23 problems

Let $\phi$ and $\psi$ be two almost strictly pure $t$-modules over $K$, and let $\phi\otimes\psi$ denote their tensor product. Choose a $K\{\tau\}$-basis of the corresponding $t$-m…

Let $M(A)$ be a non-pure t-motive defined by (1.7), where $A$ is of the form (1.8), and consider the lattice map from this set of t-motives to the set of lattices of rank $3$ in…

Let $M(A)$ be a non-pure t-motive defined by (1.7), with $A$ of the form (1.8), and let the lattice map send $M(A)$ to its lattice. The set of such $M(A)$ has dimension $3$ over…

Let $\underline{M}$ be an Anderson $t$-motive, let $v$ be a finite place, and let $L_v(\underline{M},T)$ be its $v$-adic $L$-series. Write $P_v(\underline{M},T)$ for the polynomial…

Let $x_0,x_1,\dots$ be a holonomic sequence, and let … be the ideal of operators annihilating its associated generating series. Non-principality conjecture. The ideal $I_x$ can be…

Let $M$ be a t-motive of rank $r$, and let $h^1(M)$ denote its first cohomological invariant. Generic $h^1$ conjecture. For almost all t-motives $M$ of rank $r$, one has … This pre…

Galois-representation conjecture. The standard properties of $obreakgoth p$-adic representations should hold for $rho$. The source poses this as a direction for further research a…

Let $M$ be an Anderson $t$-motive, and let the small rank of solutions to the associated system of affine equations be understood as the rank of its module of small solutions. Smal…

Generic maximality conjecture. There exists a subset $\mathcal U$ of $(\mathbb Q\cup\{+\infty\})^4$, complementary to a union of countably many, possibly finitely many, linear subs…

Lattice-map conjecture. The image of this map from the set of pure uniformizable t-motives to the set of lattices is open, and its fibre at a generic point is discrete.

Reduction conjecture. The equations (1.6) and (1.7) can be transformed in this way into an equation of type (2.2a) having the same $r$ and $n$.

Finiteness conjecture for simple equations. For given $r$ and $n$ there are only finitely many such relations; equivalently, “almost all” equations are simple.

Linear-independence conjecture. $Fq[T]$-linearly independent sets of small solutions to (2.1) are $Fq[[T]]$-linearly independent. Moreover, they are linearly independent also ove…

Let $M$ be a pure uniformizable T-motive, and let $L(M)$ denote its associated lattice. Consider the lattice map … from the set of pure uniformizable T-motives to the set of lattic…

Let $K:=\mathbb{F}_{q}(\theta)$ be the rational function field, let $K_{v(\theta)}$ be its completion at the place $v(\theta)$, and let $K(t)_{v}$ and $K^{\mathrm{sep}}[t]_{v}$ be…

Let $K$ be a finite separable extension of $\mathbf{F}_q(t)$, let $A=\mathbf{F}_q[t]$, and let $K_\infty$ and $O_K$ denote the completions and the ring of integers used in the defi…

Let $\goth K$ and $\Phi$ be as in the paper, let $\goth K^{\mathrm{ref}}$ be the reflex field, and assume that there is only one point over $\infty\in\mathbb P^1(F_q)$ in…

Let $M_1$ and $M_2$ be isomorphic standard-3 abelian T-motives of the same type $(\varphi,k,\succ)$, with $\mathbb C_\infty[\tau]$-bases $e_{1*}$ and $e_{2*}$, and write an isomorp…

Let $M$ be a standard-2 abelian T-motive of type $(\varphi,k)$, where $\varphi$ is written as $(\alpha_0+1,\ldots,\alpha_1)\cdots(\alpha_{i-1}+1,\ldots,\alpha_i)$, and let…

A self-dual T-motive is a T-motive admitting a duality with itself. The existence conjecture for other self-dual T-motives. There exist self-dual T-motives $M$, distinct from those…

Let $M$ be the T-motive defined by $Te_* = \theta e_*+A\tau e_*+\tau^2e_*$ with $A\in M_2(\mathbb C_\infty)$, and let $\alpha:M\to M'$ be a constant duality map satisfying…

Let $M$ be the dimension-$n$, rank-$2n$ T-motive defined by $Te_* = \theta e_*+A\tau e_*+\tau^2e_*$, where $A\in M_n(\mathbb C_\infty)$ is skew-symmetric, and let…

Let $M$ be a uniformizable abelian T-motive whose $goth C$-dual $M'$ exists. Let $L(M)$ and $L(M')$ be their lattices, let $goth C$ be a fixed rank-one Anderson T-motive over…