Book Reviews
Emil Artin and Helmut Hasse.
The Correspondence 1923–1958
Günther Frei, Franz Lemmermeyer
and Peter J. Roquette (Eds.)
Springer Basel, 2014
xx, 484 p.
ISBN 978-3-0348-0715-9

Reviewer: Javier Fresán
If Parallel Lives featuring mathematicians were to be
written, Emil Artin and Helmut Hasse would certainly
deserve one of the chapters. Both born in 1898, they completed their education at around the same time and weresoon recognised as rising stars of German number theory.
Already, their dissertations contained groundbreaking results: Hasse proved the local-global principle for quadratic
forms over the rationals and Artin investigated hyperelliptic curves over finite fields, in particular the analogue of
the Riemann hypothesis, which he was the first to consider.
They did not stop there: Hasse generalised his results to
any number field in his Habilitation and Artin developed
the theory of L-functions that is nowadays named afterhim. Within a few years, not only were they the youngest
professors in Germany but they also belonged to what
Weyl once called the “honours class” of mathematicians
who had solved one of Hilbert’s problems.
Artin and Hasse probably met for the first time at theannual conference of the German Mathematical Societyin 1922. This would be the beginning of a lifelong friendship that was to overcome the dark days of the Third
Reich, when Artin was dismissed from university andforced to go into exile, whereas Hasse had – to say theleast – ambiguous feelings about Nazism. Although they
only co-authored two papers, they maintained an extensive scientific correspondence during the emergence ofmodern class field theory, sometimes with high exchange
frequency. This could come as a surprise, since Hasse wasan “ardent letter writer” (p. 10) but Artin was not fond ofletter writing: he only wrote in reply and often had to begin his letters “with a long litany of apologies, accusations[…] and promises to better myself” (p. 81). He preferredteaching and conversation, at which he excelled. His
lectures have been described as “polished diamonds”:
even when he approached classical topics such as Galois
theory or the gamma function, his clarity of presentation
was so remarkable that textbooks are still based upon hisideas today.1

The correspondenceThe volume under review assembles 73 letters and postcards, mainly from Artin to Hasse, written between 1923
and 1958. Most of them had already been published in
1

To learn more about Artin’s teaching, we refer the reader toZassenhaus’s obituary, reprinted on pages 18–24.

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German, first in a small edition by Günther Frei in 1981,
then in collaboration with Peter Roquette and Franz
Lemmermeyer in 2008. This second edition was supplemented by voluminous comments, so detailed that itwill remain an “introduction to class field theory on ahistorical basis” written by the leading experts on thesubject. In particular, their detective work almost gives
the reader the impression that they would be able to reconstruct the contents of Hasse’s letters, which seem tobe lost. One can only welcome the completion of such
an ambitious project, now available to a wider audience
thanks to an English translation, which also includes
some new letters from 1937 onward, although these are
significantly less relevant than the previous ones.
The correspondence opens with an intense exchangeof letters, dated July 1923, which led to Artin and Hasse’s
first joint paper on the so-called second supplementary
law for odd prime exponents. Thirty years later, Hasse
would melancholically refer to that period as “the olddays when we bombed each other rapidly with postcards
about reciprocity formulas” (p. 429). Already in thesefirst letters, the influence of Takagi’s memoirs on classfield theory is manifest. After a decade of “utter scientific solitude”, the Japanese mathematician had presented
his theorem that all abelian extensions of number fields
are class fields during the ICM held in Strasbourg in
1920. Unfortunately, Germans were not allowed to attend and Takagi’s contribution went unnoticed. Back inJapan, he had the idea of sending reprints of his workto Siegel, from whom Artin borrowed them.2 He wasdeeply impressed by the potential of the techniques andurged Hasse to study them. This would result in Hasse’sreport on class field theory (Klassenkörperbericht), astreamlined presentation of Takagi’s work with proofs“reduced to their skeletons”, through which a whole
new generation of mathematicians encountered classfield theory.

Artin L-functions
From 1926 to 1930, Artin’s L-functions occupy a central
place in the correspondence, but the reader will not need
to wait that long to witness their birth: as early as in thefirst letter, Artin announces that he has found “the general L-series attached to Frobenius group characters whichaccomplish for general fields exactly what the usual Lseries accomplish for abelian fields” (p. 50). At the timeof writing, the paper Über eine neue Art von L-Reihen 3
was already in press. To prove that this new kind of Lfunction was well-behaved, Artin reduced to the abeliancase and compared them to Dirichlet L-series. For this heneeded to show the “general reciprocity law” that there isa canonical isomorphism between the Galois group andthe ray class group under which every unramified prime
ideal corresponds to its Frobenius substitution.
2One can still find the original reprints at the Sammlung Siegelin the MPIM Library in Bonn. I thank Anke Völzmann forthis information.
3 The title echoes Hecke’s Eine neue Art von Zetafunktion,
where L-functions of grössencharacters were defined.

