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Let f \colon X \to B be a nonisotrivial complex elliptic surface and let \mathcal {D} \subset X be an integral divisor dominating B. We study finiteness related properties of generalized (S, \mathcal {D})-integral sections \sigma \colon B \to X of X where S \subset B is arbitrary. These integral sections \sigma are algebraic and satisfy the geometric condition f (\sigma (B) \cap \mathcal {D})\subset S. For S \subset B finite, the set of (S, \mathcal {D})-integral sections of X is finite by the well-known Siegel theorem. In this article, we establish a general quantitative finiteness result for large unions of (S, \mathcal {D})-integral sections in which both the subset S and the divisor \mathcal {D} are allowed to vary in families where notably S is not necessarily finite or countable. Some applications to generalized unit equations over function fields are also obtained. [ABSTRACT FROM AUTHOR]

If K/k is a function field in one variable of positive characteristic, we describe a general algorithm to factor one-variable polynomials with coefficients in K. The algorithm is flexible enough to find factors subject to additional restrictions, e.g., to find all roots that belong to a given finite dimensional k-subspace of K, more efficiently. For bounded characteristic, it runs in polynomial time, relative to factorizations over the constant field k and also provides a deterministic polynomial time irreducibility test. We also discuss applications to places of reducible reduction, when k is a global field, and to list decoding of Reed-Solomon codes. [ABSTRACT FROM AUTHOR]

Let f (x) and g (x) be polynomials in $\mathbb F_{2}[x]$ with ${\rm deg}\text{ } f=n$. It is shown that for $n\gg 1$ , there is an $g_{1}(x)\in \mathbb F_{2}[x]$ with ${\rm deg}\text{ } g_{1}\leqslant \max\{{\rm deg}\text{ } g, 6.7\log n\}$ and $g(x)-g_{1}(x)$ having $ \lt 6.7\log n$ terms such that $\gcd(f(x), g_{1}(x))=1$. As an application, it is established using a result of Dubickas and Sha that given $f(x)\in \mathbb F_{2}[x]$ of degree $n\geqslant 1$ , there is a separable $g(x)\in 2[x]$ with ${\rm deg}\text{ } g= {\rm deg}\text{ } f$ and satisfying that $f(x)-g(x)$ has $\leqslant 6.7\log n$ terms. As a simple consequence, the latter result holds in $\mathbb Z[x]$ after replacing 'number of terms' by the L 1-norm of a polynomial and $6.7\log n$ by $6.8\log n$. This improves the bound $(\log n)^{\log 4 +\operatorname{\varepsilon}}$ obtained by Filaseta and Moy. [ABSTRACT FROM AUTHOR]

We establish the quantum unique ergodicity conjecture for Eisenstein series over function fields in the level aspect. Adapting the machinery of Luo and Sarnak (1995), we employ the spectral decomposition and handle the cuspidal and Eisenstein contributions separately. [ABSTRACT FROM AUTHOR]

Let $G$ be a split semisimple group over a global function field $K$. Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$ , there is a cyclic base change lifting of $\Pi$ along any $\mathbb {Z}/\ell \mathbb {Z}$ -extension of $K$. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$ , and almost all primes $\ell$ , any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb {Z}/\ell \mathbb {Z}$ -extension of $F$. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi. [ABSTRACT FROM AUTHOR]

In this paper, we show that ∞-adic multiple zeta values associated to the function field of an algebraic curve of higher genus over a finite field are not zero, under certain assumption on the gap sequence associated to the rational point ∞ on the given curve. Using arguments and results of Sheats and Thakur for the case of the projective line, we calculate the absolute values of power sums in the series defining multiple zeta values, and show that the calculation implies the non-vanishing result. [ABSTRACT FROM AUTHOR]

We give an axiomatic characterization of multiple Dirichlet series over the function field F q (T) , generalizing a set of axioms given by Diaconu and Pasol. The key axiom, relating the coefficients at prime powers to sums of the coefficients, formalizes an observation of Chinta. The existence of multiple Dirichlet series satisfying these axioms is proved by exhibiting the coefficients as trace functions of explicit perverse sheaves and using properties of perverse sheaves. The multiple Dirichlet series defined this way include, as special cases, many that have appeared previously in the literature. [ABSTRACT FROM AUTHOR]

