Abstract
During the 1880s, Frobenius published several papers investigating diverse aspects of the theory of abelian and theta functions. In this and the following chapter, three of these works from 1880 to 1884 will be discussed in detail. (Some other papers from this era dealing with theta functions with half-integer characteristics are considered more tangentially in Chapter 12.)
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Notes
- 1.
Functions satisfying (1) but not (2) are sometimes called trivial or degenerate abelian functions. They include constant functions (h = 0) and have the property that for every \(\varepsilon > 0\), they have a period ω with \(\parallel \omega \parallel <\varepsilon\).
- 2.
This according to Weierstrass [594, p. 9].
- 3.
A version of Weierstrass’ lectures was finally published in 1902, five years after his death [594], as part of his collected works.
- 4.
Riemann’s work is discussed in Section 11.4. Weierstrass’ more general conditions (now commonly known as Riemann’s conditions) are discussed in Section 11.2.
- 5.
The material under discussion here is now part of the theory of principally polarized abelian manifolds. For a clear modern account, including a simple example of a period matrix with no abelian functions, see Rosen’s expository article [508, pp. 96ff.].
- 6.
Since \(C = \frac{i} {2}(\Omega _{1}\Omega _{2}^{h} - \Omega _{2}\Omega _{1}^{h}) = \frac{i} {2}\Omega _{1}(\overline{T} - T)\Omega _{1}^{h} = \Omega _{1}\Psi \Omega _{1}^{h}\), C is positive definite if and only if Ψ is.
- 7.
For example, if [M Ω 1 ′] β denotes the βth column of M Ω 1 ′, then to say that [M Ω 1 ′] β is a \(\mathbb{Z}\)-linear combination of the columns of \(\Omega = \left (\begin{array}{cc} \Omega _{1} & \Omega _{2}\\ \end{array} \right )\) means that integers a α β and γ α β , α = 1, …, g, exist such that \([M\Omega _{1}^{\prime}]_{\beta } =\sum _{ \alpha =1}^{g}(a_{\alpha \beta }[\Omega _{1}]_{\alpha } +\gamma _{\alpha \beta }[\Omega _{2}]_{\alpha })\) for β = 1, …, g. These relations are equivalent to \(M\Omega _{1}^{\prime} = \Omega _{1}\mathrm{A} + \Omega _{2}\Delta \), where A = (a α β ) and Γ = (γ α β ).
- 8.
That is, using \(\Omega ^{\prime} = {M}^{-1}\Omega \tilde{A}\) again, the Hermitian symmetric matrix C ′ associated to Ω ′ is \(C^{\prime} = (i/2)\Omega ^{\prime}J{\Omega }^{h} = {M}^{-1}(i/2)\Omega (\tilde{A}J\tilde{{A}}^{t}){\Omega }^{h}{({M}^{-1})}^{h} = n({M}^{-1})C{({M}^{-1})}^{h},\) and so C ′ is positive definite when C is, provided that n > 0.
- 9.
- 10.
The nonsingularity of A + T Γ for every abelian \(\tilde{A}\) follows from the fact that \(T \in \mathfrak{H}_{g}\), as does the fact that \(T^{\prime} \in \mathfrak{H}_{g}\) [188, p. 105].
- 11.
Hermite presented his results in the Comptes rendus of the Paris Academy, and although the three-page limit was not yet in force, his substantial results were presented in outline.
- 12.
The reason Kronecker made these results known in 1866 was that Clebsch and Gordan had published a book on abelian functions that year in which they used similar elementary matrices to reduce a first-degree abelian matrix, not to I 2g but to several simple canonical forms [100, §86]. They also attributed the idea of using elementary matrices to reduce unimodular matrices to Kronecker [100, p. 308n], but evidently Kronecker wanted the world to know that he had actually applied his ideas to first-order abelian matrices and had obtained a better reduction than that of Clebsch and Gordan.
- 13.
In the case g = 2, Kronecker’s four matrices correspond to the elementary row operations (1) − row 1 → row 3 and row 3 → row 1; (2) add row 1 to row 3; (3) switch rows 1 and 2 and rows 3 and 4; (4) add row 2 to row 4. For more on generators for degree-one abelian matrices, see [350, pp. 148ff.]. It turns out that the minimal number of generators is 2 for g = 2 and 3 for g ≥ 3.
- 14.
