哈塞Helmut Hasse

哈塞的父亲是一名法官。他的母亲出生在美国威斯康星州密尔沃基，但从五岁起住在卡塞尔。哈塞在卡塞尔附近的几所中学接受教育，直到1913年，他15岁时，他的父亲被任命到柏林的一个重要职位，全家搬到了那里。哈塞在柏林的Fichte-文理中学（Gymnasium）学习了两年，然后在第一次世界大战期间自愿参加海军服役。

在1917/18学年，哈塞驻扎在基尔执行海军任务，他能够参加Otto 奥托·特普利茨的讲座。离开海军后，他进入哥廷根大学。他在那里的老师包括埃德蒙·朗道、大卫·希尔伯特、埃米·诺特和埃里希·赫克。事实上，他受埃里希·赫克的影响最大，尽管埃里希·赫克在哈塞到达哥廷根仅几个月后就离开哥廷根去汉堡任职。

人们可能会认为哈塞会跟随埃里希·赫克去汉堡，但他没有走这条路，而是在1920年去马尔堡跟随库尔特·亨泽尔学习。库尔特·亨泽尔关于$p$进数的工作对哈塞的研究方向产生了重大影响。1920年10月，哈塞发现了“局部-整体”原理，该原理表明，一个二次型如果在每个prime $p$的$p$进数上非平凡地表示0，并且在实数上非平凡地表示0，那么它在有理的上也非平凡地表示0。这个结果的重要性，现在称为哈塞原理，在于一个数能否由给定形式表示以及两个形式是否等价，都可以仅使用局部信息来决定。

“局部-整体”原理及其应用构成了他1921年的博士论文Über die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen Ⓣ（《论有理数域中二次形式对数的描述》）的重要部分，也构成了他的教授资格论文（Habilitation）论文Über die Aquivalenz quadratischer Formen im Körper der rationalen Zahlen Ⓣ（《论有理数域中二次形式的等价性》）的重要部分。

1922年，哈塞被任命为基尔大学讲师，三年后他被任命为哈勒大学教授。在基尔期间，哈塞与汉堡的数学家保持密切联系，包括埃米尔·阿廷、埃里希·赫克、亚历山大·奥斯特洛夫斯基和Schreier。他扩展了海因里希·马丁·韦伯在类域论方面的工作，撰写了几篇重要论文，并开始撰写他关于类域论的著名报告，其中包括利奥波德·克罗内克、海因里希·马丁·韦伯、大卫·希尔伯特、菲利浦·富特文勒和高木贞治的贡献。正如H M Edwards在[1]中所说：-

……[文本]像任何优秀的阐述一样，包含了大量哈塞自己对材料的重新加工。

哈塞在局部域上中心单代数的结构方面获得了基本结果。1930年，库尔特·亨泽尔从马尔堡退休，哈塞被任命填补他的讲席。在马尔堡期间，他开始与理查德·布饶尔和埃米·诺特合作研究单代数，最终完全确定了今天所谓的代数数域的理查德·布饶尔 群。他还开始研究椭圆曲线，并与Baer一起研究topological域。此时他获得了一个特别与他的名字相关的结果，当时（受路易斯·乔尔·莫德尔和哈罗德·达文波特的启发）他证明了椭圆曲线的ζ函数的黎曼假设的类似物。

1933年对全德国、尤其对哈塞而言是重要的一年。那一年纳粹上台，其对德国数学的影响是深远的。当时在哥廷根的桑德斯·麦克兰恩在[2]中写道：-

1933年4月7日，一项新法律……即刻解除了所有犹太人的职务……对[哥廷根]数学研究所的影响是剧烈的。……总共，1933年有十八位数学家离开或被逐出哥廷根数学研究所的教职。

当赫尔曼·外尔辞去他在哥廷根的教授职位时，哈塞收到了作为其继任者的聘约。正如1在[1]中所说：-

……[哈塞]看来可能为纳粹所接受，然而却是一位最高水准的数学家。

遵循赫尔曼·外尔的建议，哈塞决定接受这一职位，尽管（见[4]和[10]）他遭到了数学研究所和大学内部原教旨主义纳粹官员的激烈反对。1934年，他被任命到哥廷根。

很难确切理解哈塞在这场政治混乱中持何种观点。Edwards在[1]中按他所见总结了这一立场：-

哈塞的政治观点及其与纳粹政府的关系不易归类。一方面，他与他的老师库尔特·亨泽尔的关系极为密切，直到库尔特·亨泽尔于1941年去世，而按纳粹标准库尔特·亨泽尔无疑是犹太人……他最重的一篇论文是与埃米·诺特和理查德·布饶尔合作的，两人均为犹太人，1932年发表以庆祝库尔特·亨泽尔70岁生日。……哈塞没有因政治原因而在数学上妥协……他与那些试图（有时成功地）将数学屈从于政治教条的纳粹官员作斗争……另一方面，他毫不掩饰自己强烈的民族主义观点以及对希特勒许多政策的赞同。

