Yen-Liang Kuan

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Matilde Lalin

Université de Montréal

Matilde Lalin

Université de Montréal

Ghent University

Harish Chandra Research Institute

Andrew Jones

The University of Sheffield

Alain Connes

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Papers by Yen-Liang Kuan

On the density of abelian l-extensions

arXiv (Cornell University), Sep 4, 2015

We derive an asymptotic formula which counts the number of abelian extensions of prime degrees ov... more We derive an asymptotic formula which counts the number of abelian extensions of prime degrees over rational function fields. Specifically, let ℓ be a rational prime and K a rational function field Fq(t) with ℓ ∤ q. Let Disc f (F/K) denote the finite discriminant of F over K. Denote the number of abelian ℓ-extensions F/K with deg (Disc f (F/K)) = (ℓ − 1)αn by a ℓ (n), where α = α(q, ℓ) is the order of q in the multiplicative group (Z/ℓZ) ×. We give a explicit asymptotic formula for a ℓ (n). In the case of cubic extensions with q ≡ 2 (mod 3), our formula gives an exact analogue of Cohn's classical formula.

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On the Distribution of Primes; On the Distribution of Primes

On the density of abelian l-extensions

arXiv: Number Theory, 2015

We derive an asymptotic formula which counts the number of abelian extensions of prime degrees ov... more We derive an asymptotic formula which counts the number of abelian extensions of prime degrees over rational function fields. Specifically, let $\ell$ be a rational prime and $K$ a rational function field $\Bbb F_q(t)$ with $\ell \nmid q$. Let $\textup{Disc}_f\left(F/K\right)$ denote the finite discriminant of $F$ over $K$. Denote the number of abelian $\ell$-extensions $F/K$ with $\textup{deg}\left(\textup{Disc}_f(F/K)\right) = (\ell-1)\alpha n$ by $a_{\ell}(n)$, where $\alpha=\alpha(q, \ell)$ is the order of $q$ in the multiplicative group $\left(\Bbb Z/\ell \Bbb Z\right)^\times$. We give a explicit asymptotic formula for $a_\ell(n)$. In the case of cubic extensions with $q\equiv 2 \pmod 3$, our formula gives an exact analogue of Cohn's classical formula.

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Variation of a Theme of Landau--Shanks in Positive Characteristic

Taiwanese Journal of Mathematics, 2020

Let A := F q [t] be a polynomial ring over a finite field F q of odd characteristic and let D ∈ A... more Let A := F q [t] be a polynomial ring over a finite field F q of odd characteristic and let D ∈ A be a square-free polynomial. Denote by N D (n, q) the number of polynomials f in A of degree n which may be represented in the form u•f = A 2 −DB 2 for some A, B ∈ A and u ∈ F × q , and by B D (n, q) the number of polynomials in A of degree n which can be represented by a primitive quadratic form of a given discriminant D ∈ A, not necessary square-free. If the class number of the maximal order of F q (t, √ D) is one, then we give very precise asymptotic formulas for N D (n, q). Moreover, we also give very precise asymptotic formulas for B D (n, q).

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On depth 2 zeta-like families

Journal of Number Theory, 2018

Abstract Multizeta values for F q [ θ ] were initially studied by Thakur, who defined them as ana... more Abstract Multizeta values for F q [ θ ] were initially studied by Thakur, who defined them as analogues of classical multiple zeta values of Euler. In this present paper we give certain depth 2 families of zeta-like multizeta values, namely those whose ratio to the zeta value of the same weight is rational.

Distribution of Primes

Wolfram Demonstrations Project, 2007

A function field analogue of Romanoff's theorem

International Journal of Number Theory, 2015

We prove an analogue for Drinfeld modules of a theorem of Romanoff. Specifically, let ϕ be a Drin... more We prove an analogue for Drinfeld modules of a theorem of Romanoff. Specifically, let ϕ be a Drinfeld A-module over a global function field L and denote by ϕ(L) the A-module structure on L coming from ϕ. Let Γ ⊂ ϕ (L) be a free A-submodule of finite rank. For each effective divisor [Formula: see text] of L, let fΓ(𝔇) be the cardinality of the image of the reduction map [Formula: see text] if all elements of Γ are relatively prime to the divisor 𝔇; otherwise, just set fΓ(𝔇) = ∞. We give explicit upper bounds for the series [Formula: see text] and [Formula: see text].

