Helmut Hasse

How to use this course

Helmut Hasse’s name is attached to more theorems in number theory than almost anyone of his century: the Hasse principle, the Hasse–Minkowski theorem, the Hasse bound for elliptic curves, the Hasse norm theorem, the Hasse invariant, the Hasse–Davenport relation, the Hasse–Arf theorem. One idea runs through them: study a number-theoretic problem one prime at a time, using the \(p\)-adic numbers, and then put the local answers together.

His life raises harder questions. From 1934 to 1945 he directed the Mathematical Institute at Göttingen after its Jewish and dissenting members had been driven out. He worked hard for persecuted colleagues and refused to bend mathematics to Nazi ideology. He also held strongly nationalist views, approved of many of Hitler’s policies, and applied to join the Nazi party. Lesson 11 sets out what the documents show, and says where accounts differ.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions.

Timeline

Kassel, Berlin and the navy, 1898–1918

Helmut Hasse was born in Kassel on 25 August 1898. His father, Paul Reinhard Hasse, was a judge; his mother, Margarethe Quentin, had been born in Milwaukee in the United States. In 1913 his father took up a post in Berlin, and Hasse spent two years at the Fichte-Gymnasium there.

In 1915, in the second year of the First World War, he took an emergency school-leaving examination and volunteered for the navy. He served through the war. In 1917–18 he was stationed at Kiel and used the posting to attend mathematics lectures at the university, including those of Otto Toeplitz. The navy, and the patriotism of his generation, stayed with him: he returned to naval service in the Second World War.

Hasse was also a serious pianist. He played chamber music throughout his life, and Beethoven was his favourite composer. A student later recalled him comparing the structure of number theory to Beethoven’s late sonatas.

Göttingen and Marburg, 1918–1922

After leaving the navy Hasse enrolled at Göttingen. He heard Edmund Landau, David Hilbert, Emmy Noether and Richard Courant, but the decisive influence was Erich Hecke, whose lectures on algebraic number theory drew him into the subject. Hecke left for Hamburg a few months later.

Hasse then read Kurt Hensel’s book Zahlentheorie (1913), which developed the \(p\)-adic numbers that Hensel had introduced in 1897. Most number theorists at the time treated them as a curiosity. Hasse saw them differently, and in 1920 he moved to Marburg to work with Hensel himself.

The local–global principle

In October 1920 Hasse found the theorem that made his name. A quadratic form with rational coefficients represents zero over \(\mathbb Q\) if and only if it represents zero over the real numbers and over every \(p\)-adic field \(\mathbb Q_p\). Local solvability, which can be tested by finite computation, controls global solvability. This was the first clear demonstration that \(p\)-adic numbers do real work. Hasse took his doctorate at Marburg in May 1921 on the representation of numbers by quadratic forms over \(\mathbb Q\), and habilitated in February 1922 on their equivalence. Lessons 04 and 05 present the mathematics.

Hensel was of Jewish descent; under the later Nazi laws he counted as Jewish. Hasse admired him all his life and kept close contact with him until Hensel’s death in 1941.

Kiel, Halle and Marburg, 1922–1933

In the autumn of 1922 Hasse became Privatdozent at Kiel. He married Clara Ohle; they had a daughter, Jutta, and a son, Rüdiger. From Easter 1925 he was full professor at Halle, sharing direction of the institute with Heinrich Jung. Biographers describe the Halle years as his happiest and most productive. He was elected to the Leopoldina academy in 1926.

In 1925, at the meeting of the German Mathematical Society in Danzig, Hasse reported on Teiji Takagi’s class field theory, which had gone largely unread in Europe. The written report, the Klassenkörperbericht, appeared in parts between 1926 and 1930 and taught the theory to a generation. The second part included Emil Artin’s reciprocity law of 1927. Hasse and Artin had met in Göttingen in 1921, and their letters from 1923 onward are one of the main sources for the history of class field theory. They also wrote two joint papers on the supplementary laws of reciprocity (1925, 1928).

In 1930 Hasse succeeded Hensel at Marburg. There he worked with Emmy Noether and Richard Brauer on simple algebras over number fields, leading to the Brauer–Hasse–Noether theorem, published in 1932 in a volume honouring Hensel’s seventieth birthday. In the summer of 1931 the young English mathematician Harold Davenport came to Marburg. Their friendship and arguments led Hasse to the Riemann hypothesis for elliptic curves (Lesson 08). He was an invited speaker at the International Congress in Zürich in 1932.

