K_1 of products of Drinfeld modular curves and special values of L-functions
Ramesh Sreekantan
备注：
34 pages, 2 figures. To appear in Compositio Mathematica. This is a completely revised and improved version of a earlier preprint with C.Consani (arXiv:math/0303130)
Beilinson obtained a formula relating the special value of the L-function of H^2 of a product of modular curves to the regulator of an element of a motivic cohomology group - thus providing evidence for his general conjectures on special values of L-functions. In this paper we prove a similar formula for the L-function of the product of two Drinfeld modular curves providing evidence for an analogous conjecture in the case of function fields.
We give an abstract characterization of the Satake compactification of a general Drinfeld modular variety. We prove that it exists and is unique up to unique isomorphism, though we do not give an explicit stratification by Drinfeld modular varieties of smaller rank which is also expected. We construct a natural ample invertible sheaf on it, such that the global sections of its $k$-th power form the space of (algebraic) Drinfeld modular forms of weight $k$. We show how the Satake compactification and modular forms behave under all natural morphisms between Drinfeld modular varieties; in particu…
In this paper we describe the compactification of the Drinfeld modular curve. This compactification is analogous to the compactification of the classical modular curve given by Katz and Mazur. We show how the Weil pairing on Drinfeld modules that we defined in earlier work gives rise to a map on the Drinfeld modular curve. We introduce the Tate-Drinfeld module and show how this describes the formal neighbourhood of the scheme of cusps of the Drinfeld modular curve.
By combining theorems of Drinfeld and Strauch, we show that the monodromy representation on the special fibre of a Drinfeld modular variety, with level not divisible by the characteristic, is surjective. We illustrate this result in the special case of Drinfeld $\mathbb{F}_q[t]$-modules in level $t$, and apply this to show that the Kronecker factors of a Drinfeld modular polynomial in rank $r$ are irreducible.
Let q be a prime power and E a non-isotrivial elliptic curve over Fq(T) given by a Weierstrass model. We survey the construction, with an explicit point of view, of the modular parametrization of E by the associated Drinfeld modular curve. We then prove a formula that allows us to evaluate this modular parametrization at cusps and we produce an explicit method to compute these values. Finally we illustrate our results with several examples in characteristic 2 and 3.
In this paper we give a closed form expression for the Drinfeld modular polynomial $Φ_T(X,Y) \in \mathbb{F}_q(T)[X,Y]$ for arbitrary $q$ and prove a conjecture of Schweizer. A new identity involving the Catalan numbers plays a central role.
Let X be a product of Drinfeld modular curves over a general base ring A of odd characteristic. We classify those subvarieties of X which contain a Zariski-dense set of CM points. This is an analogue of the André-Oort conjecture. As an application, we construct non-trivial families of higher Heegner points on modular elliptic curves over global function fields.
We give explicit equations for the simplest towers of Drinfeld modular curves over any finite field, and observe that they coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth.
Let $\cal A$ be a maximal (or more generally a hereditary) order in a central simple algebra over a global field $F$ of positive characteristic. We study the reduction of the modular scheme of $\cal A$-elliptic sheaves at all places of $F$. We show that some of these varieties - for different $\cal A$ - are twists of each other. This result can be viewed as a global analogue of the Cherednik-Drinfeld theorem for these varieties.
