Related papers: Counting Drinfeld modules with prescribed local co…

Let $p$ be a rational prime and $q$ a power of $p$. Let $\wp$ be a monic irreducible polynomial of degree $d$ in $\mathbf{F}_q[t]$. In this paper, we define an analogue of the Hodge-Tate map which is suitable for the study of Drinfeld…

In this short note, we answer a question raised by M. Papikian on a universal upper bound for the degree of the extension of $K_\infty$ given by adjoining the periods of a Drinfeld module of rank 2. We show that contrary to the rank 1 case…

Let $k$ be a field containing $\mathbb{F}_q$. Let $\psi$ be a rank $r$ Drinfeld $\mathbb{F}_q[t]$-module determined by $\psi_t(X) = tX+a_1X^q+\cdots+a_{r-1}X^{q^{r-1}}+X^{q^r}$, where $t,a_1,\ldots,a_{r-1}$ are algebraically independent…

Fix a nonzero level $\mathfrak{n} \in \mathbb{F}_q[T]$. In this paper, we first establish a function field analogue of Ligozat's theorem, which serves as our main result and provides a criterion for Drinfeld modular units on the Drinfeld…

We consider the question of determining the maximum number of $\mathbb{F}_q$-rational points that can lie on a hypersurface of a given degree in a weighted projective space over the finite field $\mathbb{F}_q$, or in other words, the…

We determine the asymptotic proportion of free modules over finite chain rings with good distance properties and treat the asymptotics in the code length n and the residue field size q separately. We then specialize and apply our technique…

We construct a two-level weighted TQFT whose structure coefficents are equivariant intersection numbers on moduli spaces of admissible covers. Such a structure is parallel (and strictly related) to the local Gromov-Witten theory of curves…

Let $\mathbb{F}_q$ be the finite field with $q\geq 5$ elements, $A:=\mathbb{F}_q[T]$ and $F:=\mathbb{F}_q(T)$. Assume that $q$ is odd and take $|\cdot|$ to be the absolute value at $\infty$ that is normalized by $|T|=q$. Given a pair…

We derive constraints on the existence of walls for Bridgeland stability conditions for general projective surfaces. We show that in suitable planes of stability conditions the walls are bounded and derive conditions for when the number of…

Suppose we are given a Drinfeld Module $\phi$ over $\mathbb{F}_q(t)$ of rank $r$ and a prime ideal $\mathfrak{l}$ of $\mathbb{F}_q[T]$. In this paper, we prove that the reducibility of mod $\mathfrak{l}$ Galois representation…

Towers of algebraic function fields over finite fields play a fundamental role in arithmetic geometry and coding theory. Classical examples arising from modular and Drinfeld modular curves exhibit asymptotically good behavior. In this…

Let $\mathbb{F}_q[T]$ be the polynomial ring over a finite field $\mathbb{F}_q$. We study the endomorphism rings of Drinfeld $\mathbb{F}_q[T]$-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings…

We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well as a general local duality theorem which extends, to a much broader class of rings,…

We prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space. (Nous \'etablissons un d\'eveloppement asymptotique du nombre de points rationnels de hauteur…

In this paper we study the problem of prescribing Dirichlet eigenvalues on an arbitrary compact manifold $M$ of dimension $n\geq 3$ with a non-empty smooth boundary $\partial M$. We show that for any finite increasing sequence of real…

We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by…

We provide explicit series expansions for the exponential and logarithm functions attached to a rank r Drinfeld module that generalize well known formulas for the Carlitz exponential and logarithm. Using these results we obtain a procedure…

In this paper we investigate examples of good and optimal Drinfeld modular towers of function fields. Surprisingly, the optimality of these towers has not been investigated in full detail in the literature. We also give an algorithmic…

Let $k$ be a number field and $K$ a finite extension of $k$. We count points of bounded height in projective space over the field $K$ generating the extension $K/k$. As the height gets large we derive asymptotic estimates with a…

We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.