Related papers: Drinfeld-Stuhler modules and the Hasse principle

This is a sequel to the paper [F. Breuer, H.-G. R\"uck, 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.…

After a brief introduction to the classical theory of binary quadratic forms we use these results for proving (most of) the claims made by P\'epin in a series of articles on unsolvable quartic diophantine equations, and for constructing…

We give a geometric proof that Hasse principle holds for the following varieties defined over global function fields: smooth quadric hypersurfaces in odd characteristic, smooth cubic hypersurfaces of dimension at least $4$ in characteristic…

We introduce a method for associating a chain complex to a module over a combinatorial category, such that if the complex is exact then the module has a rational Hilbert series. We prove homology--vanishing theorems for these complexes for…

We propose a conjecture, similar to Skolem's conjecture, on a Hasse-type principle for exponential diophantine equations. We prove that in a sense the principle is valid for "almost all" equations. Based upon this we propose a general…

One of the ultimate goals of the Hassett-Keel program is the determination of the log canonical models of the moduli spaces of pointed rational curves $\overline{M}_{0,n}$. In this paper, we study log canonical models of…

In the present paper, we introduce the notion of nearly holomorphic Drinfeld modular forms and study an analogue of Maass-Shimura operators in this context. Furthermore, for a given nearly holomorphic Drinfeld modular form, we show that its…

Let $F$ be a function field over $\mathbb{F}_q$, $A$ its ring of regular functions outside a place $\infty$ and $\mathfrak{p}$ a prime ideal of $A$. First, we develop Hida theory for Drinfeld modular forms of rank $r$ which are of slope…

Let $\mathbb{F}_q$ be the finite field with $q$ elements and consider the rational function field $K:=\mathbb{F}_q(\theta)$. For a Drinfeld module $\phi$ defined over $K$, we study the transcendence of special values of the Goss…

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…

We study the family of irreducible modules for quantum affine $\lie{sl}_{n+1}$ whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique…

$\Phi $ be a Drinfeld $\mathbf{F}\_{q}[T]$-module of rank 2, over a finite field $L$. Let $P\_{\Phi}(X)=$ $X^{2}-cX+\mu P^{m}$ ($c$ an element of $\mathbf{F}\_{q}[T],$ $\mu $ be a non-vanishing element of $% \mathbf{F}\_{q}$, $m$ the degree…

Using Koll\'ar's semipositivity results, we produce a number of nef and ample tautological divisors on Hassett's spaces of weighted stable pointed curves. As an application, we prove that Hassett's spaces are log canonical models of…

Twisted $L$-functions by Dirichlet characters offer deep insights into arithmetic geometry, especially in the study of elliptic curves and abelian varieties over number fields. In the function field setting, Drinfeld modules and Anderson…

We describe a practical algorithm for computing Brauer-Manin obstructions to the existence of rational points on hyperelliptic curves defined over number fields. This offers advantages over descent based methods in that its correctness does…

We study isogeny classes of Drinfeld $A$-modules over finite fields $k$ with commutative endomorphism algebra $D$, in order to describe the isomorphism classes in a fixed isogeny class. We study when the minimal order $A[\pi]$ of $D$ occurs…

We give an equivalent condition for the validity of the Hasse norm principle for finite separable extensions of prime squared degree of global fields. Our theorem recovers the result of Drakokhrust--Platonov, which claims that the Hasse…

We suggest that the principle of holographic duality can be extended beyond conformal invariance and AdS isometry. Such an extension is based on a special relation between functional determinants of the operators acting in the bulk and on…

Ch\^atelet surfaces provide a rich source of geometrically rational surfaces which do not always satisfy the Hasse principle. Restricting attention to a special class of Ch\^atelet surfaces, we investigate the frequency that such…

In this paper we generalise the notion of Drinfeld modular form for the group $\Gamma$ := GL2(Fq[$\theta$]) to a vector-valued setting, where the target spaces are certain modules over positive characteristic Banach algebras over which are…