We explore an analogue of the André-Oort conjecture for subvarieties of Drinfeld modular varieties. The conjecture states that a subvariety $X$ of a Drinfeld modular variety contains a Zariski-dense set of complex multiplication (CM) points if and only if $X$ is a "special" subvariety (i.e. $X$ is defined by requiring additional endomorphisms). We prove this conjecture in two cases. Firstly when $X$ contains a Zariski-dense set of CM points with a certain behaviour above a fixed prime (which is the case if these CM points lie in one Hecke orbit), and secondly when $X$ is a curve containing inf…
We construct a compactification of the moduli space of Drinfeld modules of rank $r$ and level $N$ as a moduli space of $A$-reciprocal maps. This is closely related to the Satake compactification, but not exactly the same. The construction involves some technical assumptions on $N$ that are satisfied for a cofinal set of ideals $N$. In the special case $A={\mathbb F}_q[t]$ and $N=(t^n)$ we obtain a presentation for the graded ideal of Drinfeld cusp forms of level $N$ and all weights and can deduce a dimension formula for the space of cusp forms of any weight. We expect the same results in gener…
This is the second of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present part, we compare the analytic theory with the algebraic one that was begun in a paper of the third author. For any arithmetic congruence subgroup and any integral weight we establish an isomorphism between the space of analytic modular forms with the space of algebraic modular forms defined in terms of the Satake compactification. From this we deduce the important result that this space is finite dimensional.
Let $Z=X_1\times...\times X_n$ be a product of Drinfeld modular curves. We characterize those algebraic subvarieties $X \subset Z$ containing a Zariski-dense set of CM points, i.e. points corresponding to $n$-tuples of Drinfeld modules with complex multiplication (and suitable level structure). This is a characteristic $p$ analogue of a special case of the André-Oort conjecture. We follow closely the approach used by Bas Edixhoven in characteristic zero, see math.NT/0302138. Note that in this paper we assume that the characteristic $p$ is odd, and we only treat the case of Drinfeld $F_q[T]$-mo…
Let $X_0(I)$ be the Drinfeld's modular curve with level $I$ structure, where $I$ is a monic square-free ideal in $\F_{q}[T]$. In this paper we show the existence of an element in the motivic cohomology group $H^3_{\M}(X_0(I) \times X_0(I),\Q(2))$ whose regulator is related to a special value of a Ranking-Selberg convolution $L$-function. This result is the function field analogue of a theorem of Beilinson for the self product of a modular curve.
In the first part, we revisit the theory of Drinfeld modular curves and $π$-adic Drinfeld modular forms for GL(2) from the perfectoid point of view. In the second part, we review open problems for families of Drinfeld modular forms for GL(n).
We compute the stable reduction of the Lubin-Tate space $X(π^2)$ in the equal characteristic case, on the basis of Coleman-McMurdy's ideas. Namely, in this paper, we actually construct a stable covering of $X(π^2).$ This paper also includes intersection multiplicity datum of the stable model of $X(π^2)$. Our method is very explicit and elementary.
This is the first of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank r. In the present part, we develop the analytic theory. Most of the work goes into defining and studying the u-expansion of a weak Drinfeld modular form, whose coefficients are weak Drinfeld modular forms of rank r-1. Based on that we give a precise definition of when a weak Drinfeld modular form is holomorphic at infinity and thus a Drinfeld modular form in the proper sense.
We show that the Drinfeld modular forms with $A$-expansion that have been constructed by the author are precisely the hyperderivatives of the subfamily of single-cuspidal Drinfeld modular forms with $A$-expansions that remain modular after hyperdifferentiation. In addition, we show that Drinfeld-Poincaré series display a similar behavior with respect to hyperdifferentiation, giving indirect evidence that the Drinfeld modular forms with $A$-expansions are Drinfeld-Poincaré series. The Drinfeld-Poincaré series that we consider generalize previous examples of such series by Gekeler, and Gerritzen…
Let $k$ be a global function field with field of constants $\Fr$ and let $\infty$ be a fixed place of $k$. In his habilitation thesis \cite{boc2}, Gebhard Böckle attaches abelian Galois representations to characteristic $p$ valued cusp eigenforms and double cusp eigenforms \cite{go1} such that Hecke eigenvalues correspond to the image of Frobenius elements. In the case where $k=\Fr(T)$ and $\infty$ corresponds to the pole of $T$, it then becomes reasonable to ask whether rank 1 Drinfeld modules over $k$ are themselves ``modular'' in that their Galois representations arise from a cusp or double…
We calculate the effect of simple Hecke operators on u-expansions of higher rank Drinfeld modular forms, the eigenvalue for the Drinfeld discriminant function $Δ_t$ and show that a certain natural class of Hecke operators is completely multiplicative.