We prove an \Omega-result for the quadratic Dirichlet L-function |L(1/2, \chi _P)| over irreducible polynomials P associated with the hyperelliptic curve of genus g over a fixed finite field \mathbb {F}_q in the large genus limit. In particular, we showed that for any \epsilon \in (0, 1/2), \[ \max _{\substack {P\in \mathcal {P}_{2g+1}}}|L(1/2, \chi _P)|\gg \exp \left (\left (\sqrt {\left (1/2-\epsilon \right)\ln q}+o(1)\right)\sqrt {\frac {g \ln _2 g}{\ln g}}\right), \] where \mathcal {P}_{2g+1} is the set of all monic irreducible polynomials of degree 2g+1. This matches with the order of magnitude of the Bondarenko–Seip bound. [ABSTRACT FROM AUTHOR]

Keating and Rudnick [The variance of the number of prime polynomials in short intervals and in residue classes, Int. Math. Res. Not.2014(1) (2014) 259–288] derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions and short intervals. Keating and Rudnick used an involution to translate short intervals into arithmetic progressions. We follow their approach but apply this involution, in addition, to the arithmetic progressions. This creates dual arithmetic progressions in the case when the modulus Q is a polynomial in q [ T ] such that Q (0) ≠ 0. The latter is a restriction which we keep throughout our paper. At the end, we discuss what is needed to relax this condition. [ABSTRACT FROM AUTHOR]

In this paper, we establish the expected order of magnitude of the k th-moment of central values of the family of Dirichlet L -functions to a fixed prime modulus over function fields for all real k ≥ 0. [ABSTRACT FROM AUTHOR]

In this paper, we study a Drinfeld module analogue of the Uniform Boundedness Conjecture on the torsion of abelian varieties. As a result, we prove the -primary Uniform Boundedness Conjecture for one-dimensional families of Drinfeld modules of arbitrary rank, which extends a result of Poonen. This result can be regarded as a Drinfeld module analogue of the Cadoret–Tamagawa's result on the p -primary Uniform Boundedness Conjecture for one-dimensional families of abelian varieties. [ABSTRACT FROM AUTHOR]

We compute the one-level density of zeros of order ℓ Dirichlet L -functions over function fields F q [ t ] for ℓ = 3 , 4 in the Kummer setting (q ≡ 1 (mod ℓ)) and for ℓ = 3 , 4 , 6 in the non-Kummer setting (q ≢ 1 (mod ℓ)). In each case, we obtain a main term predicted by Random Matrix Theory (RMT) and lower order terms not predicted by RMT. We also confirm the symmetry type of the families is unitary, supporting Katz and Sarnak's philosophy. For a video summary of this paper, please visit https://youtu.be/0tGmXHvVKF0. [ABSTRACT FROM AUTHOR]

We prove an exact formula for the variance of the divisor function over short intervals in A : = F q [ T ] , where q is a prime power; and for correlations of the form d (A) d (A + B) , where we average both A and B over certain intervals in A . We also obtain an exact formula for correlations of the form d (K Q + N) d (N) , where Q is prime and K and N are averaged over certain intervals with deg N ≤ deg Q - 1 ≤ deg K ; and we demonstrate that d (K Q + N) and d(N) are uncorrelated. We generalize our results to σ z defined by σ z (A) : = ∑ E ∣ A | A | z for all monics A ∈ A . Our approach is to use the orthogonality relations of additive characters on F q to translate the problems to ones involving the ranks of Hankel matrices over F q . We prove several results regarding the rank and kernel structure of these matrices, thus demonstrating their number-theoretic properties. We also discuss extending our method to other divisor sums, such as those involving d k . [ABSTRACT FROM AUTHOR]

For B ∈ F q [ T ] of degree 2 n ≥ 2 , consider the number of ways of writing B = E 2 + γ F 2 , where γ ∈ F q ⁎ is fixed, and E , F ∈ F q [ T ] with deg E = n and deg F = m < n. We denote this restricted sum-of-squares function by S γ ; m (B). We obtain an exact formula for the variance of S γ ; m (B) over short intervals in F q [ T ] where q is an odd prime power. Interestingly, when m + 1 ≤ n ≤ 2 m , the variance vanishes on some (not all) short intervals, which is in contrast to the standard function that counts representations as a sum of two squares. We use the method of additive characters and Hankel matrices that we previously used for the variance and correlations of the divisor function. In Section 2, we give a short overview of our approach. Section 3 gives new results relating to values of quadratic forms, over F q , that are defined from Hankel matrices. [ABSTRACT FROM AUTHOR]

