The sup-norm problem in analytic number theory asks for the largest value taken by a given automorphic form. We observe that the function-field version of this problem can be reduced to the geometric problem of finding the largest dimension of the $i$th stalk cohomology group of a given Hecke eigensheaf at any point. This problem, in turn, can be reduced to the intersection-theoretic problem of bounding the ‘polar multiplicities’ of the characteristic cycle of the Hecke eigensheaf, which in known cases is the nilpotent cone of the moduli space of Higgs bundles. We solve this problem for newforms on $G{L}_2({\mathbb{A}}_{{\mathbb{F}}_q(T)})$ of squarefree level, leading to bounds on the sup-norm that are stronger than what is known in the analogous problem for newforms on $G{L}_2({\mathbb{A}}_{\mathbb{Q}})$ (that is, classical holomorphic and Maaß modular forms.)

中文翻译：

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自守形式的超范数问题的几何方法：GL2(Fq(T)) 上的新形式的情况与 squarefree 水平

解析数论中的超范数问题要求给定自守形式所取的最大值。我们观察到这个问题的函数域版本可以简化为寻找最大维度的几何问题$\mathrm{一世}$给定 Hecke eigensheaf 在任何点的第 th 茎上同调群。反过来，这个问题可以简化为限制 Hecke 特征环的“极性多重性”的相交理论问题，在已知情况下，它是希格斯丛的模空间的幂零锥。我们为 newforms 解决了这个问题$G{升}_2({\mathbb{一种}}_{{\mathbb{F}}_q(吨)})$ 的平方自由水平，导致超范数的界限比新形式的类似问题中已知的更强 $G{升}_2({\mathbb{一种}}_{\mathbb{问}})$ （即经典的全纯和 Maaß 模形式。）