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摘要翻译：
设$P$是素数，$N$是对$P$的正整数素数，$K$是整数。设$P_k(t)$是Atkin的$u$运算符的特征级数，它是驯服水平$n$和权值$k$的$p$-adic超收敛模形式的自同态。在Gouvea和Mazur的猜想的推动下，我们加强了Wan关于$P_k$和$P_{k'}$的系数之间的同余性，使$k'$close$P$-基本为$k$。对于$p-112$，$n=1$，$k=0$，我们计算了一个以二元有理函数幂级数为项的矩阵。我们应用这个计算来证明对于$P=3$，在$P_0$的牛顿多边形$N_0$下面有一个抛物线，它经常与$N_0$无限重合。结果，我们发现在$n_0$以上有一条多边形曲线。对于大的3-adic估值的$k$，在$n_0$上的这个最紧的界产生了$p_0$和$p_k$系数之间最强的同余。
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英文标题：
《Bounding slopes of $p$-adic modular forms》
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作者：
Lawren Smithline
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最新提交年份：
2007
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
Let $p$ be prime, $N$ be a positive integer prime to $p$, and $k$ be an integer. Let $P_k(t)$ be the characteristic series for Atkin's $U$ operator as an endomorphism of $p$-adic overconvergent modular forms of tame level $N$ and weight $k$. Motivated by conjectures of Gouvea and Mazur, we strengthen Wan's congruence between coefficients of $P_k$ and $P_{k'}$ for $k'$ close $p$-adically to $k$. For $p-1 | 12$, $N = 1$, $k = 0$, we compute a matrix for $U$ whose entries are coefficients in the power series of a rational function of two variables. We apply this computation to show for $p = 3$ a parabola below the Newton polygon $N_0$ of $P_0$, which coincides with $N_0$ infinitely often. As a consequence, we find a polygonal curve above $N_0$. This tightest bound on $N_0$ yields the strongest congruences between coefficients of $P_0$ and $P_k$ for $k$ of large 3-adic valuation.
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PDF链接：
https://arxiv.org/pdf/0705.3614
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