A divisibility sequence is a sequence of integers { d n } such that d m divides d n if m divides n. Results of Bugeaud, Corvaja, Zannier, among others, have shown that the gcd of two divisibility sequences corresponding to subgroups of the multiplicative group grows in a controlled way. Silverman conjectured that a similar behaviour should appear in many algebraic groups. We extend results by Ghioca–Hsia–Tucker and Silverman for elliptic curves and prove an analogue of Silverman's conjecture over function fields for abelian and split semiabelian varieties and some generalizations of this result. We employ tools coming from the theory of unlikely intersections as well as properties of the so-called Betti map associated to a section of an abelian scheme. [ABSTRACT FROM AUTHOR]

In order to consider j-wise relative r-primality conditions that do not necessarily require all j-tuples of elements in a Dedekind domain to be relatively r-prime, we define the notion of j-wise relative r-primality with respect to a fixed j-uniform hypergraph H. This allows us to provide further generalisations to several results on natural densities not only for a ring of algebraic integers O, but also for the ring Fq[x]. [ABSTRACT FROM AUTHOR]

We show that for a variety which admits a quasi-finite period map, finiteness (resp. non-Zariski-density) of S-integral points implies finiteness (resp. non-Zariski-density) of points over all Z -finitely generated integral domains of characteristic zero. Our proofs rely on foundational results in Hodge theory due to Deligne, Griffiths, and Schmid, and Bakker-Brunebarbe-Tsimerman. We give straightforward applications to arithmetic locally symmetric varieties, the moduli space of smooth hypersurfaces in projective space, and the moduli of smooth divisors in an abelian variety. [ABSTRACT FROM AUTHOR]

We point out an error in [2]. [ABSTRACT FROM AUTHOR]

This work addresses the prime number races for non-constant elliptic curves E over function fields. We prove that if \mathrm {rank}(E) > 0, then there exist Chebyshev biases towards being negative, and otherwise there exist Chebyshev biases towards being positive. The key input is the convergence of the partial Euler product at the centre, which follows from the Deep Riemann Hypothesis over function fields. [ABSTRACT FROM AUTHOR]

In this paper we study the problem of constructing non-trivial subtowers and supertowers of recursive towers of function fields over finite fields. [ABSTRACT FROM AUTHOR]

In this paper, we pursue positive characteristic analogs of the results of Furusho, Komori, Matsumoto, and Tsumura on p -adic multiple L -functions. We consider ∞ -adic and v -adic multiple zeta functions concerned by Anglès, Ngo Dac, and Tavares Ribeiro. Our main results in this paper consist of: (1) integral expressions of special values of ∞ -adic multiple zeta functions, (2) integral expressions of v -adic multiple zeta functions themselves that are similar to those of p -adic multiple L -functions, (3) Kummer-type congruences for the special values of v -adic multiple zeta functions at integers, (4) relationships between special values of ∞ -adic and those of v -adic multiple zeta functions at negative integers, and (5) orthogonal properties of multiple zeta functions and multiple zeta star functions. [ABSTRACT FROM AUTHOR]

We study inhomogeneous Diophantine approximation over the completion K v of a function field K (over a finite field) for a discrete valuation v , with affine algebra R v. We obtain an effective upper bound for the Hausdorff dimension of the set Bad A (ϵ) = { θ ∈ K v m : lim inf (p , q) ∈ R v m × R v n , ‖ q ‖ → ∞ lim inf (p , q) ∈ R v m × R v n , ‖ q ‖ → ∞ ‖ q ‖ n ‖ A q − θ − p ‖ m ⩾ ϵ } , of ϵ -badly approximable targets θ ∈ K v m given A ∈ M m , n (K v) , using an effective version of entropy rigidity in homogeneous dynamics for some diagonal action on the space of R v -grids. We characterize matrices A for which Bad A (ϵ) has full Hausdorff dimension for some ϵ > 0 by a Diophantine condition of singularity on average. Our methods work for the approximation using weighted ultrametric distances. [ABSTRACT FROM AUTHOR]

