Erich Hecke

How to use this course

Erich Hecke’s name is on some of the most used objects in number theory: Hecke characters, Hecke L-functions, Hecke operators, Hecke algebras, Hecke groups. Behind them is one idea, pursued for thirty years. Arithmetic questions about primes, class numbers and quadratic forms are encoded in analytic functions: zeta functions, theta functions and modular forms. Their symmetries, in the form of functional equations and transformation laws, carry the arithmetic information. Hecke made this dictionary systematic.

Each technical lesson opens with an italic paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow the chronology of Hecke’s work, from the analytic theory of number fields (1917–1923) to modular forms and Dirichlet series (1925–1940).

Timeline

Posen and Breslau, 1887–1907

Erich Hecke was born on 20 September 1887 in Buk, a small town in the Prussian province of Posen, now in western Poland. His father, Heinrich Hecke, was a master builder; he died in 1893, when Erich was five. His mother was Hermine Hecke, née Schubert. Hecke went to primary school in Buk and secondary school in the provincial capital, Posen, graduating in 1905.

That year he entered the University of Breslau to study mathematics and the natural sciences. Like many German students of the time he moved between universities. In Berlin he came under the influence of Edmund Landau, then a young lecturer and already the leading German analytic number theorist. Landau’s precise, complete style of analysis, and his interest in zeta functions of number fields, shaped Hecke’s early work. From Berlin he went on to Göttingen.

Hilbert’s Göttingen, 1907–1915

At Göttingen Hecke became a student of David Hilbert. Hilbert’s twelfth problem asked for analytic functions that generate the abelian extensions of an arbitrary number field, as the exponential function does for \(\mathbb Q\) and elliptic modular functions do for imaginary quadratic fields, a programme known as Kronecker’s Jugendtraum. Hilbert proposed that Hecke try the next case: real quadratic fields, using modular functions of two complex variables, the functions now called Hilbert modular functions.

Hecke’s dissertation of 1910, Zur Theorie der Modulfunktionen von zwei Variablen und ihrer Anwendung auf die Zahlentheorie, took the first steps. It studied these functions and their use in constructing class fields over real quadratic fields. The theme of modular functions attached to number fields stayed with him all his life.

Hecke served as assistant to Hilbert and to Felix Klein, and habilitated at Göttingen in 1912. In 1913 he married Helga Unruh in Leipzig. They had one son, who died young.

Basel, Göttingen, Hamburg, 1915–1919

In 1915 Hecke was called to the University of Basel in neutral Switzerland, first as extraordinary and from 1916 as full professor. The Basel years, during the First World War, were the most productive of his early career. In 1917 he proved that the Dedekind zeta function of every number field continues to the whole complex plane and satisfies a functional equation (Lesson 04). In 1918 he introduced a new kind of character on the ideals of a number field and the L-functions attached to them (Lesson 07).

In 1918 he returned to Göttingen as professor. He stayed only a year. In 1919 the city of Hamburg founded a university, and Hecke accepted its first chair of mathematics, turning down an offer from Berlin to do so. With the geometer Wilhelm Blaschke, who arrived soon after, he built a new department and founded its journal, the Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. Within a few years it was one of the leading centres of mathematics in Europe.

Hecke never left Hamburg, despite several further offers. His lecture course there in 1920 on analysis and number theory survives in notes and was published in 1987.

Theta functions and the Dedekind zeta function

For the general reader. Riemann showed that the zeta function, built from the primes, has a hidden mirror symmetry, and he proved it with a theta function, a sum of Gaussian exponentials. Dedekind had defined a zeta function for every number field, but nobody could prove it had the same symmetry. In 1917 Hecke did, by building a theta function for the number field.

Riemann’s proof

Let \(\vartheta(t)=\sum_{n\in\mathbb Z}e^{-\pi n^2t}\) for \(t>0\). Poisson summation applied to the Gaussian \(e^{-\pi x^2t}\), whose Fourier transform is \(t^{-1/2}e^{-\pi\xi^2/t}\), gives

For \(\operatorname{Re}s>1\), integrating term by term,

Split the integral at \(t=1\) and use the transformation formula on \((0,1)\):

The integral converges for all \(s\), and the right side is visibly symmetric under \(s\mapsto1-s\). So \(\xi\) continues meromorphically to \(\mathbb C\), with simple poles at \(0\) and \(1\), and \(\xi(s)=\xi(1-s)\).

