论文标题
降低等级2通用Drinfeld模块的内态判别环的生长
The growth of the discriminant of the endomorphism ring of the reduction of a rank 2 generic Drinfeld module
论文作者
论文摘要
令$ψ：a \ to f \ {τ\} $为$ f $等级2的$ f $ a $ a $ module，而没有复杂的乘法，其中$ a = {\ mathbb {f}} _ q [t] _ q [t] $，$ f = = {\ mathbb {f} {f} {f}} _ q（q q（q q（q）对于Prime $ \ Mathfrak {p} = P A $ $ a $ a $ a $的良好减少，并带有残留范围$ {\ Mathbb {f}} _ {\ Mathfrak {\ Mathfrak {p}} $ $ {\ mathbb {f}} _ {\ Mathfrak {p}} $ - 还原的$ψ$ modulo $ \ mathfrak {p} $的内晶环。我们证明，对于所有$ \ mathfrak {p} $，$ |δ_ {\ mathfrak {p}}} | $用$ | p | $生长。此外，我们证明，对于Primes $ \ Mathfrak {P} $，$ |δ_ {\ Mathfrak {p}}} | $的密度为1，与其上限$ | A _ {\ Mathfrak {\ Mathfrak {p}}} $ x^2+a _ {\ mathfrak {p}} x+μ_ {\ mathfrak {p}} p \ in [x] $是$τ^{\ text {deg} \ p} $的特征多项式。
Let $ψ: A \to F\{τ\}$ be a Drinfeld $A$-module over $F$ of rank 2 and without complex multiplication, where $A = {\mathbb{F}}_q[T]$, $F = {\mathbb{F}}_q(T)$, and $q$ is an odd prime power. For a prime $\mathfrak{p} = p A$ of $A$ of good reduction for $ψ$ and with residue field ${\mathbb{F}}_{\mathfrak{p}}$, we study the growth of the absolute value $|Δ_{\mathfrak{p}}|$ of the discriminant of the ${\mathbb{F}}_{\mathfrak{p}}$-endomorphism ring of the reduction of $ψ$ modulo $\mathfrak{p}$. We prove that for all $\mathfrak{p}$, $|Δ_{\mathfrak{p}}|$ grows with $|p|$. Moreover, we prove that for a density 1 of primes $\mathfrak{p}$, $|Δ_{\mathfrak{p}}|$ is as close as possible to its upper bound $|a_{\mathfrak{p}}^2 - 4 μ_{\mathfrak{p}}p|$, where $X^2+a_{\mathfrak{p}}X+μ_{\mathfrak{p}} p \in A[X]$ is the characteristic polynomial of $τ^{\text{deg} \ p}$.
