The Hecke eigenvalue of the Eisenstein period polynomial

Let w be even and n ≠ 0. Every solution ξ of Popa–Zagier's period relation (A) in ℛₙ = R[ℳₙ] acts on the Eisenstein period polynomial X^w - Y^w by the scalar σ_{w+1}(|n|):

(X^w - Y^w) ∣ ξ = σ_{w+1}(|n|) (X^w - Y^w)

(TauCeti.TraceFormulaMatrixModule.PeriodRelation.periodAction_eisensteinPeriodPolynomial). For n > 0 and positive even w, this is the eigenvalue of the Hecke operator Tₙ on the Eisenstein series E_{w+2}, whose extended even period polynomial is a multiple of X^w - Y^w: the solutions of (A) act on period polynomials as Tₙ on the Eisenstein part, as they do on the period polynomials of cusp forms (TauCeti.TraceFormulaMatrixModule.PeriodRelation.periodAction_periodPolynomial). This is the Eisenstein contribution σ_{k-1}(n) to the trace of Tₙ on the period-polynomial space, in weight k = w + 2 ≥ 4. At w = 0 the polynomial X^0 - Y^0 is zero and E₂ is not a level-one modular form, so the identity holds trivially and has no such interpretation.

The proof is algebraic. The binary form attached to the functional P ↦ P(1, 0) - P(0, 1) is Y^w - X^w, and this functional is the evaluation of the modular symbols {∞, 0} ⊗ P at the cusps (TauCeti.ModularSymbols.boundaryEval). By Popa–Zagier's Proposition 2 on modular symbols (TauCeti.TraceFormulaMatrixModule.PeriodRelation.symbolAction_eq), the action of ξ on it is the evaluation of ∑ {a∞, a0} ⊗ (P ∣ adj a) at the cusps, the sum over the upper-triangular representatives a = (a b; 0 d) of Γ \ ℳₙ. The values at a∞ = ∞ are P(adj a (1, 0)) = P(d, 0) = dʷ P(1, 0), and there are |d| representatives for each factor pair (a, d), so they sum to σ_{w+1}(|n|) P(1, 0) (TauCeti.TraceFormulaMatrixModule.boundaryEval_comp_upperTriangularSymbol). The values at a0 = (aS)∞ are the values at ∞ along the representatives aS. The functional M ↦ (P ∣ adj M)(v_{M∞}) is invariant under left multiplication by SL(2, ℤ), and Tₙ^∞ S is the formal sum of another set of representatives of Γ \ ℳₙ, so these sum to σ_{w+1}(|n|) (P ∣ S)(1, 0) = σ_{w+1}(|n|) P(0, 1). No count of the cusps b/d is needed.

Main results

• TauCeti.binaryFormDual_evalCusp_infty_sub_evalCusp_zero: the binary form attached to P ↦ P(1, 0) - P(0, 1) is -(X^w - Y^w), for even w.
• TauCeti.TraceFormulaMatrixModule.boundaryEval_comp_upperTriangularSymbol: the evaluation at the cusps of ∑ {a∞, a0} ⊗ (P ∣ adj a) is σ_{w+1}(|n|) (P(1, 0) - P(0, 1)).
• TauCeti.TraceFormulaMatrixModule.PeriodRelation.periodAction_eisensteinPeriodPolynomial: every solution of the period relation acts on X^w - Y^w by σ_{w+1}(|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, §§1–2.
• W. Kohnen and D. Zagier, Modular forms with rational periods, in Modular Forms (R. A. Rankin, ed.), Ellis Horwood, 1984, 197–249, §1.

The binary form of the evaluation at ∞ and 0

theorem TauCeti.binaryFormDual_evalCusp_infty_sub_evalCusp_zero {R : Type u_1} [CommRing R] {w : ℕ} (hw : Even w) : (binaryFormDual R w) (ModularSymbols.evalCusp R w OnePoint.infty - ModularSymbols.evalCusp R w ↑0) = -eisensteinPeriodPolynomial R w

The Eisenstein polynomial is the binary form of the evaluation at ∞ and 0. For even w, the binary form attached to P ↦ P(1, 0) - P(0, 1) is Y^w - X^w.

The values at the cusps a∞

The values at the cusps a0

The Hecke eigenvalue

theorem TauCeti.TraceFormulaMatrixModule.boundaryEval_comp_upperTriangularSymbol {R : Type u_1} [CommRing R] {n : ℤ} {w : ℕ} (hn : n ≠ 0) (hw : Even w) : ModularSymbols.boundaryEval ⊤ hw ∘ₗ upperTriangularSymbol R n w = ↑((ArithmeticFunction.sigma (w + 1)) n.natAbs) • (ModularSymbols.evalCusp R w OnePoint.infty - ModularSymbols.evalCusp R w ↑0)

The evaluation at the cusps of the Hecke sum over ℳₙ^∞. For n ≠ 0 and even w, the evaluation at the cusps of ∑ {∞, b/d} ⊗ (P ∣ adj (a b; 0 d)), the sum over the upper-triangular representatives of Γ \ ℳₙ, is σ_{w+1}(|n|) (P(1, 0) - P(0, 1)).

theorem TauCeti.TraceFormulaMatrixModule.PeriodRelation.periodAction_eisensteinPeriodPolynomial {R : Type u_1} [CommRing R] {n : ℤ} {w : ℕ} (hn : n ≠ 0) (hw : Even w) {ξ : MonoidAlgebra R (TraceFormulaMatrixModule n)} (h : PeriodRelation R n ξ) : ((periodAction hw) ξ) (eisensteinPeriodPolynomial R w) = ↑((ArithmeticFunction.sigma (w + 1)) n.natAbs) • eisensteinPeriodPolynomial R w

The Hecke eigenvalue of the Eisenstein period polynomial. For n ≠ 0 and even w, every solution ξ of Popa–Zagier's period relation (A) acts on the Eisenstein polynomial X^w - Y^w by σ_{w+1}(|n|): (X^w - Y^w) ∣ ξ = σ_{w+1}(|n|) (X^w - Y^w).