Noam D. Elkies
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.
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This article is the first one of a series aiming to construct an isomorphism between the p-adic Lubin-Tate and Drinfeld towers, describe this isomorphism and give applications. We construct a p-adic equivariant integral model of the Lubin-Tate tower with infinite level. This formal scheme will be later compared to another one associated to the Drinfeld tower.
This article is the last one about the isomorphism between Lubin-Tate and Drinfeld towers. We prove the existence of an isomorphism between the compactly supported etale cohomology of the Lubin-Tate and Drinfeld towers, and more generally their equivariant cohomology complex. We also prove the existence of a geometric local Jacquet-Langlands correspondence between some equivariant rigid etale sheaves on Gross-Hopkins period space $\mathbb{P}^{n-1}$ and Drinfeld one $Ω$.
This is the fourth article about the isomorphism between Lubin-Tate and Drinfeld towers. We prove the final result concerning the isomorphism that is to say the existence of an equivariant isomorphism between some blow-up of the formal scheme associated to Lubin-Tate space with infinite level constructed cellularly before and a blow-up of another formal scheme associated to Drinfeld tower. We follow the strategy of Gerd Faltings.
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.
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.
Elaborating on ideas of Elkies, we show how recursive equations for towers of Drinfeld modular curves $(X_0(P^n))_{n\ge 0}$ for $P\in \mathbb F_q[T]$ can be read of directly from the modular polynomial $Φ_P(X,Y)$ and how this naturally leads to recursions of depth two. Although the modular polynomial $Φ_T(X,Y)$ is not known in general, using generators and relations given by Schweizer, we find unreduced recursive equations over $\mathbb F_q(T)$ for the tower $(X_0(T^n))_{n\ge 2}$ and of a small variation of it (its partial Galois closure). Reducing at various primes, one obtains towers over fi…
Higher Hida theory for Drinfeld modular curves
受Boxer和Pilloni启发，为Drinfeld模曲线上的线丛上同调建立高阶Hida理论，并插值Serre对偶。
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.
In this article we give a Drinfeld modular interpretation for various towers of function fields meeting Zink's bound.
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.
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.
The Pila-Zannier strategy for Drinfeld modules and Drinfeld modular curves
研究将Pila-Zannier策略扩展到Drinfeld模，证明了相关定理类似物。在特征零中，用刚性解析Pila-Wilkie定理替代基于\(o\)-极小性的计数步骤，通过点数论证建立函数超越性输入，实现该策略在函数域的应用。
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.
In 2000, based on his procedure for constructing explicit towers of modular curves, Elkies deduced explicit equations of rank-2 Drinfeld modular curves which coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth. In 2015, Bassa, Beelen, Garcia, and Stichtenoth constructed a celebrated (recursive and good) tower (BBGS-tower for short) of curves and outlined a modular interpretation of the defining equations. Soon after that, Gekeler studied in depth the modular curves coming from sparse Drinfeld modules. In this paper, to establish a link between these …
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.
For applications in algebraic geometric codes, an explicit description of bases of Riemann-Roch spaces of divisors on function fields over finite fields is needed. We investigate the third function field $ F^{(3)} $ in a tower of Artin-Schreier extensions described by Garcia and Stichtenoth reaching the Drinfeld-Vl{ă}du{ţ} bound. We construct bases for the related Riemann-Roch spaces on $ F^{(3)} $ and present some basic properties of divisors on a line. From the bases, we explicitly calculate the Weierstrass semigroups and pure gaps at several places on $ F^{(3)} $. All of these results can b…
Factorization de la cohomologie étale p-adique de la tour de Drinfeld
本文研究Drinfeld塔的p进平展上同调分解，对$F={\mathbf Q}_p$给出类似Emerton模曲线塔完备上同调的分解，证明模p算术平展上同调的有限性定理，并指出$F\neq{\mathbf Q}_p$时相关${\rm GL}_2(F)$表示不可容许。
Rank-2 Drinfeld modules are a function-field analogue of elliptic curves, and the purpose of this paper is to investigate similarities and differences between rank-2 Drinfeld modules and elliptic curves in terms of supersingularity. Specifically, we provide an explicit formula of a supersingular polynomial for rank-2 Drinfeld modules and prove several basic properties. As an application, we give a numerical example of an asymptotically optimal tower of Drinfeld modular curves.
