Adolf Hurwitz

How to use this course

Adolf Hurwitz is a name most students meet many times without meeting the man. Engineers know the Routh–Hurwitz criterion. Number theorists know the Hurwitz zeta function, Hurwitz quaternions and Hurwitz class numbers. Geometers know the Riemann–Hurwitz formula and the bound of \(84(g-1)\) automorphisms. Algebraists know the theorem that composition algebras exist only in dimensions 1, 2, 4 and 8. This course gathers these results and places them in the life of a mathematician who published relatively little, taught superbly, and shaped two of the greatest minds of the next generation.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons run roughly in the order Hurwitz did the work, from his modular-function thesis of 1881 to his quaternion book of 1919.

Timeline

Hildesheim, 1859–1877

Adolf Hurwitz was born on 26 March 1859 in Hildesheim, a cathedral town south of Hanover. His father, Salomon Hurwitz, worked in manufacturing and was not wealthy. His mother, Elise Wertheimer, died when Adolf was about three. He had two brothers, Julius and Max.

In 1868 he entered the Realgymnasium Andreanum. A Realgymnasium stressed modern languages and science over Greek, a choice that later had consequences: some universities required Greek for the habilitation. His mathematics teacher was Hermann Schubert, the founder of enumerative geometry, whose “calculus of conditions” counted geometric configurations such as the number of conics tangent to five given conics. Schubert recognized the boy’s talent and wrote a paper with him on a theorem of Michel Chasles while Hurwitz was still at school, at about seventeen.

With Schubert’s encouragement he went on to study mathematics at Munich, where Felix Klein was then teaching.

Munich, Berlin, Leipzig, Göttingen, 1877–1884

Hurwitz entered the University of Munich in 1877 and spent a year in Klein’s lectures. In 1877–78 he moved to Berlin, then the strongest school of analysis and number theory in the world, and heard Ernst Kummer, Karl Weierstrass and Leopold Kronecker. He absorbed Weierstrass’s rigour and Kronecker’s arithmetic, and kept both for life. He returned to Munich in 1879, and when Klein moved to Leipzig in 1880 Hurwitz followed him.

The doctorate

Klein was then building his theory of elliptic modular functions. Hurwitz’s dissertation, Grundlagen einer independenten Theorie der elliptischen Modulfunktionen und Theorie der Multiplikatorgleichungen erster Stufe (Leipzig, 1881), placed the theory on foundations independent of elliptic functions, defining modular functions directly through their behaviour under the modular group. Lesson 04 describes what he built on it. A bout of typhoid as a student left him with migraines that troubled him for the rest of his life.

Habilitation

He spent 1881–82 again in Berlin, where Weierstrass suggested the topic for his habilitation. Leipzig required a classical education in Greek for the habilitation, which Hurwitz lacked, and as a Jew he faced further obstacles. Göttingen accepted him in 1882, and he spent two years there as a Privatdozent. That spring, on 25 April 1882, he opened the first of a long series of mathematical diaries. These notebooks, now in the ETH archive, record ideas, computations and drafts across almost four decades and are a major source for historians of the period.

The zeta-function paper of Lesson 05 also dates from this time: it is signed in Hildesheim on 10 October 1881.

Königsberg, 1884–1892

In 1884 Ferdinand Lindemann, who had proved the transcendence of \(\pi\) two years earlier, brought Hurwitz to Königsberg as extraordinary professor. Hurwitz was 25. Among the students were two locals of almost the same age: David Hilbert, then 22, and Hermann Minkowski, 19, already famous for a prize essay on sums of five squares.

What followed is one of the great friendships in mathematics. Over eight years the three walked together regularly and talked mathematics. Hilbert later said that in these walks they explored every corner of mathematical knowledge, and that Hurwitz, with his broad and well-founded knowledge, was the teacher. Hilbert and Minkowski both took their doctorates in Königsberg in 1885 under Lindemann, but the education that formed them came largely from Hurwitz.

In Königsberg Hurwitz met Ida Samuel, daughter of a professor of medicine at the university, and married her. They had three children, Lisbeth, Eva and Otto, born between 1894 and 1898.

The Königsberg years were mathematically rich. Hurwitz worked on class-number relations, on the arithmetic of modular equations, on the approximation of irrationals by rationals (1891), and on Riemann surfaces with prescribed branching (1891). In 1892 Georg Frobenius left the Zürich Polytechnikum for Berlin, and Hurwitz was called to succeed him.

