Exchange relations for Popa–Zagier's explicit element

Popa–Zagier's explicit element satisfies the exchange relations: its right symmetrization by 1 + S is invariant under left multiplication by U = T S, and its right symmetrization by 1 + U + U² is invariant under left multiplication by S. The finite order projection identities identify these fixed spaces with the ranges in the exchange relations.

The proof uses the integer weights of the explicit element, including the boundary fractions. Its auxiliary weights are twelve times the coefficients of Popa–Zagier's F and G; their factorizations in eq. (16) make the two invariances finite algebraic identities.

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, Theorem 4(a), eqs. (15)–(16).

theorem TauCeti.TraceFormulaMatrixModule.exchangeRelations_popaZagierElement (k : Type u_1) [DivisionRing k] (n : ℤ) : ExchangeRelations k n (popaZagierElement k n)

Popa–Zagier's explicit element satisfies the exchange relations (Theorem 4(a) of their paper). This holds over every coefficient division ring; when 12 = 0, the element defined by the totalized inverse is zero.