\([\tilde{B}]_{\gamma,1} {{(10.21)}\atop { =}} (p_{\gamma }\tilde{A})q_{1}^{t}=p_{\gamma }\tilde{A}(q_{1}^{t}) {{(10.19)}\atop{ =}} f_{1}(p_{\gamma }Jp_{g+1}^{t})=f_{1}[\tilde{P}_{1}J\tilde{P}_{1}^{t}]_{\gamma,g+1}=f_{1}[J]_{\gamma,g+1}=f_{1}\delta _{\gamma,1}\). Likewise, \([\tilde{B}]_{1,\delta }=(p_{1}\tilde{A})q_{\delta }^{t}=p_{1}(\tilde{A}q_{\delta }^{t}) {{(10.20)}\atop{ =-}}\!\!\! f_{1}q_{g+1}Jq_{\delta }^{t}= -f_{1}[\tilde{Q}_{1}J\tilde{Q}_{1}^{t}]_{g+1,\delta }= -f_{1}[J]_{g+1,\delta }=f_{1}\delta _{\delta,1}\).
- 15.
For example, the first relation in (10.23) means that \([\tilde{B}J\tilde{{B}}^{t}]_{1,\alpha } = n[J]_{1,\alpha } = n\delta _{\alpha,g+1},\) whereas if \([\tilde{B}J\tilde{{B}}^{t}]_{1\alpha }\) is computed using (10.22), one gets \([\tilde{B}J\tilde{{B}}^{t}]_{1,\alpha } =\sum _{\mu }[\tilde{B}]_{1,\mu }[J\tilde{{B}}^{t}]_{\mu,\alpha } = f_{1}[J\tilde{{B}}^{t}]_{1,\alpha } = f_{1}[\tilde{B}]_{\alpha,g+1}\). Comparison of the two expressions for \([\tilde{B}J\tilde{{B}}^{t}]_{1,\alpha }\) implies that \([\tilde{B}]_{\alpha,g+1} = (n/f_{1})\delta _{\alpha,g+1}\). In similar fashion, if the second relations in (10.23) and (10.22) are used to compute \([\tilde{{B}}^{t}J\tilde{B}]_{\beta,1}\) in two ways, the result is \([\tilde{B}]_{g+1,\beta } = (n/f_{1})\delta _{\beta,g+1}\). Thus we have (10.24).
- 16.
For a description of Kronecker’s paper and its relation to his Jugendtraum, see [572, pp. 66ff.].
- 17.
- 18.
In his memorial essay on Kronecker in 1893, Frobenius, who had recently spoken with the ailing Weierstrass about the happy years of mathematical give and take Weierstrass had enjoyed with Kronecker, wrote, “Since the investigation of elliptic functions with singular modules had led Kronecker to such extraordinarily interesting results, Weierstrass encouraged him to extend his researches to the complex multiplication of theta functions in several variables” [202, p. 719].
- 19.
On the eventual proof of sufficiency of Riemann’s conditions, see Section 11.4. It turns out that Frobenius’ work on generalized theta functions was involved in the first proof (by Wirtinger).
- 20.
Incidentally, Weber’s assumption is weaker than Kronecker’s assumption that the roots of \(\varphi (\lambda ) =\det (\lambda \tilde{B} -\tilde{ {B}}^{t})\) are distinct, i.e., \(\varphi (\lambda )\) can have multiple roots when \(\det (\lambda I -\tilde{ A})\) does not, but not conversely.
- 21.
In that case \(\tilde{A} ={\biggl ( \begin{array}{cc} a\!& \!b \\ c\! &\!d \end{array} \biggr )}\) is principal if and only if (10.26) holds, which for g = 1 (as noted above) means that the quadratic equation \({c\tau }^{2} + {(a - d)}^{2} - b = 0\) must have nonreal solutions (one of which will be in the upper half-plane), i.e., its discriminant must be negative. The condition for that may be written as \({(\mathrm{tr}\tilde{A})}^{2} - 4\det \tilde{A} < 0\), whereas the condition that \(\tilde{A}\) have real distinct characteristic roots is that \({(\mathrm{tr}\tilde{A})}^{2} - 4\det \tilde{A} > 0\).
- 22.
Of course this means that \(\Psi _{0} = (\det \Psi ){\Psi }^{-1}\), but surprisingly, Laguerre introduced no symbolic notation for an inverse. In the theory of determinants, attention was focused on the adjoined system Adj Ψ, and Laguerre apparently adhered to custom. In fact, he called the transpose of a matrix its inverse [393, pp. 223–224].