Modular functions and class numbers

For the general reader. Some functions repeat themselves under a whole group of symmetries of the upper half plane. Klein’s invariant \(j\) is the most famous. Hurwitz used these functions to count quadratic forms: the number of solutions of certain equations between values of \(j\) encodes how many essentially different quadratic forms there are with a given discriminant.

The modular equation

The group \(\mathrm{SL}_2(\mathbb Z)\) acts on the upper half plane \(\mathbb H\) by \(\tau\mapsto\frac{a\tau+b}{c\tau+d}\). Klein’s function \(j(\tau)=q^{-1}+744+196884\,q+\cdots\), with \(q=e^{2\pi i\tau}\), is invariant and identifies \(\mathrm{SL}_2(\mathbb Z)\backslash\mathbb H\) with \(\mathbb C\). For a positive integer \(n\), the functions \(j(M\tau)\), where \(M\) runs over integer matrices of determinant \(n\) modulo left multiplication by \(\mathrm{SL}_2(\mathbb Z)\), are permuted by the group. There are \(\psi(n)=n\prod_{p\mid n}(1+1/p)\) of them, and they are the roots of a polynomial

The equation \(\Phi_n(X,Y)=0\) is the modular equation of level \(n\). Hurwitz’s thesis studied the closely related multiplier equations from first principles.

Class numbers

For \(N>0\) with \(N\equiv0,3\pmod 4\), let \(H(N)\) be the number of \(\mathrm{SL}_2(\mathbb Z)\)-classes of positive definite binary quadratic forms \(ax^2+bxy+cy^2\) with \(b^2-4ac=-N\), not necessarily primitive, where the class of \(a(x^2+y^2)\) counts \(\tfrac12\) and that of \(a(x^2+xy+y^2)\) counts \(\tfrac13\). Set \(H(0)=-\tfrac1{12}\). These weighted counts are the Hurwitz class numbers.

Proof idea

Count the zeros of the one-variable polynomial \(\Phi_n(X,X)\) in two ways. On one side, its degree comes from the leading terms of \(\Phi_n\) and is governed by the divisor sum. On the other, a root \(X=j(\tau)\) means \(j(M\tau)=j(\tau)\), so \(\tau\) is fixed by some matrix of determinant \(n\): \(\tau\) is a quadratic irrationality and the lattice \(\mathbb Z+\mathbb Z\tau\) has an endomorphism of norm \(n\) and trace \(t\). Such \(\tau\) correspond to quadratic forms of discriminant \(t^2-4n\), so each trace \(t\) contributes \(H(4n-t^2)\). The weights \(\tfrac12,\tfrac13\) record the extra symmetry at \(\tau=i\) and \(\tau=e^{2\pi i/3}\), and the correction term comes from the behaviour at the cusp. Kronecker had found such relations in 1860 by other means; Hurwitz’s papers of the mid-1880s gave the modular-function proofs and new relations of the same kind. The modern descendant of this argument is the Eichler–Selberg trace formula.

The Hurwitz zeta function

For the general reader. Riemann’s zeta function adds up \(1/n^s\) over all whole numbers. Hurwitz shifted the numbers by a fixed amount and added \(1/(n+a)^s\). The shifted sums behave as well as Riemann’s, and they are the building blocks for the sums over arithmetic progressions that Dirichlet used to prove there are infinitely many primes in each progression.

For \(0<a\le1\) and \(\operatorname{Re}s>1\) define

Then \(\zeta(s,1)=\zeta(s)\), and for a Dirichlet character \(\chi\) modulo \(m\), \(L(s,\chi)=m^{-s}\sum_{b=1}^{m}\chi(b)\,\zeta(s,b/m)\). Hurwitz worked with rational \(a=b/m\), which is all that Dirichlet series need, in a paper written in Hildesheim in October 1881 and published in Schlömilch’s Zeitschrift für Mathematik und Physik in 1882.

Continuation

From \(\Gamma(s)\,(n+a)^{-s}=\int_0^\infty e^{-(n+a)x}x^{s-1}\,dx\) and summation of the geometric series,

Riemann’s contour-integral trick turns this into a representation valid for all \(s\). The result is that \(\zeta(s,a)\) is meromorphic on \(\mathbb C\) with a single simple pole at \(s=1\), of residue 1.