有一段时间，哈塞希望增加自己在当局中的影响力，因此他申请加入纳粹党。但哈塞的一位祖先是犹太人，因此申请未获批准。官方上，他的申请被搁置到战后。见[2]和[10]。

从1939年到1945年，哈塞从哥廷根休假参战，重返海军服役，在柏林研究弹道学问题。他回到哥廷根后，于1945年9月被英国占领军解除了职务。他的教学权被终止，他拒绝了仅做研究职位的提议，于1946年移居柏林，在柏林科学院担任研究职位。H W Leopoldt在[5]中写道：-

当他1948年在柏林恢复教学时，吸引了大量听众。在他的首次正式讲座中，他比较了音乐和数论中的美学原则。他关于音乐的大部分例子取自贝多芬晚期的钢琴奏鸣曲，那些年他——一直是一位热情的钢琴演奏者——深入研究了这些作品。我当时刚开始学习数学，这场讲座给我留下了持久的印象，事实上，它决定了我后来学习的方向。

1949年5月，哈塞被任命为东柏林洪堡大学的教授。他关于确定阿贝尔数域算术性质的工作Über die Klassenzahl abelscher ZahlkörperⓉ（《论阿贝尔数域的类数》）于此时发表。大约同时，他的教科书ZahlentheorieⓉ（《数论》）问世，其中首次基于局部方法对代数数论作了系统介绍。该书后来被译成英文。

1950年，哈塞被任命到汉堡，他在那里继续任教，直到1966年退休。他的影响在[1]中得到总结，Edwards在其中写道：-

作为二十世纪最重要的数学家之一，哈塞的成就涵盖研究、数学著述、教学和编辑工作。

提到编辑工作，是因为哈塞担任奥古斯都·利奥波德·克雷勒杂志的编辑达50年之久。

哈塞一生中受到许多组织的荣誉。其中包括：

哈勒的德意志自然研究者科学院利奥波尔迪纳，
哥廷根的科学院，
赫尔辛基的芬兰科学院，
柏林的德意志科学院，
美因茨的科学与文学科学院，
德意志国家科学技术一等奖，
马德里的西班牙科学院，
基尔大学名誉博士，
德意志自然研究者科学院利奥波尔迪纳的科特尼乌斯奖章。

Helmut Hasse's father was a judge. His mother was born in Milwaukee, Wisconsin, USA but lived in Kassel from the age of five. Helmut's education was at various secondary schools near to Kassel until in 1913, when he was 15 years of age, his father was appointed to an important position in Berlin and the family moved there. Helmut studied for two years at the Fichte-Gymnasium in Berlin before volunteering for naval service during World War I.

In the academic year 1917/18 Hasse was stationed at Kiel on his naval duties and he was able to attend the lectures of Otto Toeplitz. On leaving the navy he entered the University of Göttingen. His teachers there included Edmund Landau, Hilbert, Emmy Noether and Hecke. In fact he was most influenced by Hecke despite the fact that Hecke left Göttingen to take up an appointment in Hamburg only a few months after Hasse arrived in Göttingen.

It might be supposed that Hasse would have followed Hecke to Hamburg but he did not take this route, going to study under Hensel at Marburg in 1920. Hensel's work on $p$-adic numbers was to have a major influence on the direction of Hasse's research. In October 1920 Hasse discovered the 'local-global' principle which shows that a quadratic form that represents 0 non-trivially over the $p$-adic numbers for each prime $p$, and over the real numbers, represents 0 non-trivially over the rationals. The importance of this result, now known as the Hasse principle, is that both the representability of a number by a given form and whether two forms are equivalent can be decided using only local information.

The 'local-global' principle and its applications form an important part of his doctoral thesis Über die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen Ⓣ of 1921 and also of his habilitation thesis Über die Aquivalenz quadratischer Formen im Körper der rationalen Zahlen Ⓣ.

In 1922 Hasse was appointed a lecturer at the University of Kiel, then three years later he was appointed professor at Halle. During his time at Kiel, Hasse kept in close contact with the mathematicians at Hamburg including Artin, Hecke, Ostrowski and Schreier. He extended Heinrich Weber's work on class field theory writing several important papers and starting work on his famous report on class field theory which included the contributions of Kronecker, Heinrich Weber, Hilbert, Furtwängler and Takagi. As H M Edwards says in [1]:-

... [the text], like any good exposition, contained a great deal of Hasse's own reworking of the material.

