Extended affine root systems. V: Elliptic eta-products and their Dirichlet series (Q2782397)

This is the fifth of a series of papers on elliptic root systems [Part IV, cf. Publ. Res. Inst. Math. Sci. 36, 385-421 (2000; Zbl 0987.17012)]. Let \(Q\) be an even lattice of rank \(l\) with a quadratic form \(q\) of signature \((l,2,0)\), so that the radical of \(q\) is of rank 2. A marked elliptic root system is a pair \((R,G)\) of a generalized root system \(R\) belonging to \((Q,q)\) and a one-dimensional subspace \(G\) of \(\text{rad}(q) \otimes \mathbb{Q}\). The marked elliptic root systems were classified by the author. An elliptic eta-product \(\eta_{(R,G)} (\tau)\) for \((R,G)\) is the eta-product of the characteristic polynomial of a Coxeter element of \((R,G)\). NEWLINENEWLINENEWLINEIn this paper the author shows, among other things, (1) \(\eta_{(R,G)} (\tau)\) is always holomorphic and (2) the Fourier coefficients at \(\infty\) of \(\eta_{(R,G)} (\tau)\) are nonnegative integers if and only if it is not a cusp form. This is the case when \((R,G)\) is 1-codimensional and simply laced. There are exactly 4 such types of \((R,G)\). Explicit formulas for the Fourier coefficients are described. NEWLINENEWLINENEWLINEFor the proof, the Dirichlet series attached to \(\eta_{(R,G)} (\tau)\) is considered.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00029].