Taking \(m=1\) recovers Riemann’s functional equation \(\zeta(1-s)=2(2\pi)^{-s}\cos(\pi s/2)\,\Gamma(s)\,\zeta(s)\). Combining with characters, Hurwitz derived the functional equations of the Dirichlet \(L\)-functions of quadratic characters, and used them for Dirichlet’s class-number formula.

Values at negative integers

Expanding the generating function \(\frac{x e^{(1-a)x}}{e^x-1}\) in Bernoulli polynomials gives, for \(n\ge0\),

Riemann surfaces: branching and symmetry

For the general reader. A Riemann surface can be folded onto another, like a many-sheeted cover, with the sheets meeting at a few branch points. Hurwitz found the exact bookkeeping that relates the number of holes of the two surfaces. He then used it to show that a surface with at least two holes has only finitely many symmetries, and at most \(84(g-1)\) of them.

Proof sketch. Triangulate \(Y\) so that the branch values are vertices and lift the triangulation to \(X\). Faces and edges lift to \(n\) copies each, and a vertex \(Q\) has \(n-\sum_{f(P)=Q}(e_P-1)\) preimages. Hence \(\chi(X)=n\,\chi(Y)-\sum_P(e_P-1)\), and \(\chi=2-2g\). Riemann had used the formula for covers of the sphere; Hurwitz’s paper of 1891 on Riemann surfaces with given branch points stated it in general and also began the count of covers with prescribed branching, numbers now called Hurwitz numbers and central in enumerative geometry.

Proof of the bound. Hermann Schwarz had shown finiteness earlier by other means; Hurwitz gave a new proof and the bound. Let \(G\) be a finite group of automorphisms and apply the formula to the quotient \(X\to X/G\), of degree \(|G|\). If \(X/G\) has genus \(h\) and branch values with stabilizer orders \(m_1,\dots,m_r\), then

The bracket is positive because \(g\ge2\). A case analysis shows its smallest positive value is \(\tfrac1{42}\), reached only for \(h=0\) and \((m_1,m_2,m_3)=(2,3,7)\): indeed \(-2+\tfrac12+\tfrac23+\tfrac67=\tfrac1{42}\). Therefore \(|G|\le 42(2g-2)=84(g-1)\).

The bound is attained in genus 3 by Klein’s quartic \(x^3y+y^3z+z^3x=0\), whose automorphism group is \(\mathrm{PSL}_2(\mathbb F_7)\) of order \(168=84\cdot2\). Curves attaining the bound are called Hurwitz curves and their groups Hurwitz groups.

Approximating irrationals

For the general reader. Every irrational number can be approximated by fractions, and the question is how well, measured against the size of the denominator. Hurwitz found the best universal constant: \(\sqrt5\). The number that resists approximation most stubbornly is the golden ratio.

Write \(\xi=[a_0;a_1,a_2,\dots]\) with convergents \(p_n/q_n\). Dirichlet’s principle gives infinitely many fractions with \(|\xi-p/q|<1/q^2\), and every convergent satisfies \(|\xi-p_n/q_n|<1/(q_nq_{n+1})\).

Proof sketch

Hurwitz proved that of any three consecutive convergents, at least one satisfies the inequality. Write \(\xi-\frac{p_n}{q_n}=\frac{(-1)^n}{q_n^2(\xi_{n+1}+q_{n-1}/q_n)}\), where \(\xi_{n+1}=[a_{n+1};a_{n+2},\dots]\). The inequality fails at \(n\) exactly when \(\xi_{n+1}+q_{n-1}/q_n\le\sqrt5\). Put \(\lambda_n=q_{n-1}/q_n\). If it fails at \(n-1\) and at \(n\), the identities \(\xi_n=a_n+1/\xi_{n+1}\) and \(1/\lambda_n=a_n+\lambda_{n-1}\) combine to give \(\lambda_n+1/\lambda_n\le\sqrt5\), hence \(\lambda_n>\frac{\sqrt5-1}{2}\) (equality is impossible for a rational number). If it also fails at \(n+1\), the same holds for \(\lambda_{n+1}\). But \(1/\lambda_{n+1}=a_{n+1}+\lambda_n>1+\frac{\sqrt5-1}{2}=\frac{\sqrt5+1}{2}\), so \(\lambda_{n+1}<\frac{\sqrt5-1}{2}\), a contradiction.