In this paper we introduce a class of determinants "of Hankel type". We use them to compute certain remarkable families of Drinfeld quasi-modular forms.
The aim of this article is twofold: first, improve the multiplicity estimate obtained by the second author for Drinfeld quasi-modular forms; and then, study the structure of certain algebras of "almost-$A$-quasi-modular forms"
Let $p$ be a rational prime and $q$ a power of $p$. Let $\wp$ be a monic irreducible polynomial of degree $d$ in $\mathbf{F}_q[t]$. In this paper, we define an analogue of the Hodge-Tate map which is suitable for the study of Drinfeld modules over $\mathbf{F}_q[t]$ and, using it, develop a geometric theory of $\wp$-adic Drinfeld modular forms similar to Katz's theory in the case of elliptic modular forms. In particular, we show that for Drinfeld modular forms with congruent Fourier coefficients at $\infty$ modulo $\wp^n$, their weights are also congruent modulo $(q^d-1)p^{\lceil \log_p(n)\rcei…
This paper contains the written notes of a course the author gave at the VIASM of Hanoi in the Summer 2018. It provides an elementary introduction to the analytic naive theory of Drinfeld modular forms for the simplest 'Drinfeld modular group' GL 2 (Fq[$θ$]) also providing some perspectives of development, notably in the direction of the theory of vector modular forms with values in certain ultrametric Banach algebras.
In this paper we determine the explicit structure of the semisimple part of the Hecke algebra that acts on Drinfeld modular forms of full level modulo T . We use computations of the Hecke action modulo T to find Drinfeld modular forms that cannot be eigenforms. Finally, we conjecture that the Hecke algebra that acts on Drinfeld modular forms of full level is not smooth for large weight.
In this article, we explicitly construct a canonical basis for the space of certain weakly holomorphic Drinfeld modular forms for $Γ_0(T)$ (resp., for $Γ_0^+(T)$) and compute the generating function satisfied by the basis elements. We also give an explicit expression for the action of the $Θ$-operator, which depends on the divisor of meromorphic Drinfeld modular forms.
Let ${\mathbf F}_q$ denote a finite field of characteristic $p$ and let $n$ be an effective divisor on the affine line over ${\mathbf F}_q$ and let $v$ be a point on the affine line outside $n$. In this paper, we get congruences between ${\mathbb Q}_l$-valued weight two $v$-old Drinfeld modular forms and $v$-new Drinfeld modular forms of level $vn$. In order to do this, we shall first construct a cokernel torsion-free injection from a full lattice in the space of $v$-old Drinfeld modular forms of level $vn$ into a full lattice in the space of all Drinfeld modular forms of level $vn$. To get th…
The present paper deals with Atkin-Lehner theory for Drinfeld modular forms. We provide an equivalent definition of $\mathfrak{p}$-newforms (which makes computations easier) and commutativity results between Hecke operators and Atkin-Lehner involutions. As applications we show a criterion for a direct sum decomposition of cusp forms, we exibit $\mathfrak{p}$-newforms arising from lower levels and we provide $\mathfrak{p}$-adic Drinfeld modular forms of level greater than 1.
The aim of this paper is to describe some partial advances in the solution of the following two problems. Find the maximal order of vanishing at infinity of a non-zero Drinfeld quasi-modular form of given weight. Determine differential properties of a Drinfeld quasi-modular form of given weight and depth with maximal order of vanishing at infinity.
In \cite{Cho09}, Choi studied congruences of coefficients (modulo $T^q-T$) for Drinfeld modular forms of level $Γ_0(T)$, trivial type and the linear relations between the initial coefficients of those. In this article, we generalize these results for level $Γ_0(T)$, arbitrary type.
