We generalize our work on Carlitz prime power torsion extension to torsion extensions of Drinfeld modules of arbitrary rank. As in the Carlitz case, we give a description of these extensions in terms of evaluations of Anderson generating functions and their hyperderivatives at roots of unity. We also give a direct proof that the image of the Galois representation attached to the $\mathfrak{p}$-adic Tate module lies in the $\mathfrak{p}$-adic points of the motivic Galois group. This is a generalization of the corresponding result of Chang and Papanikolas for the $t$-adic case.
Anderson generating functions are generating series for division values of points on Drinfeld modules, and they serve as important tools for capturing periods, quasi-periods, and logarithms. They have been fundamental in recent work on special values of positive characteristic L-series and in transcendence and algebraic independence problems. In the present paper we investigate techniques for expressing Anderson generating functions in terms of the defining polynomial of the Drinfeld module and determine new formulas for periods and quasi-periods.
For an Anderson A-motive over a discretely valued field whose residue field has A-characteristic ε, we prove a criterion for good reduction in terms of its associated local shtuka at ε. This yields a criterion for good reduction of Drinfeld modules. Our criterion is the function-field analog of Grothendieck's and de Jong's criterion for good reduction of an abelian variety over a discretely valued field with residue characteristic p in terms of its associated p-divisible group.
We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.
We make a detailed account of sign-normalized rank 1 Drinfeld A-modules, for A the coordinate ring of an elliptic curve over a finite field, in order to provide a parallel theory to the Carlitz module for F_q[t]. Using precise formulas for the shtuka function for A, we obtain a product formula for the fundamental period of the Drinfeld module. Using the shtuka function we find identities for deformations of reciprocal sums and as a result prove special value formulas for Pellarin L-series in terms of an Anderson-Thakur function. We also give a new proof of a log-algebraicity theorem of Anderso…
For each positive characteristic multiple zeta value (defined by Thakur), the first and third authors constructed a $t$-module together with an algebraic point such that a specified coordinate of the logarithmic vector of the algebraic point is a rational multiple of that multiple zeta value. The objective of this paper is to use the Taylor coefficients of Anderson-Thakur series and $t$-motivic Carlitz multiple star polylogarithms to give explicit formulae for all of the coordinates of this logarithmic vector.
We define the Heegner--Drinfeld cycle on the moduli stack of Drinfeld Shtukas of rank two with $r$-modifications for an even integer $r$. We prove an identity between (1) The $r$-th central derivative of the quadratic base change $L$-function associated to an everywhere unramified cuspidal automorphic representation $π$ of $PGL_{2}$; (2) The self-intersection number of the $π$-isotypic component of the Heegner--Drinfeld cycle. This identity can be viewed as a function-field analog of the Waldspurger and Gross--Zagier formula for higher derivatives of $L$-functions.
We show that the module of Stark units associated to a sign-normalized rank one Drinfeld module can be obtained from Anderson's equivariant $A$-harmonic series. We apply this to obtain a class formula à la Taelman and to prove a several variable log-algebraicity theorem, generalizing Anderson's log-algebraicity theorem. We also give another proof of Anderson's log-algebraicity theorem using shtukas and obtain various results concerning the module of Stark units for Drinfeld modules of arbitrary rank.
We study the passage from Drinfeld-A'-modules to Drinfeld-A-modules for a given finite flat inclusion A \subset A'. We show that this defines a morphism from the moduli space of Drinfeld-A'-modules to the moduli space of Drinfeld-A-modules which is proper but in general not representable. For Drinfeld-Anderson shtuka and abelian sheaves instead of Drinfeld modules we obtain the same results.
We formulate and prove a log-algebraicity theorem for arbitrary rank Drinfeld modules defined over the polynomial ring F_q[theta]. This generalizes results of Anderson for the rank one case. As an application we show that certain special values of Goss L-functions are linear forms in Drinfeld logarithms and are transcendental.
We study tensor powers of rank 1 sign-normalized Drinfeld A-modules, where A is the coordinate ring of an elliptic curve over a finite field. Using the theory of A-motives, we find explicit formulas for the A-action of these modules. Then, by developing the theory of vector-valued Anderson generating functions, we give formulas for the period lattice of the associated exponential function.
