The hyperbolic trace fibres of Popa–Zagier's element

Let TauCeti.PopaZagier.weight M be twelve times the coefficient of the integral matrix M in Popa and Zagier's explicit Hecke element, their eq. (15). This file evaluates the sum of these weights over the matrices of trace t and determinant n when the discriminant t² - 4 n is positive. Together with the elliptic fibres (TauCeti.PopaZagier.finsum_weight_traceDetFiber_eq_neg_twelve_mul_hurwitzClassNumber) and the repeated-root fibres (TauCeti.PopaZagier.finsum_weight_traceDetFiber_of_discrim_eq_zero), this evaluates every trace fibre of positive determinant:

• if t² - 4 n = u² with u > 0 and n > 0, the weights sum to 12 u for t > 0 and to 0 for t < 0;
• if t² - 4 n > 0 is not a square, the weights sum to 0.

Popa and Zagier's coefficients are a twelfth of the weights, so in the split case the two fibres of traces ±t together carry coefficient u. Weighted by the binary-form trace P_{w+2}(t, n), the split fibres give the divisor-sum terms of the trace formula, and the nonsplit fibres give nothing.

Main results

• TauCeti.PopaZagier.finsum_weight_traceDetFiber_of_discrim_eq_sq: the split hyperbolic fibres.
• TauCeti.PopaZagier.finsum_weight_traceDetFiber_of_not_isSquare: the nonsplit hyperbolic fibres.

Proof

Translation M ↦ M + 1 preserves the sum over a fibre of positive determinant (TauCeti.PopaZagier.finsum_weight_traceDetFiber_add_two) and shifts both eigenvalues by 1. If t < 0, translating upwards reaches a fibre of nonpositive determinant, where every weight vanishes. If t > 0, translating downwards reaches a fibre with n + 1 ≤ t, that is, with an eigenvalue in (0, 1]. There only the sum T₁ contributes, and only through the matrices (1 b; 0 n) with 0 ≤ b < n, which occur exactly when t = n + 1, that is when the eigenvalues are 1 and n.

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).

Fibres of nonpositive determinant and of negative trace

Fibres with an eigenvalue in (0, 1]

The hyperbolic fibres

theorem TauCeti.PopaZagier.finsum_weight_traceDetFiber_of_discrim_eq_sq {t n u : ℤ} (hn : 0 < n) (hu : 0 < u) (h : t ^ 2 - 4 * n = u ^ 2) : ∑ᶠ (M : ↥(traceDetFiber (Fin 2) t n)), weight ↑M = if 0 < t then 12 * u else 0

The split hyperbolic trace fibres of Popa–Zagier's element. If n > 0 and t² - 4 n = u² with u > 0, the weights of the integral matrices of trace t and determinant n sum to 12 u if t > 0, and to 0 if t < 0.

theorem TauCeti.PopaZagier.finsum_weight_traceDetFiber_of_not_isSquare {t n : ℤ} (hD : 0 < t ^ 2 - 4 * n) (h : ¬IsSquare (t ^ 2 - 4 * n)) : ∑ᶠ (M : ↥(traceDetFiber (Fin 2) t n)), weight ↑M = 0

The nonsplit hyperbolic trace fibres of Popa–Zagier's element. If t² - 4 n > 0 is not a square, the weights of the integral matrices of trace t and determinant n sum to 0.