In this article, we study the logarithm of the central value L 1 2 , χ D in the symplectic family of Dirichlet L-functions associated with the hyperelliptic curve of genus g over a fixed finite field F q in the limit as g → ∞ . Unconditionally, we show that the distribution of log | L 1 2 , χ D | is asymptotically bounded above by the full Gaussian distribution of mean 1 2 log deg (D) and variance log deg (D) , and also log | L 1 2 , χ D | is at least 94.27 % Gaussian distributed. Assuming a mild condition on the distribution of the low-lying zeros in this family, we obtain the full Gaussian distribution. [ABSTRACT FROM AUTHOR]

Let F q 2 be the finite field with q 2 elements. We provide a simple and effective method, using reciprocal polynomials, for the construction of algebraic curves over F q 2 with many rational points. The curves constructed are Kummer covers or fiber products of Kummer covers of the projective line. Further, we compute the exact number of rational points for some of the curves. [ABSTRACT FROM AUTHOR]

We use the theory of Carlitz modules to define a sequence of linear polynomials L n , ϕ , n ∈ N , that have properties similar to those satisfied by the classical Laguerre polynomials. We prove in particular that for any positive integer n, the Galois group over F q (T) of L n , ϕ is the general linear group G L n (F q) . This gives us explicit polynomials realizing G L n (F q) as a Galois group over F q (T) . [ABSTRACT FROM AUTHOR]

We consider the expected value of Tr (Θ C n) where C runs over a thin family of r -cyclic covers of ℙ q 1 for any r. We obtain many lower order terms dependent on the divisors of r. We use these results to calculate the one-level density of the family and hypothesize a refined one-level density result. [ABSTRACT FROM AUTHOR]

We fix an elliptic curve E / F q (t) and consider the family { E ⊗ χ D } of E twisted by quadratic Dirichlet characters. The one-level density of their L -functions is shown to follow orthogonal symmetry for test functions with Fourier transform supported inside (− 1 , 1). As an application, we obtain an upper bound of 3/2 on the average analytic rank. By splitting the family according to the sign of the functional equation, we obtain that at least 12.5% of the family have rank zero, and at least 37.5% have rank one. The Katz and Sarnak philosophy predicts that those percentages should both be 50% and that the average analytic rank should be 1/2. We finish by computing the one-level density of E twisted by Dirichlet characters of order ℓ ≠ 2 coprime to q. We obtain a restriction of (− 1 / 2 , 1 / 2) on the support with a unitary symmetry. [ABSTRACT FROM AUTHOR]

In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence pα when α is a fixed irrational real number and p runs over the primes. In particular, he showed that the inequality | | p α | | ≤ p − 1 / 5 + ε has infinitely prime solutions p , where | |. | | denotes the distance to a nearest integer. This result has subsequently been improved by many authors. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality | | p α | | ≤ p − 1 / 3 + ε. This exponent 1/3 is considered the limit of the current technology. We prove function field analogues of this result for the fields k = F q (T) and imaginary quadratic extensions K of k. Essential in our method is the Dirichlet approximation theorem for function fields which is established in general form in the appendix authored by Arijit Ganguly. [ABSTRACT FROM AUTHOR]

We reduce the classification of finite extensions of function fields (of curves over finite fields) with the same class number to a finite computation; complete this computation in all cases except when both curves have base field F 2 and genus > 1 ; and give a conjectural answer in the remaining cases. The conjecture will be resolved in subsequent papers. [ABSTRACT FROM AUTHOR]

In this article, we introduce the extended Eulerian numbers for a large class of zeta functions, which includes the zeta functions associated to function fields, and to schemes over finite fields. This construction generalizes the extended Eulerian numbers defined by Carlitz. We give an asymptotic expansion for the summatory function associated to these numbers. Our main result generalizes the well known result on the asymptotic behavior of the extended Eulerian numbers associated to the Riemann zeta function. [ABSTRACT FROM AUTHOR]