Hecke’s theorem

Let \(K\) be a number field of degree \(n=r_1+2r_2\), with \(r_1\) real and \(r_2\) pairs of complex embeddings, discriminant \(d_K\), class number \(h\), regulator \(R\), and \(w\) roots of unity. The Dedekind zeta function is \(\zeta_K(s)=\sum_{\mathfrak a}N\mathfrak a^{-s}=\prod_{\mathfrak p}(1-N\mathfrak p^{-s})^{-1}\), summed over nonzero ideals of \(\mathcal O_K\).

Proof idea. Split \(\zeta_K\) into partial zeta functions, one per ideal class. For a class \(C\), fix an ideal \(\mathfrak b\) in the inverse class; ideals in \(C\) correspond to principal ideals \((\alpha)\subset\mathfrak b\), that is, to nonzero \(\alpha\in\mathfrak b\) modulo units. Hecke wrote the partial zeta function as a Mellin transform of a theta function in \(n\) variables, \(\sum_{\alpha\in\mathfrak b}\exp\big(-\pi\sum_j t_j|\alpha^{(j)}|^2\big)\), and applied Poisson summation on the lattice \(\mathfrak b\subset\mathbb R^n\). The dual lattice is \((\mathfrak b\mathfrak d)^{-1}\), with \(\mathfrak d\) the different, which is where \(|d_K|\) enters. The units act on the variables \(t_j\), and integrating over a fundamental domain for this action produces the regulator.

Gauss sums and reciprocity

For the general reader. The law of quadratic reciprocity, which Gauss called the golden theorem, relates whether \(p\) is a square modulo \(q\) to whether \(q\) is a square modulo \(p\). Gauss gave several proofs. A later one, going back to Cauchy and Kronecker, derives it from the same theta transformation that gives the functional equation of zeta. Hecke carried that proof over to every number field, and his 1923 textbook ends with it.

Theta functions near the boundary

Extend the theta function to the upper half-plane, \(\theta(\tau)=\sum_n e^{\pi in^2\tau}\), so that \(\theta(-1/\tau)=\sqrt{\tau/i}\,\theta(\tau)\). Letting \(\tau\) tend to a rational point \(2q/p\) on the real axis, the theta function blows up at a rate governed by a finite sum. Comparing both sides of the transformation formula gives the reciprocity formula for Gauss sums.

With \(q=1\) the right side is \(\frac{e^{\pi i/4}}{\sqrt2}(1+e^{-\pi ip/2})\), which gives Gauss’s evaluation of \(g(p)=\sum_{n=0}^{p-1}e^{2\pi in^2/p}\): \(\sqrt p\), \(0\), \(i\sqrt p\), \((1+i)\sqrt p\) according as \(p\equiv1,2,3,0\pmod 4\). For an odd prime \(p\), \(g(p)=\sum_{a}\big(\tfrac ap\big)e^{2\pi ia/p}\), so \(g(p)^2=\big(\tfrac{-1}{p}\big)p\). Quadratic reciprocity follows by computing \(g(p)^{q}\) modulo \(q\) in two ways.

The book, Vorlesungen über die Theorie der algebraischen Zahlen (1923), became a classic. André Weil wrote in the preface to Basic Number Theory that no one could hope to improve on Hecke’s treatment of classical algebraic number theory.

Zeros and class numbers

For the general reader. Gauss conjectured that only finitely many imaginary quadratic fields have any given class number. Hecke saw that the question is really about where the zeros of an L-function lie. If no zero sits too close to \(s=1\), the class number must be large. His theorem connecting the two set the agenda for Hans Heilbronn, Carl Ludwig Siegel and everyone after them.

For a fundamental discriminant \(-d<-4\), Dirichlet’s class number formula is \(h(-d)=\frac{\sqrt d}{\pi}L(1,\chi_{-d})\). A lower bound for \(h\) is a lower bound for \(L(1,\chi_{-d})\). If \(L(s,\chi_{-d})\) had a real zero \(\beta\) very near 1, then by the mean value theorem \(L(1,\chi_{-d})=(1-\beta)L'(\sigma,\chi_{-d})\) for some \(\sigma\in(\beta,1)\), and since \(L'\ll\log^2d\) near 1, \(L(1,\chi_{-d})\) would be small.

Under the generalized Riemann hypothesis the hypothesis always holds, so \(h(-d)\to\infty\). Without it, the story continued in a remarkable way. Max Deuring (1933) showed that if the Riemann hypothesis for \(\zeta\) is false then \(h(-d)\ge2\) for large \(d\); Louis Mordell strengthened this, and Heilbronn (1934) showed that the falsity of the generalized Riemann hypothesis also implies \(h(-d)\to\infty\). Since one hypothesis or the other holds, \(h(-d)\to\infty\) unconditionally. Siegel’s 1935 theorem then gave \(h(-d)\gg_\varepsilon d^{1/2-\varepsilon}\), ineffectively.

