in \mathrm {M}_2(D)\) is a diagonal matrix.
Proposition 5.4
For each \(\theta \in \mathbb {Q}^{\times }\), 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 \(P\nu P^{1}\).
Lemma 5.5
Suppose we can choose \(\lambda ,\mu \in \mathcal {O}\) such that \(N(\lambda )= p1\), \(N(\mu ) = p\) and \(\text {tr}(r) = 0\) (where \(r = \lambda \overline{\mu }\)). Then
are valid choices for g and P.
Further \(P_{\lambda ,\mu }^{1} = \overline{P_{\lambda ,\mu }}\).
Proof
A simple calculation shows that
and also that det\((A_{\lambda ,\mu }) = p^2  N(r) = p^2  p(p1) = p\) as required.
To prove the second claim, we note that \(r^2 = p(p1)\) by the Cayley–Hamilton theorem (since \(\text {tr}(r) = 0\) and \(N(r) = p(p1)\)). Then
and so \(P_{\lambda ,\mu }A\overline{P_{\lambda ,\mu }}^T =\text {diag}(1,p)\).
The final claim follows from the fact that \(P_{\lambda ,\mu }\overline{P_{\lambda ,\mu }} = I\) (which again uses the fact that \(\text {tr}(r) = 0\)). \(\square \)
It is in fact always possible to find some maximal order \(\mathcal {O}\) of D where such \(\lambda ,\mu \) exist. For proof of this I refer to an online discussion with John Voight [31], of which the author is grateful. We fix such a choice from now on.
Corollary 5.6
Let \(\nu \in \mathrm {M}_2(\mathcal {O})\). Then the bottom left entries of \(\nu \) and \(P_{\lambda ,\mu }\nu \overline{P_{\lambda ,\mu }}\) are equal (in particular this entry remains in \(\mathcal {O}\)).
Proof
Let \(\nu = \left( \begin{array}{cc} \alpha &{} \beta \\ \gamma &{} \delta \end{array}\right) \) with \(\alpha ,\beta ,\gamma ,\delta \in \mathcal {O}\). Then a simple calculation shows that
\(\square \)
The matrix \(\eta = \left( \begin{array}{cc}x &{} y\\ z &{} w\end{array}\right) \in \mathrm {M}_2(D)\) belongs to \(Z_{\theta }\) if and only if
Equivalently,
Clearly, these equations can have no solutions for \(\theta < 0\) and so we only consider \(\theta \ge 0\).
A quick calculation shows that \(N(x) = N(w)\) and \(N(z) = p^2N(y)\) (a fact we will use soon).
Corollary 5.7
Let \(\theta \ge 0\). Then \(W_{\theta }\) consists of all matrices \(\nu = \left( \begin{array}{cc} \alpha &{} \beta \\ \gamma &{} \delta \end{array}\right) \in \mathrm {M}_2(\mathcal {O})^{\times }\) such that
The following algorithm allows us to compute \(W_{\theta }\) for \(\theta \in \mathbb {N}\). Denote by \(X_i\) the subset of \(\mathcal {O}\) consisting of norm i elements.
Algorithm 1
Step 0: Set \(j:=0\). For each integer \(0\le i\le \theta p\), generate the norm lists \(X_i, X_{p(\theta p  i)}, X_{p^2 i}\).
Step 1: Loop through all \((\gamma ,\gamma ')\in X_j\times X_{p(\theta pj)}\) and test whether \(\delta := \frac{\gamma '\gamma r}{p}\in \mathcal {O}\). Whenever successful store the pair \((\gamma ,\delta )\).
Step 2: Loop through each pair from Step 1 and each \(\gamma ''\in X_{p(\theta p  j)}\), testing whether \(\alpha := \frac{\gamma ''  \overline{r}\gamma }{p}\in \mathcal {O}\). Whenever successful store the tuple \((\alpha ,\gamma ,\delta )\).
Step 3: Loop through each triple from Step 2 and each \(\gamma '''\in X_{p^2 j}\), testing whether \(\beta := \frac{\gamma '''  (\overline{r}(\gamma r + p\delta ) + p\alpha r)}{p^2}\in \mathcal {O}\). Whenever successful store the tuple \((\alpha ,\beta ,\gamma ,\delta )\).
Step 4: For each tuple from Step 3, check whether the entries satisfy the third equation of Corollary 5.7.