Introduced by Anglès, Pellarin, and Tavares Ribeiro, Drinfeld modules over Tate algebras are closely connected to Anderson log-algebraicity identities, Pellarin $L$-series, and Taelman class modules. In the present paper we define the de Rham map for Drinfeld modules over Tate algebras, and we prove that it is an isomorphism under natural hypotheses. As part of this investigation we determine further criteria for the uniformizability and rigid analytic triviality of Drinfeld modules over Tate algebras.
In the first half of the paper, we translate in the geometric situation of Drinfeld varieties, the principal results of the Harris and Taylor's book. We give in particular the restriction to the open strata of the vanishing cycles sheaves in terms of some local systems named Harris-Taylor's local systems which we calculate the alternated sum of the cohomology group with compact supports. In the last half of the paper, we describe the monodromy filtration of the vanishing cycles perverse sheaf and the spectral sequence associated to it. Thanks to the Berkovich-Fargues' theorem, we obtain the de…
We develop the analog of crystalline Dieudonné theory for p-divisible groups in the arithmetic of function fields. In our theory p-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson's abelian t-modules and t-motives, and Drinfeld shtukas. Moreover, we discuss the existence of …
We study tensor powers of rank 1 sign-normalized Drinfeld A-modules, where A is the coordinate ring of an elliptic curve over a finite field. Using the theory of vector valued Anderson generating functions, we give formulas for the coefficients of the logarithm and exponential functions associated to these A-modules. We then show that there exists a vector whose bottom coordinate contains a Goss zeta value, whose evaluation under the exponential function is defined over the Hilbert class field. This allows us to prove the transcendence of Goss zeta values and periods of Drinfeld modules as wel…
We study the group of extensions in the category of Drinfeld modules and Anderson's t-modules, and we show in certain cases that this group can itself be given the structure of a t-module. Our main result is a Drinfeld module analogue of the Weil-Barsotti formula for abelian varieties. Extensions of general t-modules are also considered, in particular extensions of tensor powers of the Carlitz module. We motivate these results from various directions and compare to the situation of elliptic curves.
As a generalization of Drinfeld modules, Greg Anderson introduced abelian t-modules and t-motives over a perfect field. In this article we study relative versions of these over rings. We investigate isogenies among them. Our main results state that every isogeny possesses a dual isogeny in the opposite direction, and that a morphism between abelian t-modules is an isogeny if and only if the corresponding morphism between their associated t-motives is an isogeny. We also study torsion submodules of abelian t-modules which in general are non-reduced group schemes. They can be obtained from the a…
In the arithmetic of function fields Drinfeld modules play the role that elliptic curves take on in the arithmetic of number fields. As higher dimensional generalizations of Drinfeld modules, and as the appropriate analogues of abelian varieties, G. Anderson introduced pure t-motives. In this article we study the arithmetic of the later. We investigate which pure t-motives are semisimple, that is, isogenous to direct sums of simple ones. We give examples for pure t-motives which are not semisimple. Over finite fields the semisimplicity is equivalent to the semisimplicity of the endomorphism al…
Pure t-motives were introduced by G. Anderson as higher dimensional generalizations of Drinfeld modules, and as the appropriate analogs of abelian varieties in the arithmetic of function fields. In order to construct moduli spaces for pure t-motives the second author has previously introduced the concept of abelian τ-sheaf. In this article we clarify the relation between pure t-motives and abelian τ-sheaves. We obtain an equivalence of the respective quasi-isogeny categories. Furthermore, we develop the elementary theory of both structures regarding morphisms, isogenies, Tate modules, and loca…
For arithmetic applications, we extend and refine our results in \cite{YZ} to allow ramifications in a minimal way. Starting with a possibly ramified quadratic extension $F'/F$ of function fields over a finite field in odd characteristic, and a finite set of places $Σ$ of $F$ that are unramified in $F'$, we define a collection of Heegner--Drinfeld cycles on the moduli stack of $\mathrm{PGL}_{2}$-Shtukas with $r$-modifications and Iwahori level structures at places of $Σ$. For a cuspidal automorphic representation $π$ of $\mathrm{PGL}_{2}(\mathbb{A}_{F})$ with square-free level $Σ$, and $r\in\m…