- 23.
In the 1850s and 1860s, when Hermitian symmetric matrices and forms were introduced and studied as analogues of real symmetric matrices and quadratic forms, the focus was exclusively on the reality of the characteristic roots. No interest was shown in generalizing the principal axes theorem, which would have led naturally to the notion of a unitary matrix as the “Hermitian” analogue of a real orthogonal matrix. That Frobenius was the first to have found a use for unitary matrices in his 1883 paper [188] on principal transformations is suggested by a remark by Hurwitz in 1897. In his seminal paper on invariant integrals on Lie groups, Hurwitz had occasion to introduce unitary transformations (and the special unitary group) in order to perform what Weyl later called the “unitarian trick” (see [276, p. 393]). Apparently because unitary transformations (or substitutions) were still sufficiently novel in 1897, Hurwitz pointed out to his readers that “these substitutions also come into consideration in other investigations” [304, p. 556, n. 1] and referred them to Frobenius’ paper [188].
- 24.
Frobenius did not utilize the notion of a dot product. I have used it for succinctness of expression.
- 25.
On Hurwitz’s paper and related work by him, see [411, pp. 332–333, 344–345].
- 26.
I am grateful to J.-P. Serre for calling my attention to the work of Humbert.
- 27.
For detailed references to the literature, see [411, p. 380].
- 28.
See Section 11.2 for a discussion of why Weierstrass introduced conditions (I)–(II) in unpublished work. In Section 11.4, the circumstances surrounding the publication of (I)–(II) are described.
- 29.
In particular, this result follows from Frobenius’ theory of Jacobian functions, as noted in Section 11.4 in the discussion of Wirtinger’s work.
- 30.
For example, \(\Omega (A_{1}A_{2}) = (\Omega A_{1})A_{2} = (M_{1}\Omega )A_{2} = M_{1}(\Omega A_{2}) = (M_{1}M_{2})\Omega \), which shows that \((M_{1}M_{2},A_{1}A_{2})\) is a multiplication.
- 31.
The corresponding abelian varieties are then in algebraic correspondence [408, pp. 79ff.].
- 32.
Lefschetz, who was born in Moscow, had spent most of the first two decades of his life living in Paris, where his parents (Turkish citizens) resided. (For further details on Lefschetz’s life and work see [564] and the references cited therein.) He was educated at the École Centrale in Paris and was graduated in 1905 as ingénieur des arts et manufactures. That same year, he emigrated to the United States to pursue a career as an engineer, but in 1907, an accident that resulted in the loss of both hands caused him to turn instead to a career in mathematics. (In Paris, he had studied mathematics under the instruction of Picard and Appell, both of whom had done important work in the theory of abelian functions.) He spent 1910–1911 at Clark University, in Worcester, Massachusetts, where he earned a doctoral degree with a dissertation on a topic in algebraic geometry. After two years teaching at the University of Nebraska, he moved to the above-mentioned positions at the University of Kansas.
- 33.
Lefschetz’s memoir was an English translation (with minor modifications) of an essay in French that was awarded the Prix Bordin of the Paris Academy of Sciences for the year 1919. In 1923, the English version was awarded the Bôcher Memorial Prize of the American Mathematical Society.
- 34.
Lefschetz refers to results in Frobenius’ paper that are not treated by Scorza [408, p. 133].
- 35.
Following terminology introduced in 1916 by Rosati, Lefschetz referred to f(α) = 0 as the “minimum equation of Frobenius” [408, p. 109n].
- 36.
The Riemann matrix Ω is pure if it is not isomorphic in Scorza’s above-defined sense to a Riemann matrix Ω ′ that is a direct sum of two Riemann matrices: Ω ′ = Ω 1 ′ ⊕ Ω 2 ′.
- 37.
For example, for x t in τ, \({y}^{t} = {x}^{t}A = {z}^{t}\Omega A = ({z}^{t}M)\Omega = z_{1}^{t}\Omega \in \tau\).
- 38.
