The trace reduction to the ambient space of binary forms

Let K be a field of characteristic zero, w a positive even natural number and ξ ∈ K[ℳₙ] an element satisfying Popa and Zagier's exchange relations (B). On the space V_w of binary forms of degree w, the action of ξ maps each of the subspaces A = ker (1 + S) and B = ker (1 + U + U²), with U = T S, into the other, and A + B = V_w. Its trace on the period-polynomial space W_w = A ∩ B is therefore its trace on V_w. This is Popa and Zagier's reduction tr(ξ | W_w) = tr(ξ | V_w) of the trace on period polynomials to the trace on all binary forms.

Main results

• TauCeti.TraceFormulaMatrixModule.ExchangeRelations.trace_periodActionRestrict_eq_trace: the trace of ξ on W_w is its trace on V_w.

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, §2, Proposition 3.

theorem TauCeti.TraceFormulaMatrixModule.ExchangeRelations.trace_periodActionRestrict_eq_trace {K : Type u_1} [Field K] {n : ℤ} {w : ℕ} [CharZero K] (hw : Even w) (hw₀ : w ≠ 0) {ξ : MonoidAlgebra K (TraceFormulaMatrixModule n)} (hξ : ExchangeRelations K n ξ) : (LinearMap.trace K ↥(periodPolynomials K w)) (periodActionRestrict hw hξ) = (LinearMap.trace K ↥(MvPolynomial.homogeneousSubmodule (Fin 2) K w)) ((periodAction hw) ξ)

The trace reduction (Popa–Zagier, §2, Proposition 3): over a field of characteristic zero and for positive even w, the action of an element ξ satisfying the exchange relations has the same trace on the period polynomials W_w as on the binary forms V_w.