Ramified-level extension of the Tu–Yang correspondence

Extend the Tu–Yang correspondence to quaternionic orders with levels at primes ramified in the quaternion algebra, in particular to the orders corresponding to the lattices L_2 and L_4, whose level includes the ramified prime 3, so that their Hecke data and newform correspondences are covered by a general theorem.

Background

The paper identifies four ternary lattices with four orders in the quaternion algebra B_6. The classical Jacquet–Langlands and Tu–Yang correspondences apply directly to the maximal order and one even-trace order, but the orders associated with L_2 and L_4 have level at the ramified prime 3 and therefore fall outside the hypotheses of the cited results.

Exact Hecke computations provide evidence that the newform level equals the reduced discriminant for all four orders, including the two ramified-level cases. A proof extending the correspondence to these levels would place the observed Hecke data and multiplicity pattern on a rigorous theoretical foundation.