We study algebraic cycles in the moduli space of $\mathrm{PGL}_2$-shtukas, arising from the diagonal torus. Our main result shows that their intersection pairing with the Heegner-Drinfeld cycle is the product of the $r$-th central derivative of an automorphic $L$-function $L(π,s)$ and Waldspurger's toric period integral. When $L(π,\frac12) \neq 0$, this gives a new geometric interpretation for the Taylor series expansion. When $L(π,\frac12) = 0$, the pairing vanishes, suggesting higher order analogues of the vanishing of cusps in the modular Jacobian, as well as other new phenomena. Our proof …
Anderson generating functions have received a growing attention in function field arithmetic in the last years. Despite their introduction by Anderson in the 80s where they were at the heart of comparison isomorphisms, further important applications e.g. to transcendence theory have only been discovered recently. The Anderson-Thakur special function interpolates L-values via Pellarin-type identities, and its values at algebraic elements recover Gauss-Thakur sums, as shown by Anglès and Pellarin. For Drinfeld-Hayes modules, generalizations of Anderson generating functions have been introduced b…
Pink has given a qualitative answer to the Mumford-Tate conjecture for Drinfeld modules in the 90s. He showed that the image of the v-adic Galois representation is v-adically open in the motivic Galois group for any prime v. In contrast to this result, we provide a family of uniformizable Anderson t-modules for which the Galois representations of their t-adic Tate-modules are "far from" having t-adically open image in their motivic Galois groups. Nevertheless, the image is still Zariski-dense in the motivic Galois group which is in accordance to the Mumford-Tate conjecture. For the proof, we e…
Abelian $t$-modules and the dual notion of $t$-motives were introduced by Anderson as a generalization of Drinfeld modules. For such Anderson defined and studied the important concept of uniformizability. It is an interesting question, and the main objective of the present article to see how uniformizability behaves in families. Since uniformizability is an analytic notion, we have to work with families over a rigid analytic base. We provide many basic results, and in fact a large part of this article concentrates on laying foundations for studying the above question. Building on these, we obt…
Elliptic sheaves (which are related to Drinfeld modules) were introduced by Drinfeld and further studied by Laumon--Rapoport--Stuhler and others. They can be viewed as function field analogues of elliptic curves and hence are objects "of dimension 1". Their higher dimensional generalisations are called abelian sheaves. In the analogy between function fields and number fields, abelian sheaves are counterparts of abelian varieties. In this article we study the moduli spaces of abelian sheaves and prove that they are algebraic stacks. We further transfer results of Cerednik--Drinfeld and Rapoport…
Pure t-motives were introduced by G. Anderson as higher dimensional generalizations of Drinfeld modules, and as the appropriate analogs of abelian varieties in the arithmetic of function fields. In this article we develop their theory regarding morphisms, isogenies, Tate modules, and local shtukas. The later are the analog of p-divisible groups. We investigate which pure t-motives are semisimple, that is, isogenous to direct sums of simple ones. We give examples for pure t-motives which are not semisimple. Over finite fields the semisimplicity is equivalent to the semisimplicity of the endomor…
We review the analog of Fontaine's theory of crystalline $p$-adic Galois representations and their classification by weakly admissible filtered isocrystals in the arithmetic of function fields over a finite field. There crystalline Galois representations are replaced by the Tate modules of so-called local shtukas. We prove that the Tate module functor is fully faithful. In addition to this étale realization of a local shtuka we discuss also the de Rham and the crystalline cohomology realizations and construct comparison isomorphisms between these realizations. We explain how local shtukas and …
In 1997 Richard Pink has clarified the concept of Hodge structures over function fields in positive characteristic, which today are called Hodge-Pink structures. They form a neutral Tannakian category over the underlying function field. He has defined Hodge realization functors from the uniformizable abelian $t$-modules and $t$-motives of Greg Anderson to Hodge-Pink structures. This allows one to associate with each uniformizable $t$-motive a Hodge-Pink group, analogous to the Mumford-Tate group of a smooth projective variety over the complex numbers. It further enabled Pink to prove the analo…