Step 5: Set \(j:=j+1\) and repeat steps 1–4 until \(j>\theta p\).
Of course, once the elements of \(W_{\theta }\) have been found, it is straightforward to generate the elements of \(Y_{\theta }\) by inverting the bijection \(\Phi _{\theta }\) 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 \(Y_1 = \Gamma ^{(2)}\) as Ibukiyama does on [19, p. 592].
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 \(U\subset \text {GU}_2(D_{\mathbb {A}_f})\) as follows:
where \(\Gamma _m = \text {GU}_2(D) \cap z_m U z_m^{1}\) for representatives \(z_1, z_2, \ldots , z_m\in \text {GU}_2(D_{\mathbb {A}_f})\) of \(\text {GU}_2(D)\backslash \text {GU}_2(D_{\mathbb {A}_f})/ U\).
Ibukiyama provides the following formulae for \(M(U_1)\) and \(M(U_2)\) in [19].
Theorem 5.8
If D is ramified at p and \(\infty \), then
This formula is analogous to the Eichler mass formula and is also a special case of the mass formula of Gan et al. [10].
Proposition 5.9
\(h_1=1\) if and only if \(\Gamma ^{(1)} = \frac{5760}{(p1)(p^2+1)}\). Similarly \(h_2=1\) if and only if \(\Gamma ^{(2)} = \frac{5760}{p^21}\).
Corollary 5.10
\(h_1 = 1\) if and only if \(p=2,3\). Similarly \(h_2=1\) if and only if \(p=2,3,5,7,11\).
Proof
A quick calculation shows that the only primes to satisfy \(\frac{5760}{(p1)(p^2+1)}\in \mathbb {N}\) are \(p=2,3\). Recall \(\Gamma ^{(1)} = 2\mathcal {O}^{\times }^2\). For \(p=2,3\), we have \(\mathcal {O}^{\times } = 24, 12\), respectively, and one checks that both values satisfy the equation.
The primes satisfying \(\frac{5760}{p^21}\in \mathbb {N}\) are \(p=2,3,5,7,11,17,19,31\). Using Algorithm 1, one finds that \(\Gamma ^{(2)} = \frac{5760}{p^21}\) for the cases \(p=2,3,5,7,11\). \(\square \)
Ibukiyama and Hashimoto have produced formulae in [17] and [18] that give the values of \(h_1\) and \(h_2\) for any ramified prime. Their formulae agree with this result.
Finding the Hecke representatives
Now that we have found an algorithm to generate the elements of \(\Gamma ^{(2)}\), we consider the same question for the Hecke representatives for the \(T_{u,q}\) operator on \(A_{j,k3}(D)\) (where \(q\ne p\) is a fixed prime).
Theorem 5.11
Let D be a quaternion algebra over \(\mathbb {Q}\) ramified at \(p,\infty \) for some \(p\in \{2,3,5,7,11\}\). Suppose \(u\in \mathrm {GU}_2(D_{\mathbb {A}_f})\) is chosen as in Definition 4.9. Then
Proof
Consider an arbitrary decomposition:
By Proposition 3.5, we may take \(x_i\in \text {GU}_2(D)\) for each i. For the rest of the proof, we embed \(\text {GU}_2(D) \hookrightarrow \text {GU}_2(D_{\mathbb {A}_f})\) diagonally.
Note that for any prime \(l\ne q\) we have
Thus, \(x_i\in \text {GU}_2(D_l) \cap g_l^{1} \mathrm {M}_2(\mathcal {O}_l)^{\times } g_l\) and \(\mu (x_i) \in \mathbb {Z}_l^{\times }\) for all i.
It remains to study the double coset locally at q. Note that \(h_q g_q^{1}\in \text {GL}_2(\mathcal {O}_q)\) so that \(h_q = k_q g_q\) for some \(k_q\in \text {GL}_2(\mathcal {O}_q)\subseteq \mathrm {M}_2(\mathcal {O}_q)^{\times }\).
Conjugation by \(h_q\) gives a bijection between \(U_{2,q} u_q U_{2,q}\) and \(G v_q G\), where \(G = \text {GU}_2(D_q)\cap \text {GL}_2(\mathcal {O}_q)\). If we fix an isomorphism as in Lemma 4.4, then this is in bijection with \(\text {GSp}_4(\mathbb {Z}_q) M_q \text {GSp}_4(\mathbb {Z}_q)\) (where \(M_q = \text {diag}(1,1,q,q)\)).