Criterion for Deciding Zeta-Like Multizeta Values in Positive Characteristic

Experimental Mathematics, 2016

ABSTRACT Multizeta values for were initially studied by Thakur, who defined them as analogues of ... more ABSTRACT Multizeta values for were initially studied by Thakur, who defined them as analogues of classical multiple zeta values of Euler. In the present article, we give an effective algorithm for determining whether a given multizeta value is zeta-like, namely, those whose ratio with the zeta value of the same weight is rational. Moreover, we state conjectures on certain families of zeta-like multizeta values. We provide the data in support of these guesses.

On counting polynomials over finite fields

Proceedings of the American Mathematical Society, 2015

We give an asymptotic formula for the proportion of polynomials of a given degree over a finite f... more We give an asymptotic formula for the proportion of polynomials of a given degree over a finite field which do not have odd degree irreducible factors. In the case of odd characteristic, this leads to an asymptotic formula for certain weighted partition function which describes the major proportion of the fundamental discriminants where the "negative" Pell equation cannot be solved. We also extend the results to counting positive divisors over an arbitrary global function field.

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On the Distribution of Torsion Points Modulo Primes

Bulletin of the Australian Mathematical Society, 2012

Let $\Bbb A$ be a commutative algebraic group defined over a number field K. For a prime ℘ in K w... more Let $\Bbb A$ be a commutative algebraic group defined over a number field K. For a prime ℘ in K where $\Bbb A$ has good reduction, let N℘,n be the number of n-torsion points of the reduction of $\Bbb A$ modulo ℘ where n is a positive integer. When $\Bbb A$ is of dimension one and n is relatively prime to a fixed finite set of primes depending on $\Bbb A_{/K}$, we determine the average values of N℘,n as the prime ℘ varies. This average value as a function of n always agrees with a divisor function.

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The Mordell-Weil theorem for certain 𝑡-modules

Proceedings of the American Mathematical Society

Poonen proved an analogue for Drinfeld modules of the Mordell-Weil theorem. We shall generalize h... more Poonen proved an analogue for Drinfeld modules of the Mordell-Weil theorem. We shall generalize his results to the case of t t -modules which have the height property. Moreover, we also prove a Mordell–Weil theorem for certain t t -modules which lack the height property. In particular, let A A be the polynomial ring in one variable θ \theta over the finite field F q \mathbb {F}_q with the quotient field K K . Let ( E , ϕ ) (E, \phi ) be an iterated extension of the tensor powers of the Carlitz module defined over K K . We show that the F q [ t ] \mathbb {F}_q[t] -module E ( K ) E(K) is isomorphic to the direct sum of its torsion module, which is finite, with a free F q [ t ] \mathbb {F}_q[t] -module of rank ℵ 0 \aleph _0 .

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On the density of abelian l-extensions

arXiv (Cornell University), Sep 4, 2015

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On the Distribution of Primes; On the Distribution of Primes

On the density of abelian l-extensions

arXiv: Number Theory, 2015

We derive an asymptotic formula which counts the number of abelian extensions of prime degrees ov... more We derive an asymptotic formula which counts the number of abelian extensions of prime degrees over rational function fields. Specifically, let $\ell$ be a rational prime and $K$ a rational function field $\Bbb F_q(t)$ with $\ell \nmid q$. Let $\textup{Disc}_f\left(F/K\right)$ denote the finite discriminant of $F$ over $K$. Denote the number of abelian $\ell$-extensions $F/K$ with $\textup{deg}\left(\textup{Disc}_f(F/K)\right) = (\ell-1)\alpha n$ by $a_{\ell}(n)$, where $\alpha=\alpha(q, \ell)$ is the order of $q$ in the multiplicative group $\left(\Bbb Z/\ell \Bbb Z\right)^\times$. We give a explicit asymptotic formula for $a_\ell(n)$. In the case of cubic extensions with $q\equiv 2 \pmod 3$, our formula gives an exact analogue of Cohn's classical formula.

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Variation of a Theme of Landau--Shanks in Positive Characteristic

Taiwanese Journal of Mathematics, 2020

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On depth 2 zeta-like families

Journal of Number Theory, 2018