Grössencharaktere and Hecke L-functions

For the general reader. Dirichlet proved that primes are evenly spread among residue classes by attaching an L-function to each character of the multiplicative group modulo \(q\). Hecke asked what plays this role in a general number field, where primes also have a ‘direction’, for example the angle of a Gaussian prime \(a+bi\). His answer, characters that see both residues and angles, gave the most general abelian L-functions. In modern language they are the characters of the idele class group, and they are the one-dimensional case of the Langlands programme.

Definition

Let \(\mathfrak m\) be an ideal of \(\mathcal O_K\) and \(I^{\mathfrak m}\) the group of fractional ideals prime to \(\mathfrak m\). A Grössencharakter (Hecke character) mod \(\mathfrak m\) is a homomorphism \(\lambda:I^{\mathfrak m}\to\mathbb C^\times\), \(|\lambda|=1\) in the unitary case, such that on principal ideals \((\alpha)\) with \(\alpha\equiv1\bmod^\times\mathfrak m\),

for fixed integers \(u_j\) and real \(v_j\) (with \(u_j\in\{0,1\}\) at real places), compatible with the units. When all \(u_j,v_j\) vanish, \(\lambda\) is a character of a ray class group, of finite order. Hecke’s point was to allow infinite order.

Example. For \(K=\mathbb Q(i)\), every ideal is principal, \((\alpha)\), with \(\alpha\) determined up to the units \(\pm1,\pm i\). Then \(\lambda_m\big((\alpha)\big)=(\alpha/|\alpha|)^{4m}\) is well defined for each integer \(m\), and it measures the angle of \(\alpha\).

The proof again uses theta functions, now weighted by the character. In 1950 John Tate’s thesis, written under Emil Artin, reproved Hecke’s theorem by Fourier analysis on adeles, in which a Grössencharakter is simply a continuous character of \(\mathbb A_K^\times/K^\times\).

Primes in sectors

Non-vanishing on \(\operatorname{Re}s=1\) for all \(\lambda_m\), \(m\ne0\), gives through Weyl’s criterion and a Tauberian argument:

The Hecke correspondence

For the general reader. A modular form is a function on the upper half-plane with a symmetry under \(\tau\mapsto-1/\tau\). A Dirichlet series is a sum like the zeta function, with a functional equation. Hecke proved in 1936 that these are two descriptions of the same object. Given one, the Mellin transform produces the other, and the symmetry of one becomes the symmetry of the other. Converse theorems built on this idea are a basic tool of the Langlands programme.

For \(\lambda>0\) let \(G(\lambda)\) be the group generated by \(\tau\mapsto\tau+\lambda\) and \(\tau\mapsto-1/\tau\). It is discrete exactly when \(\lambda\ge2\) or \(\lambda=2\cos(\pi/q)\) for an integer \(q\ge3\); these are the Hecke groups. For \(\lambda=1\) one gets \(\mathrm{SL}_2(\mathbb Z)\) modulo \(\pm1\).

Proof idea. For \(y>0\), \(\Lambda(s)=\int_0^\infty\big(f(iy)-a_0\big)y^{s}\frac{dy}{y}\), the Mellin transform. Split at \(y=1\) and use (1) with \(\tau=iy\) on \((0,1)\), exactly as in Lesson 04. Conversely, Mellin inversion recovers \(f(iy)\) from \(\Lambda\), and shifting the contour across the strip, using the functional equation and the boundedness, gives (1) on the imaginary axis, hence everywhere by analytic continuation.

Hecke then counted solutions. For \(\lambda=2\cos(\pi/q)\) the space of solutions with given \(k,\gamma\) is finite-dimensional, with an explicit dimension; for \(\lambda>2\) it is infinite-dimensional. For \(\lambda=1\) and \(k=12\) the space of cusp solutions is spanned by Ramanujan’s \(\Delta\). André Weil’s 1967 converse theorem added twists by characters to characterize modular forms on \(\Gamma_0(N)\), a step towards the modularity conjecture for elliptic curves.

Hecke operators and Euler products

For the general reader. In 1916 Srinivasa Ramanujan noticed that the coefficients \(\tau(n)\) of a certain modular form behave multiplicatively, like the divisors of \(n\). Louis Mordell proved it in 1917 with a clever trick. Twenty years later Hecke explained why: a family of commuting linear operators acts on modular forms, and forms that are eigenvectors of all of them have multiplicative coefficients and L-functions that factor over primes. These Hecke operators are now the main tool of the theory.