Since by definition \(v_q \mapsto M_q\in \text {GSp}_4(\mathbb {Q}_q)\cap \mathrm {M}_4(\mathbb {Z}_q)\), we see that \(v_q \in M_2(\mathcal {O}_q)^{\times }\) and so \(u_q \in \text {GU}_2(D_q)\cap h_q^{1} \mathrm {M}_2(\mathcal {O}_q)^{\times } h_q\).
However,
Thus, \(u_q\in \text {GU}_2(D_q)\cap g_q^{1} \mathrm {M}_2(\mathcal {O}_q)^{\times } g_q\) and the same can be said about \(x_i\).
Also since both the conjugation and our chosen isomorphism respect similitude, we find that \(\mu (u_q) = \mu (M_q) = q\) and so \(\mu (U_{2,q} u_q U_{2,q})\subseteq q\mathbb {Z}_q^{\times }\). In particular \(\mu (x_i) \in q\mathbb {Z}_q^{\times }\).
Globally, we now see that
for each i. We also observe that \(\mu (x_i)\in \mathbb {Z}\cap \left( q\mathbb {Z}_q^{\times }\prod _{l\ne q}\mathbb {Z}_l^{\times }\right) = \{\pm q\}\). However, in our case, the similitude is positive definite so that \(\mu (x_i) = q\).
Thus, \(x_i\) 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 \(x_i U_2 = x_j U_2\) if and only if \(x_j^{1} x_i \in U_2\). But also \(x_i, x_j\in \text {GU}_2(D)\), hence \(x_j^{1} x_i\in \text {GU}_2(D) \cap U_2 = \Gamma ^{(2)}\). So equivalence of left cosets is up to right multiplication by \(\Gamma ^{(2)}\). \(\square \)
We have a nice formula for the degree of \(T_{u,q}\), found in the work of Ihara [24].
Proposition 5.12
For \(q\ne p\), we have that deg\((T_{u,q}) = (q+1)(q^2+1)\).
Employing similar arguments to Theorem 5.11, we get the following:
Theorem 5.13
Let D be a quaternion algebra over \(\mathbb {Q}\) ramified at \(p,\infty \) for some \(p\in \{2,3\}\). Suppose \(u\in \mathrm {GU}_2(D_{\mathbb {A}_f})\) is chosen as in Definition 4.9. Then
Since \(\Gamma ^{(1)}\) is given explicitly, it is possible to write down explicit representatives in this case.
Corollary 5.14
Let \(n\in \mathbb {N}\). For each \(k\in \mathbb {N}\), let \(X_k = \{\alpha \in \mathcal {O}\,\,N(\alpha ) = k\}\), \(t_k = X_k/\mathcal {O}^{\times }\), and \(x_{1,k}, x_{2,k}, \ldots , x_{t_k,k}\) be a set of representatives for \(X_k/\mathcal {O}^{\times }\). For such a choice of k, define
The following matrices are representatives for \((\mathrm {GU}_2(D)_n\cap \mathrm {M}_2(\mathcal {O})^{\times })/ \Gamma ^{(1)}\):
The finite subset \(R_{m}'\subset R_m\) is to be constructed in the proof.
Proof
Let \(\nu = \left( \begin{array}{cc} \alpha &{} \beta \\ \gamma &{} \delta \end{array}\right) \in \mathrm {M}_2(\mathcal {O})^{\times }\). In order for \(\nu \in \text {GU}_2(D)_n\) to hold, we must satisfy the equations:
In a similar vein to previous discussion, these equations imply that \(N(\alpha ) = N(\delta )\) and \(N(\beta ) = N(\gamma )\). Note that the first equation implies that \(0\le N(\alpha ) \le n\).
We wish to study equivalence of these matrices under right multiplication by \(\Gamma ^{(1)}\).
Case 1: \(N(\alpha ) \ne N(\beta )\).
We may assume that \(N(\alpha ) > \frac{n}{2}\) since for \(x,y\in \mathcal {O}^{\times }\)
and \(N(\beta y) = N(\beta ) = n  N(\alpha ) > n  \frac{n}{2} = \frac{n}{2}\).
Under this assumption, there are no antidiagonal equivalences so it remains to check for diagonal equivalences.