The Geometry of Drinfeld Modular Forms
本文为同余子群的Drinfeld模形式代数提供几何视角，通过建立线丛截面环与模形式代数的同构实现几何计算，并推广了代数分解结果。
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.
Over all non-prime finite fields, we construct some recursive towers of function fields with many rational places. Thus we obtain a substantial improvement on all known lower bounds for Ihara's quantity $A(\ell)$, for $\ell = p^n$ with $p$ prime and $n>3$ odd. We relate the explicit equations to Drinfeld modular varieties.
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.
New optimal function field towers over finite fields of quartic power
本文引入两种源自特定域 $\mathcal{A}$ 的新型 Drinfeld 模曲线塔，证明其极小塔的 $I_η$-约化在有限域 $\mathbb{F}_{q^4}$ 上渐近最优。
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).
A Ramanujan bound for Drinfeld modular forms
本文证明了有限型栈上 Böckle-Pink 晶体的 Lefschetz 迹公式，结合 Eichler-Shimura 理论导出了 Drinfeld 模形式上 Hecke 算子的迹公式，并由此推导出 Hecke 算子迹的 Ramanujan 界。
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.
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…
De Rham cohomology of Lubin-Tate spaces and Drinfeld spaces
本文借助 Lue Pan 的方法，证明 Lubin-Tate 塔与 Drinfeld 塔的紧支集上同调在相应群模结构下同构。
We study modular polynomials classifying cyclic isogenies between Drinfeld modules of arbitrary rank over the ring F_q[T].
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…
Traces of Hecke operators on Drinfeld modular forms for $\mathrm{GL}_2(\mathbb{F}_q[T])$
本文研究了Hecke算子在Drinfeld模形式上的踪，推导了闭式表达式并改进了Ramanujan界，同时探讨了cusp形式的分解问题。
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.
Nearly holomorphic Drinfeld modular forms and their special values at CM points
本文提出近全纯Drinfeld模形式概念，研究Maass-Shimura算子类似物，证明CM点特殊值在自同态代数不同时代数独立，并应用于Drinfeld拟模形式的结构分析。
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…
Drinfeld Modular Curves Subordinate to Conjugacy Classes of Nilpotent Upper-Triangular Matrices
本文引入归一化Drinfeld模曲线，参数化与幂零上三角矩阵共轭类相关的$T$-挠结构，建立矩阵分类与曲线分解的树状联系，推广塔结构，并证明$(3,2)$型曲线的几何不可约性。
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.
In this paper, we will give an overview of known and new techniques on how one can obtain explicit equations for candidates of good towers of function fields. The techniques are founded in modular theory (both the classical modular theory and the Drinfeld modular theory). In the classical modular setup, optimal towers can be obtained, while in the Drinfeld modular setup, good towers over any non-prime field may be found. We illustrate the theory with several examples, thus explaining some known towers as well as giving new examples of good explicitly defined towers of function fields.
On the partial derivatives of Drinfeld modular forms of arbitrary rank
本文研究了Drinfeld模形式在任意秩下的偏导数性质，提出了一种推广Gekeler在秩二情况下的微分算子，并展示了包含所有满模群Drinfeld模形式的有限生成代数在偏导数下的稳定性。
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.
On special values of meromorphic Drinfeld modular forms of arbitrary rank at CM points
本文研究了任意秩解析狄利克雷模形式在CM点处的特殊值，并在特定条件下证明其代数独立性，扩展了Chang在秩为二情况下的研究成果。
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.
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.
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 give a new way to study recursive towers of curves over a finite field, defined from a bottom curve $\Cun$ and a correspondence $\Cdeux$ on $\Cun$.In particular, we study their asymptotic behavior. A close examination of singularities leads to a necessary condition for a tower to be asymptotically good. Then, spectral theory on a directed graph and considerations on the class of $\Cdeux$ in $\NS (\Cun \times \Cun)$ lead to the fact that, under some mild assumptions, a recursive tower which does not reach Drinfeld-Vladut bound cannot be optimal in Tsfasmann-Vladut sense. Results are applied …
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 …
Drinfeld Quasi-Modular Forms of Higher Level
该研究探讨同余子群的Drinfeld拟模形式向量空间结构，通过E展开和超导数表示其结构，引入双斜线算子定义Hecke算子，刻画本征形式并给出Γ₀(𝔫)子群下Hecke作用于E展开的显式公式。
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.
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