Newton's identities provide a way to express elementary symmetric polynomials in terms of power polynomials over fields of characteristic zero. In this article, we study the failure of this relation in positive characteristic and what can be recovered. In particular, we show how one can write the elementary symmetric polynomials as rational functions in the power polynomials over any commutative unital ring.

We prove a Lefschetz trace formula for Böckle–Pink crystals on tame Deligne–Mumford stacks of finite type over Fq and apply it to the crystal associated to the universal Drinfeld module. Combined with the Eichler–Shimura theory developed by Böckle, this leads to a trace formula for Hecke operators on Drinfeld modular forms. As an application, we deduce a Ramanujan bound on the traces of Hecke operators.

In this paper, we study traces of Hecke operators on Drinfeld modular forms of level 1 in the case . We deduce closed-form expressions for traces of Hecke operators corresponding to primes of degree at most 2 and provide algorithms for primes of higher degree. We improve the Ramanujan bound and deduce the decomposition of cusp forms of level into oldforms and newforms, as conjectured by Bandini–Valentino, under the hypothesis that each Hecke eigenvalue has multiplicity less than p.

We study the structure of the vector space of Drinfeld quasi-modular forms for congruence subgroups. We provide representations as polynomials in the false Eisenstein series with coefficients in the space of Drinfeld modular forms (the E-expansion), and, whenever possible, as sums of hyperderivatives of Drinfeld modular forms.

Moreover, we introduce and study the double-slash operator, and use it to provide a well-posed definition for Hecke operators on Drinfeld quasi-modular forms. We characterize eigenforms and, for the special case of Hecke congruence subgroups $\Gamma_0(\mathfrak{n})$, we give explicit formulas for the Hecke action on E-expansions.

We study traces of Hecke operators on spaces of elliptic cusp forms and Drinfeld cusp forms and show that, modulo any prime power, these traces are periodic in the weight.