Continuing the work of \cite{7} and \cite{8}, we derive an analogue of the classical "$k/12$-formula" for Drinfeld modular forms of rank $r \geq 2$. Here the vanishing order $ν_ω(f)$ of one modular form at some point $ω$ of the complex upper half-plane is replaced by the intersection multiplicity $ν_{\bo}(f_1,\ldots,f_{r-1})$ of $r-1$ independent Drinfeld modular forms at some point $\bo$ of the Drinfeld symmetric space $\OM^r$. We apply the formula to determine the common zeroes of $r-1$ consecutive Eisenstein series $E_{q^{i}-1}$, where $n-r<i<n$ for some $n \geq r$.
This is a sequel to the paper [F. Breuer, H.-G. Rück, Drinfeld modular polynomials in higher rank, J. Number Theory 129 (2009), 59-83.], in which we introduced Drinfeld modular polynomials of higher rank, using an analytic construction. These polynomials relate the isomorphism invariants of Drinfeld F_q[T]-modules of rank r\geq 2 linked by isogenies of a specified type. In the current paper, we give an algebraic construction of greater generality, and prove a generalization of the Kronecker congruences relations, which describe what happens when modular polynomials associated to P-isogenies ar…
We prove that the groups PSL(r,q^d) can be realized F_q(T)-regularly as Galois groups over the purely transcendental field F_q(T)(t_1,...,t_{r-1}) if r is even and r/2 is coprime to q^d-1. The method is to use twisted moduli schemes of Drinfeld modules (a la Shih). If one has sufficient control over modular forms, it can be made into an algorithm to find explicit equations. We do this for PSL(2,q^2) using Drinfeld modular forms.
This is the third part of a series of articles providing a foundation for the theory of Drinfeld modular forms of arbitrary rank. In the present article we construct and study some examples of Drinfeld modular forms. In particular we define Eisenstein series, as well as the action of Hecke operators upon them, coefficient forms and discriminant forms. In the special case A=F_q[t] we show that all modular forms for GL_r(Γ(t)) are generated by certain weight one Eisenstein series, and all modular forms for GL_r(A) and SL_r(A) are generated by certain coefficient forms and discriminant forms. We …
The modular invariant of rank 1 Drinfeld modules is introduced and used to formulate and prove an exact analog of the Weber-Fueter theorem for global function fields. The main ingredient in the proof is a version of Shimura's Main Theorem of Complex Multiplication for global function fields, which is also proved here.
In this article, we describe the structure of the $R$-algebra of Drinfeld modular forms $M(Γ_0(T))_R$ (resp., $M^0(Γ_0(T))_R$) of level $Γ_0(T)$ and the structure of mod-$\p$ reduction of $M_{\mfp}^0(Γ_0(T))$ for $\p \neq (T)$. As a result, we are able to study the properties of the weight filtration for $M_{k,l}(Γ_0(T))$. Finally, we prove a result on mod-$\p$ congruences for Drinfeld modular forms of level $Γ_0(\p T)$ for $\p \neq (T)$.
We introduce and study certain deformations of Drinfeld quasi-modular forms by using rigid analytic trivialisations of corresponding Anderson's t-motives. We show that a sub-algebra of these deformations has a natural graduation by the group Z^2 x Z/(q-1)Z and an homogeneous automorphism, and we deduce from this and other properties multiplicity estimates.
In the present paper, we introduce a special function on the Drinfeld period domain $Ω^{r}$ for $r\geq 2$ which gives the false Eisenstein series of Gekeler when $r=2$. We also study its functional equation and relation with quasi-periodic functions of a Drinfeld module as well as transcendence of its values at CM points.
In~\cite{CS04}, Calegari and Stein studied the congruences between classical cusp forms $S_k(Γ_0(p))$ of prime level and made several conjectures about them. In~\cite{AB07} (resp., ~\cite{BP11}) the authors proved one of those conjectures (resp., their generalizations). In this article, we study the analogous conjecture and its generalizations for Drinfeld modular forms.