Now
Letting \(k = N(\alpha )\), choose \(x,y\in \mathcal {O}^{\times }\) so that \(\alpha x = x_{i,k}\) and \(\delta y = x_{j,k}\) for some \(1\le i,j\le t_{k}\). Then \(\nu \) is equivalent to \(\left( \begin{array}{cc} x_{i,k} &{} v\\ w &{} x_{j,k}\end{array}\right) \). Clearly the matrices of this form are inequivalent.
It is now clear that \(R_k\) gives representatives for the particular subcase \(N(\alpha )=k > \frac{n}{2}\).
Case 2: \(N(\alpha ) = N(\beta )=\frac{n}{2} = m\).
The matrices \(\left( \begin{array}{cc} x_{i,m} &{} v\\ w &{} x_{j,m}\end{array}\right) \) may now have extra antidiagonal equivalences.
Suppose
Then x, y are uniquely determined as follows:
Thus, each such matrix \(\left( \begin{array}{cc} x_{i,m} &{} v\\ w &{} x_{j,m}\end{array}\right) \) with \(v,w\in X_m\) can only be equivalent to at most one other matrix:
where \(v\sim x_{s,m}\) and \(w\sim x_{t,m}\) 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 \(X_m/\mathcal {O}^{\times }\) and v, w run through elements of \(X_m\) satisfying \(x_{i,m}\overline{w} + v\overline{x}_{j,m}=0\)). Then it is now clear that \(R_{m}'\) is a set of representatives for this subcase. \(\square \)
In the subcase \(k=n1\), it is often easier to use antidiagonal equivalence (since \(X_1/\mathcal {O}^{\times } = \{1\}\)). 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 [19, p. 594]. \(\square \)
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 \(u\in \mathrm {GU}_2(D_{\mathbb {A}_f})\) be chosen to form the \(T_{u,q}\) operator with respect to both \(U_1\) and \(U_2\) (for prime \(q\ne p\)). 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 \(u_q\notin U_{2,q}\) so we need only to check the local condition that \(U_{2,q} = U_{1,q}\).
This is clear since
\(\square \)
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 \(p\in \{2,3\}\). Then we may use the representatives from Corollary 5.14 as Hecke representatives for \(T_{u,q}\) with respect to \(U_2\) (for \(q \ne p\)).
Proof
Since \(p\in \{2,3\}\), 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 \(U_2\). \(\square \)
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 \(\chi _{j,k3}\) the character of the representation \(V_{j,k3}\).
Given \(g = \left( \begin{array}{cc} \alpha &{} \beta \\ \gamma &{} \delta \end{array}\right) \in \text {GU}_2(D)\), we may produce a matrix \(A\in \text {GSp}_4(\mathbb {C})\) via the embedding
where \(a = i^2\) and \(b=j^2\) in D.
This embedding is the composition of the standard embedding \(D^{\times }\hookrightarrow \mathrm {M}_2(K(\sqrt{a}))\) and the isomorphism \(\text {GU}_2(\mathrm {M}_2(K(\sqrt{a}))) \cong \text {GSp}_4(K(\sqrt{a})) \subseteq \text {GSp}_4(\mathbb {C})\) given in Lemma 4.2.
We know that the image of \(\text {GU}_2(\mathbb {H})_1\cap \text {GU}_2(D)\) under this embedding is a subgroup of \(\text {USp}(4)\), so that the matrix \(B = \frac{A}{\sqrt{\mu (A)}}\in \text {USp}(4)\). By writing \(A = (\sqrt{\mu (A)}I)B\), it follows that
In order to find \(\chi _{j,k3}(B)\), we first find the eigenvalues of B. This is equivalent to conjugating into the maximal torus of diagonal matrices. Since \(B\in \text {USp}(4)\), these eigenvalues will come in two complex conjugate pairs \(z,\overline{z}, w, \overline{w}\) 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.
Finding the trace contribution for the new subspace
Let \(\text {tr}(T_{u,q})^{\text {new}}\) and \(\text {tr}(T_{u,q})^{\text {old}}\) be the traces of the action of \(T_{u,q}\) on \(A_{j,k3}^{\text {new}}(D)\) and \(A_{j,k3}^{\text {old}}(D)\), respectively. Then \(\text {tr}(T_{u,q})^{\text {new}} = \text {tr}(T_{u,q})  \text {tr}(T_{u,q})^{\text {old}}\).