At Halle Hasse obtained fundamental results on the structure of central simple algebras over local fields. In 1930 Hensel retired from Marburg and Hasse was appointed to fill his chair. While in Marburg he began joint work with Brauer and Emmy Noether on simple algebras, culminating in the complete determination of what is today called the Brauer group of an algebraic number field. He also started work on elliptic curves and, with Baer, on topological fields. At this time he obtained a result that is particularly associated with his name, when (inspired by Mordell and Davenport) he proved the analogue of the Riemann Hypothesis for zeta functions of elliptic curves.

The year 1933 was to be significant for all of Germany and for Hasse in particular. In that year the Nazis came to power and its effect on mathematics in Germany was profound. Mac Lane, who was at Göttingen at this time writes in [2]:-

On April 7, 1933, a new law ... summarily dismissed all those who were Jewish ...The effect on the Mathematical Institute [at Göttingen] was drastic. ... All told, in 1933 eighteen mathematicians left or were driven out from the faculty at the Mathematical Institute in Göttingen.

When Weyl resigned from his professorship in Göttingen, Hasse received an offer as his successor. As Edwards says in [1]:-

... [Hasse] appeared to be potentially acceptable to the Nazis and yet was a mathematician of the highest calibre.

Following the advice of Weyl, Hasse decided to accept the offer, although (see [4] and [10]) he met with fierce opposition from the fundamentalist Nazi functionaries within the Mathematics Institute and the University. In 1934 he was appointed to Göttingen.

It is hard to understand exactly what Hasse's views were in the middle of this political mess. Edwards sums up the position in [1] as he saw it:-

Hasse's political views and his relations with the Nazi government are not easily categorised. On the one hand, his relations with his teacher Hensel, who was unambiguously Jewish by Nazi standards, were extremely close, right up to Hensel's death in 1941 ... One of his most important papers was a collaboration with Emmy Noether and Richard Brauer, both Jewish, published in 1932 in honour of Hensel's 70th birthday. ... Hasse did not compromise his mathematics for political reasons ... he struggled against Nazi functionaries who tried (sometimes successfully) to subvert mathematics to political doctrine ... On the other hand, he made no secret of his strong nationalistic views and his approval of many of Hitler's policies.

At one point, Hasse hoped to increase his influence with the authorities and so he applied for membership in the Nazi Party. But one of Hasse's antecendents was a Jew and, therefore, membership was not granted. Officially his application was put on hold till after the war. See [2] and [10].

From 1939 until 1945 Hasse was on war leave from Göttingen and he returned to naval duty, working in Berlin on problems in ballistics. He returned to Göttingen where, in September 1945 he was dismissed from his post by the British occupation forces. His right to teach was terminated and he refused the offer of a research only position, moving to Berlin in 1946 when he took up a research post at Berlin Academy. H W Leopoldt writes in [5]:-

When he resumed teaching in 1948 in Berlin he attracted a large audience. In his first official lecture he compared aesthetic principles working in music and in number theory. Most of his examples regarding music he took from the late piano sonatas of Beethoven, which he - always an ardent piano player - intimately studied during those years. I had just begun to study mathematics, and this lecture made a lasting impression on me, in fact, it decided the further course of my studies.

In May 1949, Hasse was appointed professor at the Humboldt University in East Berlin. His work on determining arithmetical properties of abelian number fields Über die Klassenzahl abelscher Zahlkörper Ⓣ was published at this time. At about the same time his textbook Zahlentheorie Ⓣ appeared which contained, for the first time, a systematic introduction to algebraic number theory based on the local method. It was later translated into English.

In 1950 Hasse was appointed to Hamburg where he continued to teach until he retired in 1966. His influence is summed up in [1] where Edwards writes:-

One of the most important mathematicians of the twentieth century, Helmut Hasse was a man whose accomplishments spanned research, mathematical exposition, teaching and editorial work.

This reference to editorial work is made because Hasse was editor of Crelle's Journal for 50 years.

Hasse was honoured by many organisations during his life. Among them were:

Deutsche Akademie der Naturforscher Leopoldina zu Halle,
Akademie der Wissenschaften zu Göttingen,
Finnische Akademie der Wissenschaften in Helsinki,
Deutsche Akademie der Wissenschaften zu Berlin,
Akademie der Wissenschaften und der Literatur in Mainz,
Deutscher Nationalpreis 1.Klasse für Wissenschaft und Technik,
Spanische Akademie der Wissenschaften zu Madrid,
Doctor honoris causa Universität Kiel,
Cothenius Medaille der Deutschen Akademie der Naturforscher Leopoldina.