EMS Newsletter December 2014

Book Reviews
In 1923, Artin was only able to prove the statement
for composita of cyclotomic extensions or cyclic extensions of prime degree. However, he formulated it as atheorem instead of a conjecture. From this he could derive what two years later would become Chebotarev’s
density theorem, a former conjecture of Frobenius (alsostated as a theorem). Amusingly enough, Artin could
only prove his reciprocity law using Chebotarev’s results.
All these developments are well documented in the correspondence: on 10 February 1926, Artin asks Hasse if
he knows whether Chebotarev’s theorem is right,4 since“if it is correct, we surely will have pocketed the generalabelian reciprocity laws” (p. 82). In the following months,
Artin and Schreier thoroughly studied the article in aseminar, which led to some simplifications. Once convinced of the validity of the result, it took some time forArtin to work out the details of the proof, but on 17 July
1927 he could inform Hasse that “this semester I gave a
two hours course on class field theory and finally proved
the “general reciprocity law” in the version that I havegiven it in my article on L-series” (p. 107).
There followed a second “bombardment of letters”
in which the implications of Artin’s reciprocity law werediscussed. These included Hilbert’s conjecture that eachideal of a number field becomes principal in the Hilbertclass field (proved by Furtwängler in 1928), the question of whether a tower of successive Hilbert class fields
always terminates (finally answered in the negative by
Shafarevich and Golod in 1964) and possible generalisations of class field theory to arbitrary Galois extensions.
Hasse was particularly interested in how to derive explicit reciprocity formulas for power residues,5 which was
the subject of Part II of his Klassenkörperbericht. WhenArtin achieved his proof, he had almost completed it butdecided to rewrite it taking into account the new reciprocity law. “I am sorry that you now have to rewrite thewhole report,” Artin says, adding right after that: “I believe, however, that it will be worth the trouble” (p. 137).
In the years to follow, Hasse’s report will come up regularly in the correspondence, until the moment when Artin acknowledges the reception of the galley proofs inAugust 1930.
The letter dated 18 September 1930 is another remarkable historical document. The presentation of thetheory of L-series in Hasse’s report “tempted [Artin] tothink about […] things that [he had] put aside for sucha long time” (p. 268). The 1923 paper still suffered fromsome defects: first, Artin could only define the local factors of the L-functions at unramified primes, the maindifficulty being that Frobenius automorphisms are notuniquely determined in the ramified case. This led to an
unnatural detour: to get the whole L-series he needed toreduce to the abelian case, where the complete definition
was known, then use the functional equation and a “wellknown argument due to Hecke” that was not presented
in full detail before Hasse’s report. Now, prompted byhis friend’s remarks, Artin could give a uniform definition at all places – essentially the same as that used today– so that “all relations and theorems 6 […] hold exactly
right from the start (von vornherein)” (p. 273). This also
allowed him to define the shape of the functional equation. To do so, he introduces the gamma factor at each
infinite prime and gives the definition and the main properties of the conductor, including the “deep” fact that it
is an integral ideal. In a subsequent letter he will confess:“I believe that everything depends on guessing. In thecase of the conductor as well as that of the functional
equation I had to guess everything” (p. 291). So here wehave a letter where the local contributions to Artin Lfunctions, the gamma factors at infinity, the Artin conductor and the Artin root numbers are introduced! Ifthat was not enough, he concludes with the remark that“the new results very much support the conjecture that
the L(χ, s) are entire” (p. 277), a still unproven statementthat is considered to be “one of the great challenges ofnumber theory”.

An invitation
It should be clear from the above how precious these
documents are for the history of number theory in thelast century. The Artin-Hasse correspondence contains
many other topics that have not been addressed in thisshort overview. They vary from highly technical pages,
where local class field theory emerges, to concrete questions concerning the distribution of the argument of
cubic Gauss sums or the existence of unramified icosahedral extensions. Hasse’s theory of complex multiplication and his proof of the Riemann hypothesis for elliptic
curves are briefly mentioned as well. It only remains forme to invite the reader to accompany the masters on thisfascinating journey through class field theory.
Javier Fresán wrote his thesis in arithmetic geometry at the University Paris
13 under the supervision of Christophe
Soulé and Jörg Wildeshaus. After a year
at the MPIM in Bonn, he is currently an
SNF post-doctoral fellow at the ETH inZürich. His research interests include
periods, special values of L-functions and the arithmeticof flat vector bundles. He is also very engaged in science
popularisation.

4This seems to contradict Emmy Noether’s information that
Artin was the referee of Chebotarev’s article, since it was already published by the Mathematische Annalen when Artin
asked Hasse about it.
5 Artin himself had considered his result as “somewhat strange”
since, before him, a reciprocity law was a statement about
power residues in number fields containing roots of unity.

EMS Newsletter December 2014

6As the editors explain, Artin refers here to the functoriality
properties of L-series with respect to inflation and inductionof representations.

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