Recall that each eigenform in \(\mathcal {A}_{j,k3}^{\text {old}}(D)\) is given by a special pair of eigenforms \(F_1\in \mathcal {A}(D^{\times },U,V_j)\) and \(F_2\in \mathcal {A}(\text {GU}_2(D),U_2,V_{j,k3})\). If \(j>0\), then \(F_1\) corresponds to a unique eigenform in \(S_{j+2}^{\text {new}}(\Gamma _0(p))\) by Eichler’s correspondence. Attached to the pair \((F_1,F_2)\) is an eigenform \(\theta _{F_1\otimes F_2} \ne 0\) in \(M_{j+2k2}(\text {SL}_2(\mathbb {Z}))\) (it is a cusp form if \(j+2k6 \ne 0\)).
Let \(\alpha _n, \beta _n, \gamma _n\) be the Hecke eigenvalues of \(F_1,F_2, \theta _{F_1\otimes F_2}\), respectively. Ibukiyama links the eigensystems as follows.
Theorem 5.18
For \(q\ne p\), we have the following identity in \(\mathbb {C}(t)\):
Corollary 5.19
For \(q\ne p\), we have \(\beta _q = \gamma _q + q^{k2}\alpha _q\).
Ibukiyama conjectures that there is a bijection between pairs of eigenforms \((F_1, \theta _F)\) and eigenforms \(F_2\). With this in mind, it is now possible to calculate the oldform trace contribution.
Corollary 5.20
Suppose \(j+2k6 \ne 0\). Let \(g_1, g_2, \ldots , g_m\in S_{j+2k2}(\mathrm {SL}_2(\mathbb {Z}))\) and \(h_1, h_2, \ldots , h_n\in S_{j+2}^{\text {new}}(\Gamma _0(p))\) be the bases of normalized eigenforms with Hecke eigenvalues \(a_{q,g_i}\) and \(a_{q,h_i}\), respectively.
Then, for \(q\ne p\) and \(j>0\),
Examples and summary
The following table highlights the choices for \(D, \mathcal {O}, \lambda , \mu \) that were used.
p  D  \(\mathcal {O}\)  \(\lambda \)  \(\mu \) 
2  \(\left( \frac{1,1}{\mathbb {Q}}\right) \)  \(\mathbb {Z}\oplus \mathbb {Z} i\oplus \mathbb {Z} j\oplus \mathbb {Z}\left( \frac{1+i+j+k}{2}\right) \)  1  \(ik\) 
3  \(\left( \frac{1,3}{\mathbb {Q}}\right) \)  \(\mathbb {Z}\oplus \mathbb {Z} i\oplus \mathbb {Z} \left( \frac{1+j}{2}\right) \oplus \mathbb {Z}\left( \frac{i+k}{2}\right) \)  \(1+i\)  j 
5  \(\left( \frac{2,5}{\mathbb {Q}}\right) \)  \(\mathbb {Z}\oplus \mathbb {Z}\left( \frac{2i+k}{4}\right) \oplus \mathbb {Z}\left( \frac{2+3i+k}{4}\right) \oplus \mathbb {Z}\left( \frac{1+i+j}{2}\right) \)  2  j 
7  \(\left( \frac{1,7}{\mathbb {Q}}\right) \)  \(\mathbb {Z}\oplus \mathbb {Z} i\oplus \mathbb {Z} \left( \frac{1+j}{2}\right) \oplus \mathbb {Z}\left( \frac{i+k}{2}\right) \)  \(2+\frac{1}{2}i  \frac{1}{2}k\)  j 
11  \(\left( \frac{1,11}{\mathbb {Q}}\right) \)  \(\mathbb {Z}\oplus \mathbb {Z} i\oplus \mathbb {Z} \left( \frac{1+j}{2}\right) \oplus \mathbb {Z}\left( \frac{i+k}{2}\right) \)  \(1+3i\)  j 
Using these choices along with the algorithms and results mentioned previously, one can calculate the groups \(\Gamma ^{(1)},\Gamma ^{(2)}\) for each such p, hence generating tables of dimensions of the spaces \(A_{j,k3}^{\text {new}}(D)\). These tables are given in Appendix 1.
