$\wp$-adic continuous families of Drinfeld eigenforms of finite slope
$\wp$-adic continuous families of Drinfeld eigenforms of finite slope
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有限斜率的 Drinfeld 特征形的 $wp$-adic 连续族
DOI：
10.1016/j.aim.2021.107594
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发表时间：
2021
影响因子：
1.7
通讯作者：
Shin Hattori
中科院分区：
文献类型：
--
作者：
Yuya Mizuno;那須弘和;Shin Hattori
设p是有理数素数，v p是Z上的正规p-adic赋值，q> 1是p的幂，A= Fq [t].设n ∈ A是一个不可约多项式，n∈ A是一个与n互素的非零元.设k≥ 2，r≥ 1为整数。我们用Sk（Γ 1（n <$r））表示Fq（t）的水平为Γ 1（n <$r），权为k的Drinfeld尖形空间.设n≥ 1为整数，a≥ 0为有理数.设n <$r有一个一次不可约因子，且Sk（Γ 1（n <$r））中斜率为a的广义本征空间是一维的.本文在a充分小的假设下，构造了一个族{Fk ′| vp（k′− k）≥ logp <$（pn + a）}的Hecke特征形Fk ′∈ Sk ′（Γ 1（n <$r））的斜率为a，使得对任意Q∈ A，Fk和Fk ′在Q处的Hecke特征值模<$κ全等，且有某个κ> pvp（k′− k）− pn − a.
Let p be a rational prime, v p the normalized p-adic valuation on Z, q> 1 a power of p and A= F q [t]. Let℘∈ A be an irreducible polynomial and n∈ A a non-zero element which is prime to℘. Let k≥ 2 and r≥ 1 be integers. We denote by S k (Γ 1 (n℘ r)) the space of Drinfeld cuspforms of level Γ 1 (n℘ r) and weight k for F q (t). Let n≥ 1 be an integer and a≥ 0 a rational number. Suppose that n℘ has an irreducible factor of degree one and the generalized eigenspace in S k (Γ 1 (n℘ r)) of slope a is one-dimensional. In this paper, under an assumption that a is sufficiently small, we construct a family {F k′| v p (k′− k)≥ log p(p n+ a)} of Hecke eigenforms F k′∈ S k′(Γ 1 (n℘ r)) of slope a such that, for any Q∈ A, the Hecke eigenvalues of F k and F k′ at Q are congruent modulo℘ κ with some κ> p v p (k′− k)− p n− a.