The elliptic trace fibres of Popa–Zagier's explicit 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). If t² - 4 n = -D < 0, then

∑_{tr M = t, det M = n} weight M = -12 H(D),

the sum running over all integral matrices of trace t and determinant n, where H is the Hurwitz class number TauCeti.hurwitzClassNumber. In Popa and Zagier's normalisation: the coefficients of their element sum to -H(4 n - t²) over every elliptic trace fibre. The trace of a matrix on the binary forms of degree w only depends on its trace and determinant (TauCeti.trace_binaryFormRep_eq_dickson_eval), so this identity is the input that evaluates the contribution of the elliptic matrices to the trace of the element on binary forms, the class-number terms P_{w+2}(t, n) H(4 n - t²) of the Eichler–Selberg trace formula.

Main results

• TauCeti.PopaZagier.finsum_weight_traceDetFiber_add_two: for n > 0, the weights of the matrices of trace t + 2 and determinant n + t + 1 have the same sum as those of trace t and determinant n.
• TauCeti.PopaZagier.finsum_weight_traceDetFiber_eq_neg_twelve_mul_hurwitzClassNumber: the weights of the matrices of trace t and determinant n sum to -12 H(D) when t² - 4 n = -D < 0.

Proof

The translation M ↦ M + 1 maps the matrices of trace t and determinant n bijectively to those of trace t + 2 and determinant n + t + 1. It fixes the form Q_M = c x² + (d - a) x y - b y² of M = (a b; c d), hence the discriminant t² - 4 n, and it fixes the first line of each of the four sums of eq. (15). So it changes the weight of M only where one of the second-line inequalities c < a, b < d ≤ 0, a ≤ 0 and d ≤ 0 changes, that is at the matrices with a = c, b = d, d = 0 or a = 0. For n > 0 these are parametrised by the factorisations n = u v into positive integers, and their changes cancel under (u, v) ↦ (v, u). This gives the first result; the sum over the matrices of trace t and determinant n therefore depends only on D = 4 n - t².

For t ≤ -D only the sums T₃ and T₄ contribute, with all their second-line inequalities automatic. Through M ↦ Q_M their matrices are the forms ⟨a', b', c'⟩ of discriminant -D with 0 ≤ -b' ≤ a' ≤ c', respectively 0 ≤ -b' ≤ c' ≤ a'. Each of the two families is, with its weights, six times the weighted count of the reduced forms of discriminant -D that defines H(D).

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, §1 and §4, eq. (15).

Translation by the identity

Factorisations of n

theorem TauCeti.PopaZagier.finsum_weight_traceDetFiber_add_two {t n : ℤ} (hn : 0 < n) : ∑ᶠ (M : ↥(traceDetFiber (Fin 2) (t + 2) (n + t + 1))), weight ↑M = ∑ᶠ (M : ↥(traceDetFiber (Fin 2) t n)), weight ↑M

Translation by the identity. For n > 0, the weights of the matrices of trace t + 2 and determinant n + t + 1 have the same sum as those of trace t and determinant n.

The translation M ↦ M + 1 is a bijection between these matrices, it fixes the discriminant t² - 4 n, and its effect on the weights cancels in total.

Very negative trace

Reduced forms

The elliptic fibres

theorem TauCeti.PopaZagier.finsum_weight_traceDetFiber_eq_neg_twelve_mul_hurwitzClassNumber {t n : ℤ} {D : ℕ} [NeZero D] (h : t ^ 2 - 4 * n = -↑D) : ∑ᶠ (M : ↥(traceDetFiber (Fin 2) t n)), ↑(weight ↑M) = -12 * hurwitzClassNumber D

The elliptic trace fibres of Popa–Zagier's element. If t² - 4 n = -D < 0, then the weights of the integral matrices of trace t and determinant n sum to -12 H(D): the coefficients of Popa and Zagier's explicit element, a twelfth of the weights, sum to -H(4 n - t²) over the matrices of trace t and determinant n.

The hypothesis D ≠ 0 is needed: for t = 2 a and n = a² with a > 0 the weights sum to 2, the weight of the scalar matrix a • 1, while -12 H(0) = 1.