From these tables, one isolates 1dimensional spaces. For each possibility, the MAGMA command LRatio allows us to test for large primes dividing \(\Lambda _{\text {alg}}\) 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 2. In particular, for \(p=2\) one observes congruences provided in Bergström [1]. We finish with some new examples for \(p=3\).
Example 6.1
By Appendix 1, we see that
Then \(j=2\) and \(k=8\) so that \(j+2k2 = 16\). Let \(F\in S_{2,8}^{\text {new}}(K(3))\) be the unique normalized eigenform.
One easily checks that dim\((S_{16}^{\text {new}}(\Gamma _0(3))) = 2\). This space is spanned by the two normalized eigenforms with qexpansions:
Indeed, MAGMA informs us that \(\text {ord}_{109}(\Lambda _{\text {alg}}(f_1, 10)) = 1\) and so we expect a congruence of the form:
for all \(q\ne 3\), 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.
The algorithms mentioned earlier then calculate the necessary \(\frac{5760}{3^21} = 720\) matrices belonging to \(\Gamma ^{(2)}\) and the \((2+1)(2^2+1) = 15\) Hecke representatives for the operator \(T_{u,2}\). Applying the trace formula, we find that \(\text {tr}(T_{u,2}) = 312\).
Now since \(A_{2,5}(D) = A_{2,5}^{\text {new}}(D)\), we have that \(\text {tr}(T_{u,2}) = \text {tr}(T_{u,2})^{\text {new}}\). Also the spaces are 1dimensional and so in fact \(b_2 = \text {tr}(T_{u,2})^{\text {new}} = 312\).
The congruence is then simple to check:
\(\square \)
Example 6.2
We see an example where we must subtract off the oldform contribution from the trace. By Appendix 1, we see that
whereas
Then \(j=8\) and \(k=5\) so that \(j+2k2 = 16\) again. Let \(F\in S_{8,5}^{\text {new}}(K(3))\) be the unique normalized eigenform.
MAGMA informs us that \(\text {ord}_{67}(\Lambda _{\text {alg}}(f_2, 13)) = 1\) and so we expect a congruence of the form
for all \(q\ne 3\).
Applying the trace formula this time gives \(\text {tr}(T_{u,2}) = 300\). However, since \(\text {dim}(A_{8,2}(D)) > \text {dim}(A_{8,2}^{\text {new}}(D))\), there is an oldform contribution to this trace. In order to find it, we need Hecke eigenvalues of normalized eigenforms for the spaces \(S_{16}(\mathrm {SL}_2(\mathbb {Z}))\) and \(S_{10}^{\text {new}}(\Gamma _0(3))\).
It is known that dim\((S_{16}(\mathrm {SL}_2(\mathbb {Z}))) = 1\) and that the unique normalized eigenform has qexpansion:
Also dim\((S_{10}^{\text {new}}(\Gamma _0(3))) = 2\) and the normalized eigenforms have the following qexpansions:
Thus, using Corollary 5.20 the oldform contribution is
Hence \(\text {tr}(T_{u,2})^{\text {new}} = \text {tr}(T_{u,2})  \text {tr}(T_{u,2})^{\text {old}} = 300  288 = 12\). Since our space of algebraic forms is 1dimensional, we must have \(b_2 = \text {tr}(T_{u,2})^{\text {new}} = 12\).
The congruence is then simple to check:
\(\square \)
Example 6.3
Our final example is a case where the Hecke eigenvalues of the elliptic modular form lie in a quadratic extension of \(\mathbb {Q}\).
By Appendix 1, we see that
Then \(j=6\) and \(k=5\) so that \(j+2k2 = 14\). Let \(F\in S_{6,5}^{\text {new}}(K(3))\) be the unique normalized eigenform.
One easily checks that dim\((S_{14}^{\text {new}}(\Gamma _0(3))) = 3\). This space is spanned by the three normalized newforms with qexpansions:
MAGMA informs us that \(\text {ord}_{47}(N_{\mathbb {Q}(\sqrt{1969})/\mathbb {Q}}(\Lambda _\text {alg}(f_2, 11))) = 1\) and so we expect a congruence of the form
for some prime ideal \(\lambda \) of \(\mathbb {Z}\left[ \frac{1+\sqrt{1969}}{2}\right] \) satisfying \(\lambda \mid 47\) (note that 47 splits in this extension).
