A prime analogue Erdős-Pomerance result for Drinfeld modules with arbitrary endomorphism rings

Kuo, Wentang; Tweedle, David

doi:10.1090/proc/15002

Cited by 2 publications

(8 citation statements)

References 26 publications

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“…We will follow a similar strategy as [16]. If ℘ is a prime of K with (℘, K /K) = c, then there is a prime ℘ of E with (℘ , K /E) = g. By focusing on the primes ℘ of E with (℘, K /E) = g, we now only need to work with C q,g .…”

Section: The Case Of Non-trivial Endomorphismsmentioning

confidence: 99%

“…We now calculate the size of

C_{q}

. Note the similarity to work in [15] and [16]. Proposition Assume

q ∤ M

.…”

Section: Conjugacy Classes Of Matricesmentioning

confidence: 99%

“…Given

W

of dimension

k

, the number of invertible

σ : V \to V

with

W \subseteq ker (σ - 1)

f false(W false) = false(q^{n} - q^{k} false) false(q^{n} - q^{k + 1} false) \dots false(q^{n} - q^{n - 1} false) .

The number of invertible

σ : V \to V

with

W = ker (σ - 1)

will be denoted

g (W)

. Then we want to calculate

N = \sum_{dim (W) ⩾ 2} g false(W false),

given that

f false(W false) = \sum_{W \subseteq W^{'}} g false(W^{'} false) .

By Möbius inversion (see [20] or [16] for a similar calculation),

g false(W false) = \sum_{W \subseteq W^{'}} μ false(W, W^{'} false) f false(W^{'} false) .

But

μ false(W, W^{'} false) = {(- 1)}^{j} q^{((), \binom{j}{2})}

, where

j = dim false(...

…”

Section: Conjugacy Classes Of Matricesmentioning

confidence: 99%

“…In the case that

{prefixEnd}_{K^{sep}} false(ϕ false) \neq A

, we have some Galois group calculations to complete. The following calculations can be found in [16, Section 3], but we will restate them here for convenience.…”

Section: The Case Of Non‐trivial Endomorphismsmentioning

confidence: 99%

“…First of all, the condition that

ϕ ({double-struckF}_{℘})

is cyclic (in general) is captured by a non‐trivial conjugacy class in

{prefixGL}_{n} false(A / frakturq false)

(whereas

(n - 1)

‐cyclicity is captured by the conjugacy class of the identity in

{prefixGL}_{n} false(A / frakturq false)

). We are able to estimate the size of this conjugacy class by using inclusion–exclusion (see [20] and the authors work [16] for a similar application). Furthermore, since the conjugacy classes are larger, the error terms coming from the Chebotarev density theorem are harder to control.…”

Section: Introductionmentioning

confidence: 99%

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Cyclicity of Drinfeld modules

Kuo

Tweedle

2021

Bull. London Math. Soc.

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Let X be a smooth, projective curve defined over a finite field Fr, and ∞ a (closed) point of X. Let κ be the function field of X and A the elements of κ regular everywhere except possibly at ∞. Let φ : A → K{τ } be a Drinfeld module defined over K, where ι : A → K is an injective map that identifies K as a finite extension of κ. Suppose that the rank of φ is n 2.For a place ℘ of good reduction for φ, reducing the coefficients of φ modulo ℘ equips the residue field F℘ with an A-module structure. We establish that the number of places ℘ of K of good reduction for φ, of degree x, and such that φ(F℘) is a cyclic A-module, has a natural density. Furthermore, this density is positive if and only if there are infinitely many primes ℘ such that φ(F℘) is cyclic as an A-module.We do not make further restrictions on either A or the ring of endomorphisms EndKsep (φ).

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Section: The Case Of Non-trivial Endomorphismsmentioning

confidence: 99%

“…We now calculate the size of

C_{q}

. Note the similarity to work in [15] and [16]. Proposition Assume

q ∤ M

.…”

Section: Conjugacy Classes Of Matricesmentioning

confidence: 99%

“…Given

W

of dimension

k

, the number of invertible

σ : V \to V

with

W \subseteq ker (σ - 1)

f false(W false) = false(q^{n} - q^{k} false) false(q^{n} - q^{k + 1} false) \dots false(q^{n} - q^{n - 1} false) .

The number of invertible

σ : V \to V

with

W = ker (σ - 1)

will be denoted

g (W)

. Then we want to calculate

N = \sum_{dim (W) ⩾ 2} g false(W false),

given that

f false(W false) = \sum_{W \subseteq W^{'}} g false(W^{'} false) .

By Möbius inversion (see [20] or [16] for a similar calculation),

g false(W false) = \sum_{W \subseteq W^{'}} μ false(W, W^{'} false) f false(W^{'} false) .

But

μ false(W, W^{'} false) = {(- 1)}^{j} q^{((), \binom{j}{2})}

, where

j = dim false(...

…”

Section: Conjugacy Classes Of Matricesmentioning

confidence: 99%

“…In the case that

{prefixEnd}_{K^{sep}} false(ϕ false) \neq A

, we have some Galois group calculations to complete. The following calculations can be found in [16, Section 3], but we will restate them here for convenience.…”

Section: The Case Of Non‐trivial Endomorphismsmentioning

confidence: 99%

“…First of all, the condition that