Related papers: Genus formulas for dormant modular curves and asym…
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…
This manuscript represents an advance in the enumerative geometry of opers that takes the subject beyond our previous work. Motivated by a counting problem of linear differential equations in positive characteristic, we investigate the…
We introduce two new types of towers of Drinfeld modular curves. These towers originate from a specific domain $\mathcal{A} $ and are analogous to the towers of rank-two Drinfeld modular curves over the polynomial ring. Specifically, the…
In this paper, we will give an overview of known and new techniques on how one can obtain explicit equations for candidates of good towers of function fields. The techniques are founded in modular theory (both the classical modular theory…
A dormant oper is a specific type of principal bundle with a flat connection, defined on an algebraic curve in positive characteristic. The moduli spaces of dormant opers and their variants, known as dormant Miura opers, have been studied…
A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion…
The Drinfeld module is a tool of the explicit class field theory for the function fields. We first observe a similarity of such modules with the noncommutative tori, and then use it to develop an explicit class field theory for the number…
We give a general recipe for explicitly constructing asymptotically optimal towers of modular curves such as {X_0(l^n): n=1,2,3,...}. We illustrate the method by giving equations for eight towers with various geometric features. We conclude…
Rank-2 Drinfeld modules are a function-field analogue of elliptic curves, and the purpose of this paper is to investigate similarities and differences between rank-2 Drinfeld modules and elliptic curves in terms of supersingularity.…
We give explicit equations for the simplest towers of Drinfeld modular curves over any finite field, and observe that they coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth.
In the first half of the present paper, we study higher-level generalizations of differential modules in positive characteristic. These objects may be regarded as ring-theoretic counterparts of vector bundles on a curve equipped with an…
The article motivates, presents and describes large computer calculations concerning the asymptotic behaviour of arithmetic properties of coefficient fields of modular forms. The observations suggest certain patterns, which deserve further…
Let $X$ be a smooth, projective curve of genus $g\geq 2$ over an algebraically closed field of characteristic $p>0$. I provide a conjectural formula for the degree of the scheme of dormant ${\rm PGL}(r)$-opers on $X$ where $r\geq 2$ (I…
This paper concerns towers of curves over a finite field with many rational points, following Garcia--Stichtenoth and Elkies. We present a new method to produce such towers. A key ingredient is the study of algebraic solutions to Fuchsian…
This thesis is concerned with the application of operadic methods, particularly modular operads, to questions arising in the study of moduli spaces of surfaces as well as applications to the study of homotopy algebras and new constructions…
In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable curves in positive characteristic. It shows that the topological structures of moduli spaces of curves can be understood from the…
We introduce normalized Drinfeld modular curves that parameterize rank $m$ Drinfeld modules compatible with a $T$-torsion structure arising from a given conjugacy class of nilpotent upper-triangular $n\times n$ matrices with rank $\geqslant…
A $\mathfrak{g}$-oper for a simple Lie algebra $\mathfrak{g}$ is a specific type of flat principal bundle on an algebraic curve. When the base field is of prime characteristic $p$, those with vanishing $p$-curvature are called dormant…
We study the moduli space of meromorphic 1-forms on complex algebraic curves having at most simple poles with fixed nonzero residues. We interpret the Bergman tau function on this moduli space as a section of a line bundle and study its…
We focus on combinatorial aspects of the Hilbert series of the cohomology ring of the moduli space of stable pointed curves of genus zero. We show its graded Hilbert series satisfies an integral operator identity. This is used to give…