The trace formula gives \(\text {tr}(T_{u,2}) = 72\) and the usual arguments show that \(b_2 = 72\). It is then observed that
This is divisible by 47 and so the congruence holds for \(q=2\). \(\square \)
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Appendices
Appendix 1: Newform dimensions
For each prime \(p=2,3,5,7,11\), the following tables give the values of dim\((A_{j,k}^{\text {new}}(D))\) for \(0\le j \le 20\) even and \(0\le k \le 15\). We use the specific quaternion algebras given in Section 6. Note that Ibukiyama conjectures that these values are equal to dim\((S_{j,k+3}^{\text {new}}(K(p)))\).
0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  

\(p=2\)  
0  1  0  0  0  1  1  1  0  2  1  2  1  3  2  3  2 
2  0  0  0  0  0  0  0  1  1  1  1  2  4  2  4  5 
4  0  0  0  0  1  1  1  1  3  3  4  4  7  7  9  9 
6  0  0  0  0  1  0  2  2  4  3  5  7  10  9  13  14 
8  0  0  0  1  2  2  2  4  7  7  9  10  15  17  20  22 
10  0  0  0  1  3  4  4  6  10  10  14  17  21  23  29  33 
12  0  0  1  1  3  5  6  8  12  14  17  21  28  30  37  41 
14  0  0  1  3  5  6  9  12  17  19  24  29  37  40  49  56 
16  0  1  2  4  8  9  13  16  23  26  32  38  48  53  63  70 
18  0  0  2  5  9  11  15  20  28  31  39  46  58  64  76  86 
20  0  2  3  7  12  16  20  26  35  41  50  58  71  81  94  106 
\(p=3\)  
0  1  0  0  1  1  1  2  1  2  3  3  3  5  4  5  8 
2  0  0  0  0  0  1  1  2  2  4  4  6  8  9  11  14 
4  0  0  0  1  0  2  3  3  5  8  8  12  15  17  22  27 
6  0  0  1  2  2  3  7  7  10  14  16  21  27  30  37  45 
8  0  0  1  3  4  6  8  12  16  20  25  31  38  46  54  64 
10  0  0  1  4  5  10  13  16  23  30  35  45  54  63  76  90 
12  0  1  4  7  8  15  20  25  32  43  49  62  75  86  102  121 
14  0  1  5  9  13  19  27  34  44  55  67  81  97  113  133  154 
16  0  2  6  13  17  25  36  44  57  72  84  104  124  142  167  194 
18  1  3  10  18  24  35  47  58  75  93  109  131  157  180  209  242 
20  0  6  12  22  31  45  58  74  92  114  136  162  189  221  254  292 
\(p=5\)  
0  1  0  1  1  2  2  3  3  5  5  7  8  10  11  14  16 
2  0  0  0  0  1  2  3  5  7  10  13  17  22  27  33  40 
4  0  1  1  3  4  7  10  14  18  25  31  39  48  59  70  84 
6  0  0  3  4  7  11  17  22  31  39  50  63  77  92  112  131 
8  0  3  5  9  15  21  28  40  51  64  81  99  119  144  169  198 
10  0  2  6  12  20  29  41  54  71  90  112  136  165  196  231  270 
12  1  6  14  22  31  48  62  81  104  130  157  193  228  269  316  366 
14  0  7  17  27  44  60  82  107  136  167  207  247  294  346  404  465 
16  3  13  24  43  61  84  113  145  180  224  269  322  381  445  514  594 
18  3  14  34  53  78  109  143  181  230  279  336  401  472  548  636  727 
20  4  26  45  72  105  143  183  236  289  352  423  500  582  680  779  890 
\(p=7\)  
0  1  1  1  2  3  4  5  6  8  10  13  15  18  22  26  31 
2  0  0  1  1  3  5  8  12  16  22  29  37  47  57  70  84 
4  0  1  1  5  7  12  18  26  34  47  59  75  93  114  136  164 
6  1  3  7  11  18  26  38  50  67  85  107  133  162  194  232  272 
8  0  6  10  19  29  43  57  80  102  130  162  199  239  289  339  398 
10  1  5  14  26  42  60  85  111  145  183  228  276  334  396  467  545 
12  4  15  29  47  67  98  128  168  212  265  321  391  463  546  638  740 
[XMLCONT]
14  4  18  38  60  93  127  171  221  280  344  422  504  599  703  819  943 