We investigate in this paper the vanishing at s = 1 of the twisted L-functions of elliptic curves E defined over the rational function field F q (t) (where F q is a finite field of q elements and characteristic ≥ 5 ) for twists by Dirichlet characters of prime order ℓ ≥ 3 , from both a theoretical and numerical point of view. In the case of number fields, it is predicted that such vanishing is a very rare event, and our numerical data seems to indicate that this is also the case over function fields for non-constant curves. For constant curves, we adapt the techniques of Li (J Number Theory 191:85–103, 2018) and Donepudi and Li (Rocky Mountain J Math 51(5):1615–1628, 2021) who proved vanishing at s = 1 / 2 for infinitely many Dirichlet L-functions over F q (t) based on the existence of one, and we can prove that if there is one χ 0 such that L (E , χ 0 , 1) = 0 , then there are infinitely many. Finally, we provide some examples which show that twisted L-functions of constant elliptic curves over F q (t) behave differently than the general ones. [ABSTRACT FROM AUTHOR]

We consider modules generated by torsion elements of a fixed normalized rank one Drinfeld module of generic characteristic. The behavior of these modules seems to be governed by a vector space whose dimension is bounded by the genus of the ground field. [ABSTRACT FROM AUTHOR]

A (projective, geometrically irreducible, non-singular) curve X defined over a finite field F q 2 is maximal if the number N q 2 of its F q 2 -rational points attains the Hasse-Weil upper bound, that is N q 2 = q 2 + 2 g q + 1 where g is the genus of X. An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order p 2 where p is the characteristic of F q 2 . Doing so we also determine the F q 2 -isomorphism classes of such curves and describe their full F q 2 -automorphism groups. [ABSTRACT FROM AUTHOR]

We prove a lower and an upper bound for the large sieve with square moduli in function fields. These bounds correspond to bounds for the classical large sieve with square moduli established in [2] and [5]. Our lower bound in the function field setting contradicts an upper bound obtained in [3]. Indeed, in [4] we pointed out an error in [3]. [ABSTRACT FROM AUTHOR]

We classify the coefficients (a 1 , ... , a n) ∈ F q [ t ] n that appear in a linear relation ∑ i = 1 n a i γ i = 0 among Galois conjugates γ i ∈ F q (t) ¯ . We call such an n-tuple a Smyth tuple. Our main theorem gives an affirmative answer to a function field analogue of a 1986 conjecture of Smyth (J Numb Theory 23, 243–254, 1986) over Q . Smyth showed that certain local conditions on the a i are necessary and conjectured that they are sufficient. Our main result is that the analogous conditions are necessary and sufficient over F q (t) , which we show using a combinatorial characterization of Smyth tuples from Smyth (J Numb Theory 23, 243–254, 1986). We also formulate a generalization of Smyth's Conjecture in an arbitrary number field that is not a straightforward generalization of the conjecture over Q due to a subtlety occurring at the archimedean places. [ABSTRACT FROM AUTHOR]

In [J. P. Keating, B. Rodgers, E. Roditty-Gershon and Z. Rudnick, Sums of divisor functions in 픽 q ⁢ [ t ] \mathbb{F}_{q}[t] and matrix integrals, Math. Z. 288 2018, 1–2, 167–198], the authors established relationships of the mean-square of sums of the divisor function d k ⁢ (f) {d_{k}(f)} over short intervals and over arithmetic progressions for the function field 픽 q ⁢ [ T ] {\mathbb{F}_{q}[T]} to certain integrals over the ensemble of unitary matrices. We consider similar problems leading to distributions over the ensemble of symplectic matrices. We also consider analogous questions involving convolutions of the von Mangoldt function. [ABSTRACT FROM AUTHOR]

In this paper, we study cyclic torsion subgroups of Drinfeld F q [ T ] -modules of rank two over F q (T) via isogeny characters associated to them. Among other things, we prove that such Drinfeld F q [ T ] -modules do not have a cyclic p -torsion subgroup defined over F q (T) under various assumptions, where p is a maximal ideal of F q [ T ] . For example, we show that any Drinfeld F q [ T ] -module of rank two over F q (T) which has good reduction at every finite place of F q (T) does not have a p -isogeny defined over F q (T) , where p is a maximal ideal of F q [ T ] with deg (p) > q . We also show that the set of K-rational points of Drinfeld modular curve X 0 (p) only consists of cusps when deg (p) is equal to four. These results partially generalize preceding works by Pál and Armana, respectively. [ABSTRACT FROM AUTHOR]