For optimality take \(\xi=\varphi=\frac{1+\sqrt5}2=[1;1,1,\dots]\), whose convergents are ratios of Fibonacci numbers \(F_{n+1}/F_n\). Here \(q_n^2|\varphi-p_n/q_n|\to1/\sqrt5\), and any good approximation must be a convergent once \(A>2\).

Removing \(\varphi\) and its equivalents, the best constant jumps to \(\sqrt8\) (attained by \(\sqrt2\)), then to \(\sqrt{221}/5\), and so on, accumulating at 3. This is the Lagrange spectrum, described by Andrei Markov’s theory of 1879–80, which Hurwitz’s theorem complements.

The stability criterion

For the general reader. A machine controlled by feedback, such as a steam turbine with a speed governor, is stable when small disturbances die away. Mathematically this means that all roots of a certain polynomial have negative real part. Hurwitz gave a test that decides this from the coefficients alone, without finding any root. Engineers still teach it in every control course.

Aurel Stodola, professor of mechanical engineering at the Zürich Polytechnikum, was designing turbine regulators and needed to know when the linearized equations were stable. He asked Hurwitz. The answer appeared in 1895 in Mathematische Annalen as Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt. Edward Routh in Cambridge had reached an equivalent test in the 1870s by a different route.

Let \(p(s)=a_0s^n+a_1s^{n-1}+\cdots+a_n\) with real coefficients and \(a_0>0\). Its Hurwitz matrix is the \(n\times n\) matrix with entries \(h_{ij}=a_{2j-i}\), where \(a_k=0\) for \(k<0\) or \(k>n\):

The first minors are \(\Delta_1=a_1\), \(\Delta_2=a_1a_2-a_0a_3\), and always \(\Delta_n=a_n\Delta_{n-1}\).

Proof idea

By the argument principle, \(p\) has all its roots in the left half plane exactly when \(\arg p(i\omega)\) increases by \(n\pi\) as \(\omega\) runs from \(-\infty\) to \(\infty\). Splitting \(p(i\omega)\) into real and imaginary parts, this is equivalent to the real zeros of the even and odd parts interlacing (the Hermite–Biehler theorem), which is in turn a statement about the Cauchy index of their ratio. Charles Hermite had expressed such indices through the signature of a quadratic form in 1856. Hurwitz reduced that form to the determinants \(\Delta_k\), and positivity of all of them is positive definiteness.

Quaternions and the 1, 2, 4, 8 theorem

For the general reader. The product of two sums of two squares is a sum of two squares; the same holds for four squares and for eight. Hurwitz proved that these, with the trivial case of one square, are the only possibilities. He also found the right “integers” among Hamilton’s quaternions, and with them gave a clean proof that every number is a sum of four squares.

Integral quaternions (1896)

Let \(N(q)=q\bar q=a^2+b^2+c^2+d^2\) for \(q=a+bi+cj+dk\). The obvious lattice of quaternions with integer coordinates (the Lipschitz quaternions) lacks a division algorithm. Hurwitz enlarged it:

From this, an odd prime \(p\) divides some \(1+x^2+y^2\) (by counting squares mod \(p\)), so \(p\) is not irreducible in \(\mathcal H\), and a factorization \(p=\pi\bar\pi\) with \(N(\pi)=p\) writes \(p\) as a sum of four squares, after adjusting \(\pi\) by a unit to have integer coordinates. Multiplicativity of \(N\) extends this to all integers. Hurwitz developed this theory fully in his book Vorlesungen über die Zahlentheorie der Quaternionen (1919), where he also derived Jacobi’s formula for the number of representations, \(r_4(n)=8\sum_{d\mid n,\ 4\nmid d}d\).

Composition of quadratic forms (1898)

Proof sketch. Write \(z=A(y)x\) with \(A(y)=\sum_j y_jA_j\). The identity says \(A(y)^{\mathsf T}A(y)=|y|^2I\), which gives \(A_j^{\mathsf T}A_j=I\) and \(A_j^{\mathsf T}A_k+A_k^{\mathsf T}A_j=0\) for \(j\ne k\). Replacing \(A_j\) by \(A_n^{\mathsf T}A_j\) we may take \(A_n=I\). Then \(B_j=A_j\) for \(j<n\) are \(n-1\) real \(n\times n\) matrices with \(B_j^2=-I\) and \(B_jB_k=-B_kB_j\). Hurwitz showed that products of distinct \(B_j\) yield too many linearly independent matrices in the \(n^2\)-dimensional space of matrices unless \(n\le 8\), and parity arguments exclude the other values. The four cases correspond to the real numbers, complex numbers, quaternions and octonions.