16  5  27  49  86  122  170  226  291  361  449  539  646  762  892  1030  1189 
18  13  37  76  116  168  228  299  377  473  573  690  818  962  1116  1291  1475 
20  13  54  94  150  214  291  373  477  585  712  852  1008  1174  1367  1567  1791 
\(p=11\)  
0  1  1  2  3  4  6  8  11  15  19  24  31  38  46  56  67 
2  0  1  2  4  9  14  21  31  43  57  75  95  119  147  178  213 
4  1  4  6  15  22  35  51  71  93  125  157  197  243  296  353  422 
6  3  5  18  27  44  66  94  124  168  212  268  332  405  484  581  681 
8  2  17  28  49  77  111  149  205  261  331  413  506  607  730  858  1005 
10  7  20  43  75  115  161  225  293  377  475  586  709  856  1012  1189  1386 
12  11  38  74  120  170  248  342  422  536  667  808  983  1163  1372  1603  1857 
14  15  53  103  159  243  329  439  567  714  875  1072  1278  1515  1778  2068  2379 
16  26  78  138  230  324  444  586  749  928  1147  1377  1642  1937  2261  2610  3008 
18  38  100  198  298  428  582  759  954  1195  1447  1738  2063  2421  2806  3246  3707 
20  44  148  252  390  554  745  954  1215  1487  1804  2157  2547  2966  3448  3951  4509 
Appendix 2: Congruences
The following table gives information on the congruences found. For simplicity, we only give the Hecke eigenvalues at \(q=3\) when \(p=2\) and \(q=2\) when \(p=3,5,7,11\).
Note that there is no congruence at level 11 (even though one is expected). This does not contradict the conjecture since \(\lambda \mid 11\) in this case.
Whenever \(a_q\) is rational, we give the Hecke eigenvalue explicitly. When it lies in a bigger number field, we give the minimal polynomial f(x) defining \(\mathbb {Q}_f\) (then the Hecke eigenvalue \(a_2\) in all of our cases is exactly a root \(\alpha \) of this polynomial).
The large primes given are the rational primes lying below the prime for which the congruence holds.
(j, k)  \(N(\lambda )\)  tr\((T_q)\)  \(b_q\)  \(a_q\) 

\(p=2\)  
(0, 14)  37  2, 223, 720  2, 223, 720  97, 956 
(2, 10)  61  18, 360  18, 360  \(13,092\) 
(2, 11)  71  \(57,528\)  \(57,528\)  59, 316 
(2, 12)  29  \(122,040\)  \(122,040\)  \(505,908\) 
(4, 10)  61  \(189,720\)  \(189,720\)  71, 604 
(6, 7)  29  1872  3240  6084 
(10, 6)  109  216  216  \(13,092\) 
(12, 5)  79  77, 544  \(7560\)  \(53,028\) 
(12, 6)  23  \(275,688\)  30, 600  71, 604 
(14, 5)  379  102, 960  63, 000  59, 316 
(16, 4)  37  \(97,488\)  \(23,400\)  71, 604 
\(p=3\)  
(2, 8)  109  \(312\)  \(312\)  \(234\) 
(4, 6)  23  \(36\)  \(36\)  \(12\) 
[XMLCONT]
(6, 5)  47  72  72  \(x^2 + 54x  16,992\) 
(8, 5)  67  300  12  \(72\) 
(10, 5)  433  120  24  \(x^2  594x  42,912\) 
(12, 4)  23  \(1716\)  132  204 
(14, 4)  617  \(240\)  72  \(x^2  702x  664,128\) 
\(p=5\)  
(2, 7)  61  \(76\)  \(76\)  \(x^3  142x^2  11,144x + 901,248\) 
\(p=7\)  
(2, 5)  263  \(44\)  \(44\)  \(x^3  21x^2  1326x + 19,080\) 
(4, 4)  101  \(2\)  \(2\)  \(x^2 + 6x  184\) 
(4, 5)  43  \(70\)  10  \(x^2 + 54x  2640\) 
\(p=11\)  
(2, 4)  11  \(20\)  \(20\)  N/A 
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Fretwell, D. Genus 2 paramodular Eisenstein congruences. Ramanujan J 46, 447–473 (2018). https://doi.org/10.1007/s1113901798847
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Keywords
 Number theory
 Automorphic forms
 Eisenstein congruences
Mathematics Subject Classification
 11F33
 11F46
 11F55