Hasse also kept mathematical diaries, and he became an editor of Crelle’s Journal, a role he held for about fifty years. Under his editing it became again one of the leading journals in mathematics.

p-adic numbers and Hensel’s lemma

For the general reader. The real numbers fill in the gaps between fractions by measuring size in the ordinary way. For each prime \(p\) there is another way to measure size, in which a number is small when it is divisible by a high power of \(p\). Filling in the gaps for this measure gives the \(p\)-adic numbers. They let one solve equations “one prime at a time”, and Hasse was the first to show how powerful this is.

Definitions

For a prime \(p\) and non-zero \(x\in\mathbb Q\), write \(x=p^{v}\frac ab\) with \(p\nmid ab\), and set \(v_p(x)=v\), \(|x|_p=p^{-v}\), \(|0|_p=0\). This absolute value is non-archimedean: \(|x+y|_p\le\max(|x|_p,|y|_p)\). The completion of \(\mathbb Q\) for \(|\cdot|_p\) is the field \(\mathbb Q_p\); its unit ball \(\mathbb Z_p=\{|x|_p\le1\}\) is the ring of \(p\)-adic integers, which equals \(\varprojlim\mathbb Z/p^n\). By Ostrowski’s theorem, every non-trivial absolute value on \(\mathbb Q\) is equivalent to some \(|\cdot|_p\) or to the usual \(|\cdot|_\infty\). One writes \(\mathbb Q_\infty=\mathbb R\), and the product formula \(\prod_{v\le\infty}|x|_v=1\) holds for \(x\in\mathbb Q^\times\).

Proof sketch. Newton’s iteration \(a_{n+1}=a_n-f(a_n)/f'(a_n)\) converges \(p\)-adically. Taylor expansion gives \(|f(a_{n+1})|_p\le|f(a_n)|_p^2/|f'(a_0)|_p^2\), so the errors shrink quadratically, and \(|f'(a_n)|_p\) stays equal to \(|f'(a_0)|_p\).

The Hasse–Minkowski theorem

For the general reader. To show an equation has no whole-number solutions, one often checks it modulo some number: \(x^2+y^2=3\) fails because squares mod 4 are 0 or 1. Hasse proved that for quadratic equations this is the only kind of obstruction. If nothing goes wrong over the reals or modulo powers of any prime, there is a rational solution.

Hermann Minkowski had classified quadratic forms with rational coefficients in 1890 by invariants that, in hindsight, are local; Hasse’s contribution was the \(p\)-adic formulation and proof, which made the principle visible. He soon extended the theorem to all number fields.

The Hilbert symbol

For \(a,b\in\mathbb Q_v^\times\), set \((a,b)_v=1\) if \(ax^2+by^2=z^2\) has a non-trivial solution in \(\mathbb Q_v\), and \(-1\) otherwise. For odd \(p\) and units \(u,w\), \((u,w)_p=1\), \((u,p)_p=\big(\frac up\big)\), and \((p,p)_p=\big(\frac{-1}p\big)\). The Hilbert reciprocity law \(\prod_{v\le\infty}(a,b)_v=1\) is equivalent to quadratic reciprocity with its supplements.

The ternary case

Any ternary form can be diagonalized and scaled to \(ax^2+by^2-z^2\), or after clearing squares to \(ax^2+by^2+cz^2\) with \(a,b,c\) square-free, pairwise coprime integers. Legendre’s theorem (1785) says such a form has a non-trivial integer zero iff \(a,b,c\) do not all have the same sign and \(-bc,-ca,-ab\) are squares modulo \(|a|,|b|,|c|\) respectively. Each of these conditions follows from local solvability at \(\infty\) or at an odd prime dividing \(abc\). So local solvability everywhere implies Legendre’s conditions, which imply a rational zero: Legendre’s theorem contains the ternary Hasse principle. The higher cases follow by induction, using Dirichlet’s theorem on primes in progressions and the fact that every form in five or more variables has a zero over every \(\mathbb Q_p\).

The principle can fail

Beyond quadratic forms the principle fails. Ernst Selmer showed in 1951 that \(3x^3+4y^3+5z^3=0\) has non-trivial solutions in \(\mathbb R\) and every \(\mathbb Q_p\) but none in \(\mathbb Q\). Measuring the failure of the Hasse principle, through Tate–Shafarevich groups and Brauer–Manin obstructions, is a central topic of modern arithmetic geometry.

Class field theory and the norm theorem

For the general reader. Class field theory describes all the abelian extensions of a number field in terms of data inside the field itself. Hasse made it a theory one could learn: his report organized Takagi’s results, his norm residue symbol made the local behaviour explicit, and in 1930 he proposed building the theory from the local fields upward. That reorganization is how class field theory is taught today.