In a posthumous paper of 1923, and independently Johann Radon in 1922, he answered the general question: with \(n=2^{4a+b}\cdot(\text{odd})\), \(0\le b\le3\), there are at most \(\rho(n)-1\) such anticommuting matrices, where \(\rho(n)=8a+2^b\). These Radon–Hurwitz numbers reappeared in 1962 when Frank Adams proved that \(\rho(n)-1\) is the maximum number of independent vector fields on the sphere \(S^{n-1}\).

Fourier series and invariant integrals

For the general reader. Among all closed curves of a given length, the circle encloses the most area. Hurwitz gave a proof in a few lines by writing the curve as a sum of circular motions. In another short paper he showed how to average over all rotations of space, an idea that became the foundation of the representation theory of compact groups.

Proof for smooth curves. Parametrize by arc length scaled to \(t\in[0,2\pi]\), so \(|z'(t)|=\frac{L}{2\pi}\) for \(z=x+iy\). Write \(z(t)=\sum_k c_ke^{ikt}\). Parseval gives \(\frac{L^2}{2\pi}=\int_0^{2\pi}|z'|^2dt=2\pi\sum k^2|c_k|^2\), and Green’s theorem gives \(A=\tfrac12\int\operatorname{Im}(\bar z z')\,dt=\pi\sum k|c_k|^2\). Hence

since \(k^2-k\ge0\) for every integer \(k\), with equality iff \(c_k=0\) for \(k\ne0,1\), that is, \(z\) is a circle. Hurwitz published the argument in a note to the Paris Academy in 1901 and in a longer paper in the Annales de l’École Normale Supérieure in 1902.

Invariant integration (1897)

To produce invariants of a finite group one averages over the group (the Reynolds operator). In Über die Erzeugung der Invarianten durch Integration (1897), Hurwitz replaced the sum by an integral over the orthogonal group \(\mathrm{SO}(n)\), using an explicit invariant volume form in Euler-angle coordinates, and also treated the special linear group. This is the first construction of what is now called Haar measure on a Lie group. Issai Schur and Hermann Weyl made it the main tool of their representation theory of compact groups in the 1920s.

Zürich, 1892–1919

In 1892 Hurwitz succeeded Georg Frobenius at the Eidgenössisches Polytechnikum in Zürich, the federal technical university that later became ETH Zürich. He remained there for 27 years. Göttingen later offered him a chair, which he declined because he had already given his word to Zürich.

He taught the full range of mathematics to engineers and to the future mathematics teachers of Switzerland, and his lectures were famous for their clarity. Among his students at the Polytechnikum in the late 1890s was Albert Einstein, who preferred physics and was not a regular attender. On 30 September 1900, after graduating, Einstein wrote from Milan to ask whether he might succeed Hurwitz’s departing assistant, Karl Matter. He did not get the post. Hurwitz supervised about twenty doctoral theses, at the University and the Polytechnikum, among them those of Ernst Meissner (1907), who later wrote the memoir for his collected papers, and Louis-Gustave Du Pasquier (1906).

In 1897 Hurwitz organized the first International Congress of Mathematicians, held in Zürich, gave one of the plenary lectures, and chaired a section. The Congress started the series that continues today. Engineering colleagues brought him problems: the stability criterion of Lesson 08 answered a question from Aurel Stodola. In 1914 the young Hungarian George Pólya joined the Polytechnikum on his recommendation and became a close colleague.

His health was poor throughout. Kidney disease led to the removal of a kidney in 1905, and from then on he worked with frequent interruptions. In 1913 the London Mathematical Society elected him an honorary member.

Death and legacy

Hurwitz died in Zürich on 18 November 1919, aged 60. Earlier that year his book on the number theory of quaternions had appeared. Hilbert wrote his obituary for Mathematische Annalen, recalling the Königsberg walks and the debt he and Minkowski owed to their older friend.

