Related papers: Kummer Theory for Drinfeld Modules

Let $K$ be an algebraic function field with constant field ${\mathbb F}_q$. Fix a place $\infty$ of $K$ of degree $\delta$ and let $A$ be the ring of elements of $K$ that are integral outside $\infty$. We give an explicit description of the…

Let $p$ be a prime number and $F$ a totally real number field. For each prime $\mathfrak{p}$ of $F$ above $p$ we construct a Hecke operator $T_\mathfrak{p}$ acting on $(\mathrm{mod}\, p^m)$ Katz Hilbert modular classes which agrees with the…

Let K be an imaginary quadratic field of discriminant -D_K<0. We introduce a notion of an adelic Maass space S_{k, -k/2}^M for automorphic forms on the quasi-split unitary group U(2,2) associated with K and prove that it is stable under the…

Let $Y$ be an abelian variety over a subfield $k \subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford-Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serre…

We construct the moduli stack of torsors over the formal punctured disk in characteristic p > 0 for a finite group isomorphic to the semidirect product of a p-group and a tame cyclic group. We prove that the stack is a limit of separated…

Given a finite category T, we consider the functor category [T,A], where A can in particular be any quasi-abelian category. Examples of quasi-abelian categories are given by any abelian category but also by non-exact additive categories as…

Let $q$ be a power of the prime number $p$, let $K={\mathbb F}_q(t)$, and let $r\ge 2$ be an integer. For points ${\mathbf a}, {\mathbf b}\in K$ which are $\mathbb{F}_q$-linearly independent, we show that there exist positive constants…

We discuss lifting properties of continuous homomorphisms from absolute Galois groups into (pro)finite groups. An analogy with the Langlands program is pointed out in the beginning of the note.

Let $A$ be the ring of elements in an algebraic function field $K$ over $\mathbb{F}_q$ which are integral outside a fixed place $\infty$. In contrast to the classical modular group $SL_2(\mathbb{Z})$ and the Bianchi groups, the {\it…

Let K be a finite extension of Q_p with residue field F_q and let P(T) = T^d + a_{d-1}T^{d-1} + ... +a_1 T, where d is a power of q and a_i is in the maximal ideal of K for all i. Let u_0 be a uniformizer of O_K and let {u_n}_{n \geq 0} be…

If K/F is a finite abelian Galois extension of global fields whose Galois group has exponent t, we prove that there exists a short exact sequence that has as a consequence that if t is square free, then Dec(K/F)=Br_{t}(K/F) which we use to…

Let $\mathbf K$ be a finite field, $X$ and $Y$ two curves over $\mathbf K$, and $Y\rightarrow X$ an unramified abelian cover with Galois group $G$. Let $D$ be a divisor on $X$ and $E$ its pullback on $Y$. Under mild conditions the linear…

We prove that for any infinite countable amenable group $G$, any $\epsilon > 0$ and any finite subset $K\subset G$, there exists a tiling (partition of $G$ into finite "tiles" using only finitely many "shapes"), where all the tiles are $(K;…

For a number field $F$ and an odd prime number $p,$ let $\tilde{F}$ be the compositum of all $\mathbb{Z}_p$-extensions of $F$ and $\tilde{\Lambda}$ the associated Iwasawa algebra. Let $G_{S}(\tilde{F})$ be the Galois group over $\tilde{F}$…

Let $X$ be an integral affine or projective hypersurface over a field $F$ of characteristic $p>0$, and let $F(X)$ denote its function field. In a recent article, Dolphin and Hoffmann obtained an explicit description of the kernel of the…

Let (k1,k2,k3,k4) be a quartet of cyclic cubic number fields sharing a common conductor c=pqr divisible by exactly three prime(power)s p,q,r. For those components k of the quartet whose 3-class group Cl(3,k) = Z/3Z x Z/3Z is elementary…

Following the philosophy of arithmetic topology, we describe a point of view which helps look at surfaces and $p$-adic fields in a "uniform way", and show that results on mapping class groups can be extended to this point of view, and thus…

Let $A$ be an absolutely simple abelian variety without (potential) complex multiplication, defined over the number field $K$. Suppose that either $\dim A=2$ or $A$ is of $\operatorname{GL}_2$-type: we give an explicit bound $\ell_0(A,K)$…

Let G be a compact p-adic analytic group. We study K-theoretic questions related to the representation theory of the completed group algebra kG of G with coefficients in a finite field k of characteristic p. We show that if M is a finitely…

Galois cohomology groups $H^i(K,M)$ are widely used in algebraic number theory, in such contexts as Selmer groups of elliptic curves, Brauer groups of fields, class field theory, and Iwasawa theory. The standard construction of these groups…