Our goal is to study a sequence (gk⋆(z, t))k≥0 of families of Drinfeld modular forms that produce, for certain values of the parameter t, several kinds of Eisenstein series that Gekeler considers. These series no longer appear in the series expansion of the "analogs of ℘": the exponential functions of rank 2Drinfeld modules. The evaluation at ζ∈C∞ of the exponential function eΛz associated with the lattice Λz is given by the series.
This series expansion can in many ways be understood as the analogue of the Laurent series expansion at 0 for the Weierstrass℘ function in Drinfeldian theory. For fixedk, this is the normalization (2) of the Eisenstein series of weight qk−1:. 10) Together with (gk)k≥0 we have the second series (mk)k≥0 of modular Drinfeld forms, called para-Eisenstein series, discussed by Gekeler in [13], which has the same range of weights and types. Similar obvious comments can also be made for the classic Eisenstein sequence or Poincar'e sequence, but this will be one of the most important observations we make in this article.
Fort=θqk, the seriesL(χt, qk) diverges and the series on the right-hand side of the above identity become conditionally convergent. The integrity of the coefficients of the normalized extremal quasi-modular form of weightqk+ 1 and type 1 supports Conjecture 2 of Kaneko and Koike in [18], asserting that if fl,w ∈Q[[q]] is theq-expansion of the quasi-form -normalized extremal modular of scale depth l≤4, thenfl,w∈Zp[[x]] for allp > w. In the Drinfeldian case, the integrity of the coefficients of Ek is a final consequence of our formula (18).
This is a classic result that can be easily proved by induction ons≥0; we recall the proof here for the convenience of the reader.
Extending to existentially closed fields
The last part of the proposition follows from a simple application of Proposition 11 which provides the operatorL. We will say that a sequence G of K is generic if there exist matrices E ∈ Mat1×s(K) and F ∈Mats×1(K), both with Kτ-linearly independent entries, such that for allk∈Z,. Then of course we have the following proposition, which contains all the properties encountered so far; later we will use it for a specific generic sequence of modular forms.
If L =A0τ0+· · ·+Asτs is the split operator of K[τ] associated to F, and ifL′=A′0τ0+· · ·+A′sτ−sis the split operator of K[τ−1]associated totE (by assertion 11), then G is at once τ-linear recurrent and τ-linearized recurrent (in both orders). Now let xi for all i ≥ 1 be a solution of Lx0xi =yi (they exist, again because K is existentially closed). The proof above requires us to also solve non-homogeneous linear equations of order 1, but this does not present any problem even if we work in a field where only linear homogeneous x n systems of τ-difference equations of order 1 can be solved, as we note , that the solution of the non-homogeneous equationτ x=ax+b, reduces to the solution of the systemτ y1=ay1+by2ogτ y2=y2, which is homogeneous.
Having described some basic facts of the theory of repeated τ-linear sequences, we return to our modular forms and now begin to deal with modular vector forms and their deformations.
Notation, tools
By section 2.1, the fields L and L∞ can be embedded in a field K equipped with an automorphism extending τ (and again denoted by τ) such that (K, τ) is existentially closed, with Kτ=Fq(t).
Basic properties of vectorial modular forms
This definition is obviously compatible with the concept of type of a Drinfeld modular form already discussed in the introduction. The set of deformations of vector modular forms of weightw, dimensions, type and radius rassociated to a representationρ is a T If all are negative, the corresponding coefficients, raised to powerqk, are holomorphic to Ω. The next proposition is a simple reproduction of the main properties described in section 2 in the framework of deformations of modular vector forms. Proposition 18 Assuming that r >1, consider F in Msw,m(ρ, r) and let E be such that. hat means we skip the corresponding term in the plural). Ifr >1and if the components of F Fq(t) are linearly independent, then Lis is divided, for any knonnegative integer, Gk is an element of Mwq. We apply theorem 11 to obtain that Lis distribution of orders and if the components of E Fq(t) are linearly independent, L′ is also divided. The part of the proposal involving properties of the formE is similar and is left to the reader. If logq|t| < 0, then the components of Eα,m,l(z, t) converge absolutely and uniformly on every compact subset of Ωeven ifα−2m >0. More precisely, let's choose an integer 0≤s≤and look at the component in places. Taking into account the first part of the proposal, we can arbitrarily replace zz+awitha∈A to verify convergence and assume, without loss of generality, that degθz=λ6∈Z. We now look for upper bounds on the absolute values of the terms of the upper series that separate the two cases in a similar way to Gerritzen and van der Put in loc. This property can be deduced from the proof of the second part because if logq|t|<0, then. According to Gekeler [11, section 3], we recall that for all α > 0 there exists a polynomial Gα(u)∈ C∞[u], called the αth Goss polynomial, so that for all z∈Ω Gα(u(z )) is equal to the sum of the convergent series. The Fq(t)-linear independence of the components of Eα,0,1 follows from analysis of the behavior atu= 0 described by Lemmas 24 and 25. This means that the sequence (Gα,0,1,k)k∈Z is generic and therefore is generic. proves the first part of the proposition. The function µ7→sCar,µ is well defined by the image inT But again from (32),Ld2= 0 and we see that all coefficients offF in the above identity contribute 0 (otherwise, we can apply Lemma 30 and the fact that d2 is a formal inu power series). We start by showing that for q6= 2 equation (44) has a unique solutionY which can be expanded as a formal series in powers ofu, with the property thatY|t=θ= 0. This means that there is one and only one solution from (44) for q 6= 2 which is a power series ofu, with the additional property that atu= 0 vanishes. Now we need to show that Y is the function we are looking for, but this is a simple task. It can be proved that F is, up to multiplication by a factor inFq(t), the only function for which we can write d1=d2F+d3, with d2,d3 formal power series of nonnegative exponents. Since we do not need this property in this paper, we will not give its proof. Computer-assisted experiments are possible and generate large tables of coefficients of the ψandd3 functions, but we will not report them here. Appendix: transcendence of F ⋆ and d 1 over formal Laurent seriesExamples
Proofs of the main theorems
