Narrow Results

Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessary and sufficient condition is given for the class group H(K) to contain a cyclic subgroup of any order n, which holds true for both real and imaginary fields K. Then several series of function fields K, including real, inertia imaginary, and ramified imaginary quadratic function fields, are given, for which the class groups H(K) are proved to contain cyclic subgroups of order n.

Algebraic approximation to points in projective spaces offers a new and more flexible approach to algebraic independence theory. When working over the field of algebraic numbers, it leads to open conjectures in higher dimension extending known results in Diophantine approximation. We show here that over the algebraic closure of a function field in one variable, the analog of these conjectures is true. We also derive transfer lemmas which have applications in the study of multiplicity estimates, for example.

We prove a formula for the limit of logarithmic derivatives of zeta functions in families of global fields with an explicit error term. This can be regarded as a rather far reaching generalization of the explicit Brauer–Siegel theorem both for number fields and function fields.

We study the moments of the Dirichlet L-function when defined over the polynomial ring over finite fields. We obtain an asymptotic formula of the fourth moment for the central value of these Dirichlet L-functions. In addition, we find a lower bound for the 2kth moment of these L-functions. These results agree up to constants with the polynomial ring analog of the Keating and Snaith Conjecture for the asymptotic of leading terms.

We study the natural analogue of the Mertens conjecture in the setting of global function fields. Building on the work of Cha, we show that most hyperelliptic curves do not satisfy the Mertens conjecture, but that if we modify the Mertens conjecture to have a larger constant, then this modified conjecture is satisfied by a positive proportion of hyperelliptic curves.

In this paper we apply methods from the number field case of Perrin-Riou [20] and Zábrádi [32] in the function field setup. In ℤ_ℓ- and GL₂-cases (ℓ ≠ p), we prove algebraic functional equations of the Pontryagin dual of Selmer group which give further evidence of the main conjectures of Iwasawa theory. We also prove some parity conjectures in commutative and non-commutative cases. As a consequence, we also get results on the growth behavior of Selmer groups in commutative and non-commutative extension of function fields.

Let ${𝔽}_q[t]$ denote the ring of polynomials over ${𝔽}_q$, the finite field of $q$ elements. We prove an estimate for fractional parts of polynomials over ${𝔽}_q[t]$ satisfying a certain divisibility condition analogous to that of intersective polynomials in the case of integers. We then extend our result to consider linear combinations of such polynomials as well.

There are two types of Belyi’s Theorems for curves defined over finite fields of characteristic $p$, namely the Wild and the Tame $p$-Belyi Theorems. In this paper, we discuss them in the language of function fields. In particular, we provide a constructive proof for the existence of a pseudo-tame element introduced in [Y. Sugiyama and S. Yasuda, Belyi’s theorem in characteristic two, Compos. Math. 156(2) (2020) 325–339], which leads to a self-contained proof for the Tame $2$-Belyi Theorem. Moreover, we provide unified and simple proofs for Belyi’s Theorems unlike the known ones that use technical results from Algebraic Geometry.

Let ${𝔽}_q[t]$ be the polynomial ring over the finite field ${𝔽}_q$ of $q$ elements. For a natural number $N,$ let ${𝔾}_N$ be the set of all polynomials in ${𝔽}_q[t]$ of degree less than $N.$ Let $h$ be a quadratic polynomial over ${𝔽}_q[t].$ Suppose that $h$ is intersective, that is, which satisfies $(A-A)\cap (h({𝔽}_q[t])\setminus \{0\})\ne \varnothing$ for any $A\subseteq {𝔽}_q[t]$ with ${limsup}_{N\to \infty }|A\cap {𝔾}_N|/q^N>0,$ where $A-A$ denotes the difference set of $A.$ Let $B\subseteq {𝔾}_N.$ Suppose that $(B-B)\cap \left(h({𝔽}_q[t])\setminus \{0\}\right)=\varnothing$ and that the characteristic of ${𝔽}_q$ is not divisible by 2. It is proved that $\left|B\right|\le C{N}^{-clogloglogN}{q}^N$ for any $0<c<1/log3,$ where $C\ge 1$ is a constant depending only on $q,h$ and $c.$

In this paper, we study a special holomorphic family (M, π, R) of closed Riemann surfaces of genus two over a fourth punctured torus R, which is a kind of a Kodaira surface and is constructed by Riera. We give two explicit defining equations for (M, π, R) by using elliptic functions, and determine all the holomorphic sections of (M, π, R). Proofs will appear elsewhere.