Local class field theory

Hasse defined the norm residue symbol in the 1920s through global reciprocity. In 1930–31, prompted by Noether, he and others began to construct it locally, using the simple algebras of Lesson 07. Claude Chevalley completed the local-to-global construction with idèles in 1936–1940.

The norm theorem

For quadratic extensions \(K=k(\sqrt b)\), being a local norm at \(v\) means \((a,b)_v=1\), so the theorem is the Hasse–Minkowski theorem for the ternary form \(x^2-by^2-az^2\). Cyclicity cannot be dropped. A standard example: in the biquadratic extension \(\mathbb Q(\sqrt{-3},\sqrt{13})/\mathbb Q\), the number 3 is a local norm everywhere but not a global norm. In modern terms, the obstruction is a Tate cohomology group that vanishes for cyclic groups.

Local invariants of simple algebras

For the general reader. The quaternions are a number system in which multiplication is not commutative. Over the rationals there are infinitely many quaternion-like systems, and more general ones of higher dimension. In 1931 Hasse classified them over \(p\)-adic fields by a single fraction, now called the Hasse invariant, and with Brauer and Noether he showed that the local fractions determine everything.

The injectivity on the left is the local–global principle for algebras, the main theorem of the joint paper with Brauer and Noether. That the sum of invariants is zero is a form of Artin reciprocity; Hasse saw this as a new, algebraic route to the reciprocity law. A. Adrian Albert proved the cyclicity result independently and the two published a joint note in 1932. The full story, with Noether’s crossed products, is in Lesson 10 of the Noether course.

The Riemann hypothesis for elliptic curves

For the general reader. Count the solutions of \(y^2=x^3+ax+b\) modulo a prime \(p\). On average you get about \(p\). Hasse proved that the error is never more than \(2\sqrt p\). This bound is the analogue, for curves over finite fields, of the Riemann hypothesis, and Hasse’s proof was the first case beyond Artin’s computations.

Background

Artin’s thesis (see the Artin course, Lesson 04) introduced zeta functions of curves over finite fields and the analogue of the Riemann hypothesis. Davenport and Louis Mordell had studied bounds for such point counts by analytic methods. During Davenport’s visit to Marburg in 1931, Davenport challenged Hasse to show what his structural methods could do. Hasse announced a proof for elliptic curves in 1933, lectured on it in Hamburg in 1934 at Artin’s invitation, and published the complete theory in three papers in Crelle’s Journal in 1936.

Proof sketch (modern form)

Let \(\pi:(x,y)\mapsto(x^q,y^q)\) be the Frobenius endomorphism. Its fixed points are exactly \(E(\mathbb F_q)\). The endomorphism \(1-\pi\) is separable, so \(\#E(\mathbb F_q)=\#\ker(1-\pi)=\deg(1-\pi)\). The degree map on \(\mathrm{End}(E)\) is a positive definite quadratic form; this is the key structural fact, and Hasse established it in his 1936 papers. Set \(a=1+q-\deg(1-\pi)\). For integers \(m,n\),

A binary quadratic form that is non-negative on \(\mathbb Z^2\), hence on \(\mathbb Q^2\) and \(\mathbb R^2\), has non-positive discriminant: \(a^2-4q\le0\), so \(|a|\le2\sqrt q\).

Gauss sums and the Hasse–Davenport relation

For the general reader. Gauss sums combine the multiplicative and additive structure of a finite field into a single complex number. They control the point counts of many curves. In 1934 Hasse and Davenport found how a Gauss sum changes when the field is enlarged. Their relation was the key to proving the Riemann hypothesis for a family of curves and a model for Weil’s later work.

Gauss sums

Let \(\chi\) be a non-trivial multiplicative character of \(\mathbb F_q^\times\) and \(\psi\) a non-trivial additive character of \(\mathbb F_q\). The Gauss sum is \(g(\chi,\psi)=\sum_{x\in\mathbb F_q^\times}\chi(x)\psi(x)\). Expanding \(|g|^2\) and using orthogonality gives \(|g(\chi,\psi)|=\sqrt q\).

The proof writes \(-g\) as a sum over monic polynomials and uses the factorization of polynomials over \(\mathbb F_q\) into irreducibles, which gives an Euler product of length one. Davenport and Hasse used it to show that for curves \(y^e=\gamma x^f+\delta\), the numerator of the zeta function is a product of Gauss and Jacobi sums, all of absolute value \(\sqrt q\), which proves the Riemann hypothesis for those curves. There is also a product relation for Gauss sums of characters of different orders, the Hasse–Davenport product formula.

