The Eichler–Selberg trace formula at level one

For even k ≥ 4 and n ≥ 1, the trace of the Hecke operator Tₙ on the cusp forms of weight k on SL(2, ℤ) is

tr(Tₙ | S_k(SL(2, ℤ))) = -½ ∑_{t² ≤ 4n} P_k(t, n) H(4n - t²) - ½ ∑_{dd' = n} min(d, d')^(k-1),

where H is the Hurwitz class number, with H(0) = -1/12, and the weight P_k(t, n) is the value at t of the second-kind Dickson polynomial of degree k - 2 and parameter n, the coefficient of x^(k-2) in (1 - t x + n x²)⁻¹.

This is Popa and Zagier's proof. Their explicit element popaZagierElement of the group ring of the integral matrices of determinant n acts on the period polynomials W_w, w = k - 2, and its trace there is the class-number expression σ_{k-1}(n) - ∑ min(d, d')^(k-1) - ∑ P_k(t, n) H(4n - t²) (the main theorem of TauCeti.NumberTheory.ModularForms.LevelOne.TraceFormula.Formula). By Popa and Zagier's trace reduction, this is also its trace on all binary forms V_w (TauCeti.TraceFormulaMatrixModule.ExchangeRelations.trace_periodActionRestrict_eq_trace). By the Eichler–Shimura theorem, W_w is isomorphic to M_k × S_k, and popaZagierElement acts there as Tₙ × Tₙ, so the same trace is 2 tr(Tₙ | S_k) + σ_{k-1}(n) (TauCeti.TraceFormulaMatrixModule.trace_heckeTCuspLevelOne_eq_trace_popaZagierElement). Comparing the two gives the formula.

Main results

• TauCeti.trace_heckeTCuspLevelOne_eq_neg_hurwitzClassNumber_sub_min: the Eichler–Selberg trace formula for Tₙ on S_k(SL(2, ℤ)).

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, Theorem 1.
• D. Zagier, The Eichler–Selberg trace formula on SL₂(ℤ), appendix to S. Lang, Introduction to Modular Forms, for the normalization H(0) = -1/12.

theorem TauCeti.trace_heckeTCuspLevelOne_eq_neg_hurwitzClassNumber_sub_min {k : ℤ} {w : ℕ} (hk : k = ↑w + 2) (hw : Even w) (hw₀ : w ≠ 0) (n : ℕ) [NeZero n] : (LinearMap.trace ℂ (CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k)) (heckeTCuspLevelOne k n) = (-∑ t ∈ Finset.Icc (-4 * ↑n) (4 * ↑n) with t ^ 2 ≤ 4 * ↑n, ↑(hurwitzClassNumber (4 * ↑n - t ^ 2).toNat) * Polynomial.eval (↑t) (Polynomial.dickson 2 (↑n) w)) / 2 - (∑ p ∈ n.divisorsAntidiagonal, ↑(min p.1 p.2) ^ (w + 1)) / 2

The Eichler–Selberg trace formula at level one. For even k = w + 2 ≥ 4 and n ≥ 1, the trace of Tₙ on the cusp forms of weight k on SL(2, ℤ) is

tr(Tₙ | S_k) = -½ ∑_{t² ≤ 4n} H(4n - t²) P_k(t, n) - ½ ∑_{dd' = n} min(d, d')^(k-1),

with Hurwitz class numbers normalized by H(0) = -1/12 and P_k(t, n) = (dickson 2 n w).eval t. The sum over t runs over the integers with t² ≤ 4n, written as a filter of [-4n, 4n].