The repeated-root trace fibres of Popa–Zagier's element

For a nonzero integer a, the sum of Popa–Zagier's integer weights on the matrices of trace 2a and determinant a² is 2 if a > 0 and 0 if a < 0. These are the discriminant-zero fibres, containing both scalar and nonscalar parabolic matrices. The result evaluates the boundary contribution in the ambient binary-form trace; it does not assert that individual nonscalar matrices have weight zero.

The asymmetry in the sign of the trace comes from the choice of representatives in the explicit element. Together, the two fibres have weight 2, corresponding to coefficient 1/6 before the passage from the ambient trace to the cusp-form trace.

References

• A. Popa and D. Zagier, An elementary proof of the Eichler–Selberg trace formula, J. Reine Angew. Math. 762 (2020), 105–122, arXiv:1711.00327, §4, eq. (15).

theorem TauCeti.PopaZagier.finsum_weight_traceDetFiber_of_discrim_eq_zero {t n : ℤ} (hn : 0 < n) (hdisc : t ^ 2 - 4 * n = 0) : ∑ᶠ (M : ↥(traceDetFiber (Fin 2) t n)), weight ↑M = if 0 < t then 2 else 0

The repeated-root trace fibres of Popa–Zagier's element. At positive determinant and zero discriminant, the total integer weight is 2 for positive trace and zero for negative trace. The sum includes scalar and nonscalar matrices, not just conjugacy representatives.