This is a survey on Anderson t-motives -- high-dimensional generalizations of Drinfeld modules. They are the functional field analogs of abelian varieties with multiplication by an imaginary quadratic field. We describe their lattices, their groups $H^1$ and $H_1$, their tensor products, the duality functor and the duality theorem, endomorphisms of Drinfeld modules in finite characteristic, and their L-functions of a certain type. Further on, we introduce the notion of affine equations, $T$-divisible $\Bbb F_q[[T]]$-modules, holonomic sequences in the functional field case, analogs of Siegel matrices as elements of flag varieties, and some other notions (to be continued). Many examples of explicit calculations are given, some elementary research problems are stated. Some results (Sections 16; 19) are apparently new.
We consider Anderson t-motives $M$ of dimension 2 and rank 4 defined by some simple explicit equations parameterized by $2\times2$ matrices. We use methods of explicit calculation of $h^1(M)$ -- the dimension of their cohomology group $H^1(M)$ ( = the dimension of the lattice of their dual t-motive $M'$) developed in our earlier paper. We calculate $h^1(M)$ for $M$ defined by all matrices having 0 on the diagonal, and by some other matrices. These methods permit to make analogous calculations for most (probably all) t-motives. $h^1$ of all Anderson t-motives $M$ under consideration satisfy the…
Let $M$ be an Anderson t-motive of dimension $n$ and rank $r$. Associated are two $\Bbb F_q[T]$-modules $H^1(M)$, $H_1(M)$ of dimensions $h^1(M)$, $h_1(M)\le r$ - analogs of $H^1(A,\Bbb Z)$, $H_1(A,\Bbb Z)$ for an abelian variety $A$. There is a theorem (Anderson): $h^1(M)=r \iff h_1(M)=r$; in this case $M$ is called uniformizable. It is natural to expect that always $h^1(M)=h_1(M)$. Nevertheless, we explicitly construct a counterexample. Further, we answer a question of D.Goss: is it possible that two Anderson t-motives that differ only by a nilpotent operator $N$ are of different uniformizab…
Analogy between Anderson T-motives and abelian varieties with multiplication by an imaginary quadratic field (MIQF) is a source of 2 results: 1. A description of abelian varieties with MIQF of dimension $r$ and signature $(n, r-n)$ in terms of "lattices" of dimension $r$ in $\Bbb C^n$; 2. A construction of exterior powers of abelian varieties with MIQF having $n=1$.
G. Anderson作为Drinfeld模的高维推广，并作为函数域算术中阿贝尔簇的适当类似物。为了构造纯t-模的模空间，第二作者在前面引入了交换τ-层的概念。本文阐明了纯t-基模与交换τ-层之间的关系。我们得到了相应的拟等价范畴。此外，我们发展的基本理论，这两种结构的态射，同源，泰特模块，和当地shtukas。后者是p-可除群的类似物。
Pure Anderson Motives and Abelian Sheaves over Finite Fields
G. Anderson作为Drinfeld模的高维推广，并作为函数域算术中阿贝尔簇的适当类似物。在这篇文章中，我们发展他们的理论，关于态射，同源，泰特模，和当地shtukas。后者类似于p-可除群。我们调查纯t动机是半单的，也就是说，同源简单的直接和。我们给出的例子，纯t-动机是不是半单。在有限域上，半单性等价于自同态代数的半单性，但在无限域上也不成立。仍然在有限域上，我们研究了纯t-动机的自同态环，并得到了类似于Tate关于阿贝尔簇的著名结果。最后，我们澄清了纯t-模与第二作者为了构造模空间而引入的交换τ-层的关系。我们得到了相应的拟等价范畴。
In this article we introduce and study a motivic category in the arithmetic of function fields, namely the category of motives over an algebraic closure $L$ of a finite field with coefficients in a global function field over this finite field. It is semi-simple, non-neutral Tannakian and possesses all the expected fiber functors. This category generalizes the previous construction due to Anderson and is more relevant for applications to the theory of $G$-Shtukas, such as formulating the analog of the Langlands-Rapoport conjecture over function fields. We further develop the analogy with the ca…
提出了一个关于正特征系数函数域中$ L $ -函数的特殊值的猜想。对于$ M $，一个处处好约化的可一致化$ t $ -模，我们猜想Goss $ L $ -函数$ L(M^\vee, s) $在$ s = 0 $的值与$ M $相关的阿贝尔$ t $ -模的一致化之间的关系.当$ M $是Carlitz $ t $ -动机的一个幂时，该猜想专门化为Anderson和Thakur关于Carlitz zeta值的一个定理。除此之外，我们还提供了数字证据。
The integral $t$-motivic cohomology and the class module of a (rigid analytically trivial) Anderson $t$-motive were introduced by the first author in [Gaz22b]. This paper is devoted to their determination in the particular case of tensor powers of the Carlitz $t$-motive, namely, the function field counterpart $\underline{A}(n)$ of Tate twists $\mathbb{Z}(n)$. We find out that these modules are in relation with fundamental objects of function field arithmetic: integral $t$-motivic cohomology governs linear relations among Carlitz polylogarithms, its torsion is expressed in terms of the denomina…
We survey Colmez's theory and conjecture about the Faltings height and a product formula for the periods of abelian varieties with complex multiplication, along with the function field analog developed by the authors. In this analog, abelian varieties are replaced by Drinfeld modules and $A$-motives. We also explain the necessary background on abelian varieties, Drinfeld modules and $A$-motives, including their cohomology theories and comparison isomorphisms and their theory of complex multiplication.