Proof of \(a(n)=\lambda_n\). The coefficient of \(q^1\) in \(T_nf\) is \(\sum_{d\mid(1,n)}d^{k-1}a(n/d^2)=a(n)\). On the other hand it is \(\lambda_na(1)=\lambda_n\). The Euler product follows from the multiplicative relations among the \(T_n\), transferred to the \(a(n)\). In 1939 his student Hans Petersson introduced an inner product on cusp forms for which the \(T_n\) are self-adjoint, so \(S_k\) has a basis of eigenforms with real eigenvalues. Hecke also extended the theory to the congruence groups \(\Gamma_0(N)\), where the operators \(T_p\) with \(p\mid N\) need separate treatment, a problem later solved by the newform theory of Atkin and Lehner.

Theta series of quadratic forms

For the general reader. How many ways can \(n\) be written as a sum of four squares? The generating function of the answer is a modular form, and modular forms come in two pieces: an Eisenstein series, whose coefficients are explicit divisor sums, and a cusp form, whose coefficients are smaller. Hecke made this decomposition into a general method for counting representations by quadratic forms, the subject of his 1938 Princeton lectures and his last great memoir.

Let \(Q(x)=\tfrac12x^{\mathsf T}Ax\) with \(A\) a positive definite even integral symmetric matrix of even size \(2k\), and let \(N\) be the level (the least \(N\) with \(NA^{-1}\) even). The theta series is \(\theta_Q(\tau)=\sum_{x\in\mathbb Z^{2k}}q^{Q(x)}=\sum_{n\ge0}r_Q(n)q^n\).

Proof. The function \(y^{k/2}|g(x+iy)|\) is invariant under the group and decays at every cusp, so it is bounded by some \(C\). Then \(b(n)=\int_0^1g(x+iy)e^{-2\pi in(x+iy)}dx\) gives \(|b(n)|\le Cy^{-k/2}e^{2\pi ny}\). Take \(y=1/n\): \(|b(n)|\le Ce^{2\pi}n^{k/2}\).

The Eisenstein coefficients typically grow like \(n^{k-1}\), which beats \(n^{k/2}\) once \(k\ge3\), that is, in at least 6 variables. In 4 variables the cusp bound \(O(n)\) is not enough on its own; Martin Eichler (for weight 2, in 1954) and Pierre Deligne (in general, 1974) improved the bound for eigenform coefficients to \(O(n^{(k-1)/2+\varepsilon})\). Hecke’s method in this lesson and Siegel’s mass formula are two views of the same object: Siegel showed that \(E_Q\) is the genus average of the theta series.

The Hamburg school and the Nazi years, 1919–1945

In the 1920s Hamburg became, with Göttingen, the leading German centre of number theory and algebra. Hecke and Blaschke brought in Emil Artin as assistant in 1922 and pressed for his promotion to a full chair in 1926. Carl Ludwig Siegel had held a teaching assignment there in 1920–21. Visitors and young researchers included Otto Schreier, Hans Zassenhaus, Ernst Witt and many others. Hecke served as president of the German Mathematical Society in 1923 and from 1929 as an editor of Mathematische Annalen.

His doctoral students included Kurt Reidemeister (1921), Heinrich Behnke (1923), Hans Petersson (1925), Bruno Schoeneberg (1931), Wilhelm Maak (1936) and Hans Maass (1937). Petersson and Maass continued his theory of modular forms; Maass introduced the non-holomorphic eigenfunctions now called Maass forms. Reidemeister became a knot theorist and Behnke a founder of the German school of several complex variables. Hecke also wrote papers on integral equations in the kinetic theory of gases. His friends in Hamburg included the physicists Otto Stern and Wolfgang Pauli and the astronomer Walter Baade.

Under National Socialism

In November 1933, like most German professors, Hecke was listed among the signatories of the professors’ declaration of loyalty to Hitler and the National Socialist state. Within the university he was known as a critic of the regime, and during the war he made his opposition so openly that he was in constant danger of arrest. Horst Tietz, a student at the time, recalled a framed letter on Hecke’s office wall from a would-be circle-squarer who, told about Lindemann’s proof that \(\pi\) is transcendental, had answered that nothing was impossible for the German spirit. Hecke kept it as a comment on the times.

The department lost Artin, whose wife Natascha was of Jewish descent, in 1937. In 1934 Hecke, Blaschke and Artin asked for Petersson’s promotion; the request was refused. In 1936 he gave a plenary address at the International Congress in Oslo on recent progress in the theory of elliptic modular functions, and in 1938 he lectured at the Institute for Advanced Study in Princeton. He returned to Germany. That year he received the Ackermann–Teubner Memorial Award, and in 1943 he was elected to the Leopoldina.