Ramification and the Hasse invariant

For the general reader. When a prime “ramifies” in an extension of number fields, the extension is singular there, and measuring how badly is subtle. Hasse found exact formulas linking ramification to the conductors of class field theory. In a different direction, he found a single number that detects a special kind of elliptic curve in characteristic \(p\).

Conductors and ramification

Hasse proved it for finite residue fields using class field theory; Cahit Arf, his doctoral student, proved the general case in 1939 by a direct argument. It implies that the Artin conductor of any representation is an integer, which was the point of departure for the theory of Swan conductors.

The Hasse invariant

An elliptic curve \(E\) over a field of characteristic \(p\) is supersingular if \(E\) has no points of order \(p\) over the algebraic closure, and ordinary otherwise. In 1934 Hasse attached to \(E\) an invariant \(A(E)\), now understood as the action of the Cartier operator on differentials, which vanishes exactly when \(E\) is supersingular. Hasse and Ernst Witt extended this in 1936 to a matrix for curves of any genus, the Hasse–Witt matrix.

Göttingen under National Socialism, 1933–1945

This lesson sets out the documented record. Historians agree on most of the facts. They differ on how to weigh them.

1933: Noether and the professors’ declaration

In the spring of 1933 the new civil-service law removed Jewish and politically suspect academics. Göttingen lost Noether, Courant, Landau, Bernays and many others; Hermann Weyl left as well. Hasse, then at Marburg, tried to keep Noether in Göttingen. On 3 June 1933 he wrote to the university’s Kurator that he had asked leading mathematicians for assessments of her work. On 31 July he sent the collected statements, from Philipp Furtwängler, Oskar Perron, Jan Schouten, Beniamino Segre, Kenjiro Shoda, Carl Ludwig Siegel, Andreas Speiser, Teiji Takagi, Bartel van der Waerden, Weyl and others, together with his own, and argued that she could at least supervise advanced students. The Kurator forwarded the file to the ministry on 7 August with a covering letter declining to support her, citing her politics. She emigrated that autumn.

On 11 November 1933 Hasse was among the roughly 900 signatories of the declaration of German university professors in support of Hitler and the National Socialist state.

1934: the Göttingen appointment

In February 1934 the Göttingen dean asked Werner Weber, the acting head of the institute and a Nazi activist, to propose a successor to Weyl. Weber named Hasse as the strongest mathematician. The ministry official Theodor Vahlen appointed him in the spring. Nazi activists in the institute were not satisfied with his politics, and questioned his ancestry; Weber at first refused to hand over the institute’s keys. Hasse took up the directorship in the summer of 1934, and Weber left Göttingen. Erhard Tornier, a convinced Nazi whose career Hasse had earlier helped, obtained the second chair and turned against him. In a report of 2 May 1935, found by Peter Roquette in the Bundesarchiv, Tornier denounced Hasse for a Jewish strain in his ancestry and for sending a wreath from the Göttingen mathematicians to Noether’s memorial. In 1936 Tornier wrote in Deutsche Mathematik calling for a “German mathematics”. Hasse never published in that journal.

Hasse’s aim, by his own account and that of his biographers, was to rebuild Göttingen on purely mathematical standards. He did not accept political criteria in appointments or examinations. In 1935 he examined and approved the thesis of Oswald Teichmüller, who as a student leader had organized the November 1933 boycott of Landau’s lectures.

Nationalism and the party application

Harold Edwards, in the Dictionary of Scientific Biography, writes that Hasse made no secret of his strongly nationalistic views and his approval of many of Hitler’s policies. In May 1937 Hasse applied to join the NSDAP, and in the application he mentioned a Jewish branch of his family. He was registered as a candidate, but admission was deferred until after the war and never completed. Roquette judges that the Jewish ancestry probably contributed. According to his son-in-law Martin Kneser, Hasse applied to increase his influence with the authorities; Hasse later explained his reasons in letters to Constance Reid. Siegel reported to Courant in a letter of March 1939 that he had seen Hasse wearing Nazi insignia.

On the other side of the ledger, Hasse kept up his friendship with Hensel until 1941, worked with the Jewish mathematicians Noether and Brauer, and continued his correspondence with Artin. When Artin was forced out of Hamburg in 1937 because of his wife’s Jewish ancestry, Hasse reportedly tried to obtain an official exemption for Artin’s children, without success.

War

During the Second World War Hasse was on leave from Göttingen for naval service in Berlin, where he led a group working on ballistics problems, using among other tools the theory of elliptic functions.

