Quinary forms and paramodular forms

Abstract

We work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local lattices with specific open compact subgroups, paramodular at split places, and with Atkin-Lehner operators. Combining this with the recent work of Rösner and Weissauer, proving conjectures of Ibukiyama on Jacquet-Langlands type correspondences (mildly generalised here), provides an effective tool for computing Hecke eigenvalues for Siegel modular forms of degree two and paramodular level. It also enables us to prove examples of congruences of Hecke eigenvalues connecting Siegel modular forms of degrees two and one. These include some of a type conjectured by Harder at level one, supported by computations of Fretwell at higher levels, and a subtly different congruence discovered experimentally by Buzzard and Golyshev.

Finding all Borcherds product paramodular cusp forms of a given weight and level

“…The congruence of eigenvalues modulo a prime ideal above 43 is also proven in [4], which also contains a proof of a congruence above 19, discovered by Buzzard and Golyshev, of f 61 to a Yoshida lift. Theorem 4.7.…”

“…In fact, Ibukiyama's conjecture has been recently proven by Mirko Rösner and Rainer Weissauer in [29]. Hence the Hecke eigenvalues of paramodular forms and orthogonal modular forms are known to agree; compare [4], where Neil Dummigan, Ariel Pacetti, Rama, and Tornaria prove the agreement for all levels N such that p N for some prime p.…”

“…The first appearance of the aforementioned f N -eigenvalues occurred in the work of Avner Ash, Paul E. Gunnells, and Mark McConnell [1], who were searching for an element of the cohomology space H 5 (Γ 0 (N ); C) that genuinely arises from SL (4). Here Γ 0 (N ) ⊆ SL 4 (Z) is the subgroup of elements having bottom row congruent to (0, 0, 0, * ) mod N .…”

“…According to a conjecture of Ibukiyama [14,16], paramodular forms of weight 3, Fricke sign −1, and prime level have the same eigenvalues as certain orthogonal modular forms on the compact twist of GSp (4). A method for computing eigenvalues of orthogonal modular forms, which computes eigenvalues faster than the previously mentioned computational methods, was developed by a collaboration of Jeffery Hein, Watson Ladd, and Gonzalo Tornaria [13,19].…”

“…Galois representations can be associated to elements of H 5 (Γ 0 (N ); C) as in [2], but computing eigenvalues using the sharbly complex is expensive. Thus the current path to proving the modularity of a Calabi-Yau threefold of Hodge type (1, 1, 1, 1) relies on associating Galois representations to both the threefold and a paramodular newform, and computing enough Hecke eigenvalues to apply the Faltings-Serre method for GSp (4). Once modularity is proven, the most efficient way to compute Euler factors will be by counting points over finite fields or by the fourth order differential equations associated to the Calabi-Yau threefolds.…”