Copenhagen, death and legacy, 1945–1947

Hamburg was heavily bombed in 1943, and after the war food was scarce. Hecke was already weakened by a long illness. In 1946, seriously ill, he accepted an invitation from Danish friends, among them Harald Bohr, and moved to Copenhagen. He died there of cancer on 13 February 1947, at the age of 59. Jakob Nielsen gave a memorial address. Pauli wrote to his widow of a friendship that had gone far beyond mathematics and physics.

His Mathematische Werke appeared in 1959, edited by his student Bruno Schoeneberg, with a critical appreciation by Carl Ludwig Siegel. His Princeton lectures were published in 1983, and his 1923 textbook appeared in English in 1981.

Legacy

• Automorphic forms: Hecke operators and Hecke algebras, with Petersson’s inner product, are the backbone of the modern theory. Eigenvalues of Hecke operators are the data that the Langlands correspondence matches with Galois representations.
• L-functions: Hecke L-functions are the \(\mathrm{GL}_1\) case of automorphic L-functions. Tate’s thesis turned them into harmonic analysis on adeles, the model for all later work.
• Converse theorems: Hecke’s correspondence, extended by Weil, Jacquet, Langlands and Piatetski-Shapiro, is a standard route to proving that a given L-function comes from an automorphic form.
• Arithmetic: modularity of elliptic curves, proved by Andrew Wiles and his collaborators, is a statement about Hecke eigenforms of weight 2.
• Teaching: his 1923 book remains a model introduction to algebraic number theory.

Connections

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

Teachers and advisors

• David Hilbert 1910

Students

• Bruno Schoeneberg 1931
• Hans Maass 1937
• Hans Petersson 1925
• Heinrich Behnke 1923
• Kurt Reidemeister 1921
• Wilhelm Maak 1936

Influenced by

• Bernhard Riemann
• Richard Dedekind
• Edmund Landau

Influenced

• André Weil
• Helmut Hasse 1918
• John Tate

Colleagues and collaborators

• Carl Ludwig Siegel 1920-1921
• Emil Artin 1922-1937
• Felix Klein
• Richard Courant
• Wilhelm Blaschke 1919-1945

Friends

• Wolfgang Pauli
• Harald Bohr 1946-1947

Also mentioned in this life

• Augustin-Louis Cauchy
• Pierre Deligne
• Peter Gustav Lejeune Dirichlet
• Leonhard Euler
• Joseph Fourier
• Georg Frobenius
• Évariste Galois
• Carl Friedrich Gauss
• Carl Gustav Jacob Jacobi
• Leopold Kronecker
• Robert Langlands
• Gottfried Wilhelm Leibniz
• Siméon Denis Poisson
• Jean-Pierre Serre
• Hermann Weyl
• Andrew Wiles

Fields

• number theory 45 lives
• algebraic number theory 9 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Erich Hecke, MacTutor History of Mathematics.
2. Hecke, Erich, Dictionary of Scientific Biography entry, via MacTutor (PDF).
3. Hecke, Erich, Neue Deutsche Biographie.
4. Erich Hecke, Wikipedia; Erich Hecke, German Wikipedia.
5. Erich Hecke, Mathematics Genealogy Project.
6. Hans Petersson, MacTutor (for the Hamburg department in the 1930s).
7. H. Petersson, Erich Hecke zum Gedächtnis, Abh. Math. Sem. Univ. Hamburg 16 (1949); W. Maak, Erich Hecke als Lehrer, same volume.

Mathematics

1. E. Hecke, Über die Zetafunktion beliebiger algebraischer Zahlkörper, Nachr. Ges. Wiss. Göttingen (1917).
2. E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen I, II, Math. Z. 1 (1918), 6 (1920).
3. E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen, Leipzig, 1923; English translation Lectures on the Theory of Algebraic Numbers, Springer, 1981.
4. E. Hecke, Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann. 112 (1936).
5. E. Hecke, Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung I, II, Math. Ann. 114 (1937).
6. E. Hecke, Lectures on Dirichlet Series, Modular Functions and Quadratic Forms (Institute for Advanced Study, 1938), Vandenhoeck & Ruprecht, 1983.
7. E. Hecke, Mathematische Werke, ed. B. Schoeneberg, Vandenhoeck & Ruprecht, 1959.
8. B. C. Berndt and M. I. Knopp, Hecke’s Theory of Modular Forms and Dirichlet Series, World Scientific, 2008; MAA review.
9. J. Neukirch, Algebraic Number Theory, Springer, 1999, chapter VII (Hecke L-series).
10. J.-P. Serre, A Course in Arithmetic, Springer, 1973, chapter VII (Hecke operators).