Open problems: Descending cohomology, geometrically
Open problems: Descending cohomology, geometrically
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开放问题：几何上的降上同调
DOI：
10.4310/iccm.2014.v2.n1.a7
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发表时间：
2014
影响因子：
0.7
通讯作者：
B. Mazur
中科院分区：
文献类型：
--
作者：
B. Mazur
2011年8月为纪念乔·哈里斯而举办的生日会议的组织者要求我在会议上就“开放问题”做一个简短的陈述。现在，当你和乔一起工作时，一件很棒的事情是，你发现自己身处一大堆鼓舞人心的问题中，这要归功于他对主题的各个方面的深度参与和强烈的好奇心。他是在索玛尼层面上阐述问题的大师，仅仅选择其中的一两个来纪念他，已经是一个开放的问题了。有时，Joe引入一个问题的范围很广，而且有些拐弯抹角，就像他曾经问过“Q上定义了多少条曲线？”当然，这是一个邀请来讨论q-有理点集的Zariski闭包的维度--作为亏格g的函数--亏格g的曲线的模空间。超椭圆曲线已经给出了模空间的2/3的维度(mod O(1)，和g→∞)，但是你能得到，比方说，更好的分数吗？当然，通过著名的朗氏猜想，这个问题立即与关于mg的代数几何结构的问题联系在一起。
I was asked by the organizers of the August 2011 birthday conference in honor of Joe Harris to give a short presentation in the session on “Open Problems” in the conference. Now, a great thing when you work together with Joe is that you find yourself in the midst of loads of inspiring problems, thanks to his deep engagement with, and intense curiosity about, all aspects of his subject. He’s a master of formulating problems on somany levels that it’s already something of an open problem simply to choose just one or two of them in his honor. Sometimes Joe introduces a problem very broadly and somewhat obliquely, as when he once asked “How many curves are there defined over Q?” Of course, this was an invitation to discuss the dimension—as a function of the genus g—of the Zariski closure of the set of Q-rational points in Mg the moduli space of curves of genus g. Hyperelliptic curves already gives you 2/3 of the dimension of that moduli space (mod O(1), and as g → ∞) but can you get, say, a better fraction than that? This question, of course, immediately connects, via celebrated conjectures of Lang, to questions regarding the algebraic geometric structure of Mg.