Three posthumous projects preserved his work. Richard Courant edited his Zürich lectures on function theory into Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen (1922), adding a part on geometric function theory; the Hurwitz–Courant volume became a standard text. Pólya, as literary executor, oversaw the Mathematische Werke (1932–33), with a memoir by Ernst Meissner. The mathematical diaries remain in the ETH library and continue to be studied.

Legacy

• Number theory: the Hurwitz zeta function, Hurwitz class numbers and the class-number relations, Hurwitz quaternions, and the \(\sqrt5\) theorem in Diophantine approximation.
• Geometry: the Riemann–Hurwitz formula, Hurwitz numbers in enumerative geometry and string theory, and Hurwitz curves and groups.
• Algebra and topology: composition algebras in dimensions 1, 2, 4, 8; Radon–Hurwitz numbers and vector fields on spheres.
• Analysis and engineering: the Routh–Hurwitz criterion; Hurwitz’s theorem on zeros; the first invariant integral on a Lie group.
• People: the mathematical education of Hilbert and Minkowski, and through them much of Göttingen mathematics.

Connections

People linked to Adolf Hurwitz across this collection. A name without a link has no page here yet. See Adolf Hurwitz in the network of all 337 lives →

Teachers and advisors

• Felix Klein 1881
• Wilhelm Scheibner 1881

Students

• Albert Einstein 1896-1900
• Ernst Amberg 1897
• Ernst Meissner 1907
• Louis-Gustave Du Pasquier 1906

Influenced by

• Andrey Markov
• Ernst Kummer 1877-1878
• Karl Weierstrass 1877-1882
• Leopold Kronecker 1877-1882
• Hermann Schubert

Influenced

• Arnold Sommerfeld
• Hermann Weyl
• Issai Schur
• Richard Courant

Colleagues and collaborators

• Georg Frobenius
• Richard Courant
• Aurel Stodola
• Ferdinand Lindemann 1884-1892
• George Pólya 1914-1919
• Johann Radon

Friends

• David Hilbert 1884-1919
• Hermann Minkowski 1884-1909

Also mentioned in this life

• Augustin-Louis Cauchy
• Michel Chasles
• Peter Gustav Lejeune Dirichlet
• Leonhard Euler
• Pierre de Fermat
• Fibonacci
• Joseph Fourier
• William Rowan Hamilton
• Charles Hermite
• Carl Gustav Jacob Jacobi
• Joseph-Louis Lagrange
• Adrien-Marie Legendre
• Sophus Lie
• Osborne Reynolds
• Bernhard Riemann
• Atle Selberg

Fields

• number theory 45 lives
• complex analysis 13 lives
• riemann surfaces 3 lives
• algebra 29 lives

On CoursesHub

• Complex Analysis · Mathematics for Physics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Adolf Hurwitz, MacTutor History of Mathematics.
2. Adolf Hurwitz, Wikipedia.
3. Adolf Hurwitz, Mathematics Genealogy Project.
4. N. Oswald and J. Steuding, Aspects of zeta-function theory in the mathematical works of Adolf Hurwitz, arXiv:1506.00856.
5. D. Hilbert, Adolf Hurwitz, Mathematische Annalen 83 (1921).
6. The Collected Papers of Albert Einstein, vol. 1, Princeton University Press, 1987: letter to Hurwitz of 30 September 1900.
7. Obituary of Hurwitz, London Mathematical Society (MacTutor scan).

Mathematics

1. A. Hurwitz, Einige Eigenschaften der Dirichlet’schen Functionen \(F(s)=\sum(D/n)\,n^{-s}\), Zeitschrift für Mathematik und Physik 27 (1882).
2. A. Hurwitz, Über die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche, Math. Ann. 39 (1891).
3. A. Hurwitz, Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891).
4. A. Hurwitz, Über algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1893).
5. A. Hurwitz, Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt, Math. Ann. 46 (1895).
6. A. Hurwitz, Über die Composition der quadratischen Formen von beliebig vielen Variablen, Nachr. Ges. Wiss. Göttingen (1898).
7. A. Hurwitz, Vorlesungen über die Zahlentheorie der Quaternionen, Springer, 1919.
8. A. Hurwitz and R. Courant, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen, Springer, 1922.
9. A. Hurwitz, Mathematische Werke, 2 vols., Birkhäuser, 1932–33.
10. J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters, 2003.
11. D. Zagier, Nombres de classes et formes modulaires de poids 3/2, C. R. Acad. Sci. Paris 281 (1975).