Artin’s conjecture for Drinfeld modules

Kuo, Wentang; Tweedle, David

doi:10.2140/ant.2022.16.1025

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Primitive Roots Over Function Fields1

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2024

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Cited by 2 publications

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“…For irreducible g(t), a proof which relies only on establishing a zero-free region of relevant L-functions, instead of the results of Weil, is given in [65,64]. Several further variations of Artin's primitive root conjecture over function fields have also been studied, for example, over Carlitz modules [36,59], rank one Drinfeld modules [60,120,67], and one dimensional tori over function fields [22].…”

Section: Primitive Roots Over Function Fieldsmentioning

confidence: 99%

Diophantine problems over global fields and a conjecture of Artin over function fields

Leonhard¹

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Section: Primitive Roots Over Function Fieldsmentioning

confidence: 99%

Diophantine problems over global fields and a conjecture of Artin over function fields

Leonhard¹

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On Artin’s Primitive Root Conjecture for Function Fields over 𝔽q

Hochfilzer,

Waxman

2024

The Quarterly Journal of Mathematics

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In 1927, E. Artin proposed a conjecture for the natural density of primes p for which g generates $(\mathbb{Z}/p\mathbb{Z})^\times$. By carefully observing numerical deviations from Artin’s originally predicted asymptotic, Derrick and Emma Lehmer (1957) identified the need for an additional correction factor, leading to a modified conjecture which was eventually proved to be correct by Hooley (1967) under the assumption of the generalized Riemann hypothesis. An appropriate analogue of Artin’s primitive root conjecture may moreover be formulated for an algebraic function field K of r variables over $\mathbb{F}_{q}$. Relying on a soon to be established theorem of Weil (1948), Bilharz (1937) provided a proof in the particular case that K is a global function field (that is, r = 1), which is correct under the assumption that $g \in K$ is a geometric element. Under the same assumptions, Pappalardi and Shparlinski (1995) established a quantitative version of Bilharz’s result. In this paper we build upon these works by both generalizing to function fields in r variables over $\mathbb{F}_{q}$ and removing the assumption that $g \in K$ is geometric, thereby completing a proof of Artin’s primitive root conjecture for function fields over $\mathbb{F}_{q}$. In doing so, we moreover identify an interesting correction factor which emerges when g is not geometric. A crucial feature of our work is an exponential sum estimate over varieties that we derive from Weil’s Theorem.

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Artin’s conjecture for Drinfeld modules

Cited by 2 publications

References 21 publications

Diophantine problems over global fields and a conjecture of Artin over function fields

Diophantine problems over global fields and a conjecture of Artin over function fields

On Artin’s Primitive Root Conjecture for Function Fields over 𝔽q

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