Berlin, Hamburg and legacy, 1945–1979

After the war Hasse returned briefly to Göttingen. In September 1945 the British occupation authorities dismissed him and revoked his right to teach. He declined a research-only position. In 1946 he took a research post at the German Academy of Sciences in Berlin, resumed teaching in 1948, and in May 1949 became professor at the Humboldt University in East Berlin. There he published Zahlentheorie (1949) and Vorlesungen über Zahlentheorie (1950).

In 1950 he moved to the University of Hamburg, where he taught until he retired in 1966. Many of his Berlin students followed him. His Hamburg colleagues included Emanuel Sperner, with whom he arranged Artin’s return in 1958. His doctoral students over his career included Cahit Arf, who became the leading Turkish mathematician of his generation, Teichmüller, Heinrich-Wolfgang Leopoldt, Peter Roquette, Paul Lorenzen and Wolfgang Franz. He remained active in research and editing after retirement until illness slowed him. He died in Ahrensburg, near Hamburg, on 26 December 1979.

Edwards notes that invitations to the United States after the war were few for a mathematician of his standing, and those that came were controversial.

Honours

Hasse was a member of the Leopoldina (1926), the Göttingen Academy (1935), the Finnish Academy (1942), the Berlin Academy (1949), the Mainz Academy (1952) and the Spanish Royal Academy (1956). He received the National Prize of the German Democratic Republic, first class, in 1953, and in 1968 an honorary doctorate from Kiel and the Cothenius Medal of the Leopoldina. He gave a plenary address at the International Congress of Mathematicians in Oslo in 1936.

Legacy

• Method: the local–global principle, and its failures, organize Diophantine geometry; the Brauer–Manin obstruction is a direct descendant of the Brauer–Hasse–Noether theorem.
• Class field theory: his report, his norm residue symbol and his local approach shaped the theory as it is taught.
• Arithmetic geometry: his bound for elliptic curves, his invariant, and the Hasse–Weil conjecture on the L-function of an elliptic curve over \(\mathbb Q\), proved through the modularity theorem, all bear his name.
• Infrastructure: fifty years of editing Crelle’s Journal, mathematical diaries and thousands of letters that are now a primary archive for twentieth-century number theory. The Hasse diagram of a partial order is named after his use of such diagrams.

Sources and further reading

Biography and history

1. Helmut Hasse, MacTutor History of Mathematics.
2. H. M. Edwards, Hasse, Helmut, Dictionary of Scientific Biography.
3. G. Frei, P. Roquette, Helmut Hasse, biographical sketch.
4. P. Roquette, note on Hasse’s ancestry and the Tornier report (2012).
5. P. Roquette (ed.), Documents on Emmy Noether, 1933: Hasse’s letters to the Kurator and the collected assessments.
6. F. Lemmermeyer, P. Roquette (eds.), Helmut Hasse und Emmy Noether: Die Korrespondenz 1925–1935, Universitätsverlag Göttingen, 2006.
7. S. Mac Lane, Mathematics at Göttingen under the Nazis, Notices AMS 42 (1995).
8. S. L. Segal, Helmut Hasse in 1934, Historia Mathematica (1980); and Mathematicians under the Nazis, Princeton University Press, 2003.
9. Werner Weber (mathematician), Wikipedia, on the 1934 appointment.
10. Helmut Hasse, Wikipedia.

Mathematics

1. H. Hasse, Über die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen, J. reine angew. Math. 152 (1923).
2. H. Hasse, Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, Jahresbericht DMV, 1926–1930.
3. H. Hasse, Über ℘-adische Schiefkörper und ihre Bedeutung für die Arithmetik hyperkomplexer Zahlsysteme, Math. Ann. 104 (1931).
4. R. Brauer, H. Hasse, E. Noether, Beweis eines Hauptsatzes in der Theorie der Algebren, J. reine angew. Math. 167 (1932).
5. H. Davenport, H. Hasse, Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen, J. reine angew. Math. 172 (1935).
6. H. Hasse, Zur Theorie der abstrakten elliptischen Funktionenkörper I–III, J. reine angew. Math. 175 (1936).
7. P. Roquette, The Riemann hypothesis in characteristic p, its origin and development.
8. H. Hasse, Zahlentheorie, Akademie-Verlag, 1949; English translation Number Theory, Springer, 1980.
9. J.-P. Serre, A Course in Arithmetic, Springer, 1973 (p-adic numbers, Hilbert symbol, Hasse–Minkowski).
10. J. W. S. Cassels, A. Fröhlich (eds.), Algebraic Number Theory, Academic Press, 1967.
11. J. H. Silverman, The Arithmetic of Elliptic Curves, Springer, 2nd ed. 2009.