This article suggests a new hypergeometric function for function fields and a new operator such that the Drinfeld logarithm is stable under it, and studies an equation satisfied by the hypergeometric function. As applications, our function is related to a supersingular polynomial of Drinfeld modules, a period of Drinfeld modules, and the Kochubei's polylogarithm, which are function-field analogues of well-known facts for the classical setting. Moreover, we generalize results for the Carlitz modules proven by Thakur to those for the Drinfeld ones. [ABSTRACT FROM AUTHOR]

We study the elliptic curves given by y 2 = x 3 + b x + t 3 n + 1 over global function fields of characteristic 3 ; in particular we perform an explicit computation of the L-function by relating it to the zeta function of a certain superelliptic curve u 3 + b u = v 3 n + 1 . In this way, using the Néron–Tate height on the Mordell–Weil group, we obtain lattices in dimension 2 · 3 n for every n ≥ 1 , which improve on the currently best known sphere packing densities in dimensions 162 (case n = 4 ) and 486 (case n = 5 ). For n = 3 , the construction has the same packing density as the best currently known sphere packing in dimension 54, and for n = 1 it has the same density as the lattice E 6 in dimension 6. [ABSTRACT FROM AUTHOR]

Let K = p (z 1 , ... , z r) be a finitely generated field over p and fix a , b ∈ K ∗ . We study the solutions of the Catalan equation a x m + b y n = 1 to be solved in x , y ∈ K and integers m , n > 1 coprime with p. Our main result corrects earlier work of Silverman. [ABSTRACT FROM AUTHOR]

We give an effective algorithm to determine the endomorphism ring of a Drinfeld module, both over its field of definition and over a separable or algebraic closure thereof. Using previous results we deduce an effective description of the image of the adelic Galois representation associated to the Drinfeld module, up to commensurability. We also give an effective algorithm to decide whether two Drinfeld modules are isogenous, again both over their field of definition and over a separable or algebraic closure thereof. Questions of efficiency are left completely out of consideration. [ABSTRACT FROM AUTHOR]

We generalize some results of Greither and Popescu to a geometric Galois cover X → Y which appears naturally for example in extensions generated by p n -torsion points of a rank 1 normalized Drinfeld module (i.e. in subextensions of Carlitz–Hayes cyclotomic extensions of global fields of positive characteristic). We obtain a description of the Fitting ideal of class groups (or of their dual) via a formula involving Stickelberger elements and providing a link (similar to the one in [1]) with Goss ζ -function. [ABSTRACT FROM AUTHOR]

We introduce multi-poly-Bernoulli–Carlitz numbers, function field analogues of multi-poly-Bernoulli numbers of Imatomi–Kaneko–Takeda. We explicitly describe multi-poly-Bernoulli Carlitz numbers in terms of the Carlitz factorial and the Stirling–Carlitz numbers of the second kind and also show their relationships with function field analogues of finite multiple zeta values. [ABSTRACT FROM AUTHOR]

We propose a function-field analog of Pisot's d -th root conjecture on linear recurrences, and prove it under some "non-triviality" assumption. Besides a recent result of Pasten-Wang on Büchi's d -th power problem, our main tool, which is also developed in this paper, is a function-field analog of a GCD estimate in a recent work of Levin and Levin-Wang. As an easy corollary of such a GCD estimate, we also obtain an asymptotic result. [ABSTRACT FROM AUTHOR]

The truncated Bernoulli-Carlitz numbers and the truncated Cauchy-Carlitz numbers are defined as analogues of hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers, and as extensions of Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers. These numbers can be expressed explicitly in terms of incomplete Stirling-Carlitz numbers. In this paper, we give several expressions of truncated Bernoulli-Carlitz numbers and truncated Cauchy-Carlitz numbers as natural extensions. One kind of expressions is in continued fractions. Another is in determinants originated in Glaisher, giving several interesting determinant expressions of numbers, including Bernoulli and Cauchy numbers.