We consider the generalized Jacobian $\widetilde{J}$ of the modular curve $X_0(N)$ of level $N$ with respect to a reduced divisor consisting of all cusps. Supposing $N$ is square free, we explicitly determine the structure of the $\mathbb{Q}$-rational torsion points on $\widetilde{J}$ up to $6$-primary torsion. The result turns out to be very different from the case of prime power level previously studied by Yang and the second author. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first …
We show that the absolute value $|f|$ of an invertible holomorphic function $f$ on the Drinfeld symmetric space $\OM^r$ $(r \geq 2)$ is constant on fibers of the building map to the Bruhat-Tits building $\MB\MT$. Its logarithm $\log|f|$ is an affine map on the realization of $\MB\MT$. These results are used to study the vanishing loci of modular forms (coefficient forms, Eisenstein series, para-Eisenstein series) and to determine their images in $\MB\MT$.
In this paper we investigate examples of good and optimal Drinfeld modular towers of function fields. Surprisingly, the optimality of these towers has not been investigated in full detail in the literature. We also give an algorithmic approach on how to obtain explicit defining equations for some of these towers and in particular give a new explicit example of an optimal tower over a quadratic finite field.
We evaluate a rigid analytical analogue of the Beilinson-Bloch-Deligne regulator on certain explicit elements in the K_2 of Drinfeld modular curves, constructed from analogues of modular units, and relate its value to special values of L-series using the Rankin-Selberg method.
In this text, we develop the theory of vectorial modular forms with values in Tate algebras introduced by the first author, in a very special case (dimension two, for a very particular representation of Γ := GL 2 (Fq[$theta$])). Among several results that we prove here, we determine the complete structure of the modules of these forms, we describe their specializations at roots of unity and their connection with Drinfeld modular forms for congruence subgroups of Γ and we prove that the modules generated by these forms are stable under the actions of Hecke operators.
We introduce the notion of Drinfeld modular forms with $A$-expansions, where instead of the usual Fourier expansion in $t^n$ ($t$ being the uniformizer at `infinity'), parametrized by $n \in \mathbb{N}$, we look at expansions in $t_a$, parametrized by $a \in A = \Fq [T]$. We construct an infinite family of eigenforms with $A$-expansions. Drinfeld modular forms with $A$-expansions have many desirable properties that allow us to explicitly compute the Hecke action. The applications of our results include: (i) various congruences between Drinfeld eigenforms; (ii) the computation of the eigensyste…
Let $\goth E(\goth p)$ denote the Eisenstein ideal in the Hecke algebra $\Bbb T(\goth p)$ of the Drinfeld modular curve $X_0(\goth p)$ parameterizing Drinfeld modules of rank two over $\Bbb F_q[T]$ of general characteristic with Hecke level $\goth p$-structure, where $\goth p\triangleleft\Bbb F_q[T]$ is a non-zero prime ideal. We prove that the characteristic $p$ of the field $\Bbb F_q$ does not divide the order of the quotient $\Bbb T(\goth p)/\goth E(\goth p)$ and the Eisenstein ideal $\goth E(\goth p)$ is locally principal.
We construct and study a natural compactification $\overline{M}^r(N)$ of the moduli scheme $M^r(N)$ for rank-$r$ Drinfeld $\F_q[T]$-modules with a structure of level $N \in \F_q[T]$. Namely, $\overline{M}^r(N) = {\rm Proj}\,{\bf Eis}(N)$, the projective variety associated with the graded ring ${\bf Eis}(N)$ generated by the Eisenstein series of rank $r$ and level $N$. We use this to define the ring ${\bf Mod}(N)$ of all modular forms of rank $r$ and level $N$. It equals the integral closure of ${\bf Eis}(N)$ in their common quotient field $\widetilde{\MF}_r(N)$. Modular forms are characterized…