Make the coordinate change \({x}^{t} = {u}^{t}\tilde{\Omega }\). In these coordinates, the homography is given by \({v}^{t} = {u}^{t}(M \oplus \overline{M})\), and the fixed point satisfies \(\alpha _{j}u_{j}^{t} = u_{j}^{t}(M \oplus \overline{M})\), which by block multiplication with \(u_{j}^{t} = \left (\begin{array}{cc} u_{1}^{t}&u_{2}^{t}\\ \end{array} \right )\) gives \(\alpha _{j}u_{1}^{t} = u_{1}^{t}M\) and \(\alpha _{j}u_{2}^{t} = u_{2}^{t}\overline{M}\) and so u 2 t = 0. Thus \(u_{j}^{t} = \left (\begin{array}{cc} u_{1}^{t}&0\\ \end{array} \right )\), and it follows that x j is in τ because in the new coordinate system, τ is characterized by \(\tilde{\Omega }u = {z}^{t}\Omega \) and \(\tilde{\Omega }u_{j}^{t} = u_{1}^{t}\Omega \) is of the requisite form.
- 39.
In that case, it follows from Gaussian elimination, since the fields \(\mathbb{Q}(\alpha _{j})\), j = 1, …, g, are isomorphic. It also follows by algebraic-topological considerations, as Lefschetz showed [408, pp. 111–112]. Whether it is true when the minimal polynomial of A is not assumed irreducible is unclear, although Scorza stated Theorem 10.14 below without any explicit assumption on the minimal polynomial of A.
- 40.
If F is to be the characteristic polynomial associated to a multiplication (M, A), then since \(\tilde{\Omega }A\tilde{{\Omega }}^{-1} = M \oplus \overline{M}\), any real roots would be multiple roots, contrary to the assumption that F is irreducible.
- 41.
Conditions (a) and (b) on p. 117 of [408].
- 42.
See papers 33, 37, and 42 of [2]. Earlier papers in [2] document Albert’s progress toward the solution. See also Jacobson’s account on pp. lvi–lvii and lii–lv of [2]. The modern theory of abelian varieties with complex multiplication was initiated in 1955 (Weil, Shimura, Taniyama), and was motivated by arithmetic considerations originating in the elliptical case g = 1 (Kronecker, Deuring, Hasse) [530, pp. x–xi]. The modern theory involves restrictions on \(\mathcal{M}\) [530, pp. x, 35].
References
A. A. Albert. Collected Mathematical Papers Part 1. Associative Algebras and Riemann Matrices. American Mathematical Society, Providence, RI, 1993.
A. Clebsch and P. Gordan. Theorie der Abelschen Functionen. Teubner, Leipzig, 1866.
G. Frobenius. Theorie der linearen Formen mit ganzen Coefficienten. Jl. für die reine u. angew. Math., 86:146–208, 1879. Reprinted in Abhandlungen 1, 482–544.
G. Frobenius. Zur Theorie der Transformation der Thetafunctionen. Jl. für die reine u. angew. Math., 89:40–46, 1880. Reprinted in Abhandlungen 2, 1–7.
G. Frobenius. Über die principale Transformation der Thetafunctionen mehrerer Variableln. Jl. für die reine u. angew. Math., 95:264–296, 1883. Reprinted in Abhandlungen 2, 97–129.
G. Frobenius. Gedächtnisrede auf Leopold Kronecker. Abhandlungen d. Akad. der Wiss. zu Berlin, pages 3–22, 1893. Reprinted in Abhandlungen 3, 707–724.
G. Frobenius and I. Schur. Über die reellen Darstellungen der endlichen Gruppen. Sitzungsberichte der Akademie der Wiss. zu Berlin, pages 186–208, 1906. Reprinted in Frobenius, Abhandlungen 3, 355–377.
T. Hawkins. Emergence of the Theory of Lie Groups. An Essay on the History of Mathematics 1869–1926. Springer, New York, 2000.
C. Hermite. Remarque sur un théorème de M. Cauchy. Comptes Rendus, Acad. Sci. Paris, 41:181–183, 1855. Reprinted in Oeuvres 1, 459–481.
G. Humbert. Sur les fonctions abéliennes singulières (Premier Mémoire). Jl. de math. pures et appl., (5)5:233–350, 1899.
G. Humbert. Sur les fonctions abéliennes singulières (Deuxième Mémoire). Jl. de math. pures et appl., (5)6:279–386, 1900.
A. Hurwitz. Ueber diejenigen algebraische Gebilde, welche eindeutige Transformationen in sich zulassen. Math. Ann., 32:290–308, 1888. Reprinted in Werke 1, 241–259.
A. Hurwitz. Über die Erzeugung der Invarianten durch Integration. Nachrichten von der Königlichen Gesellschaft der Wissenschaften und der Georg-Augustus-Universität zu Göttingen, pages 71–90, 1897. Reprinted in Werke 2, 546–564.
A. Krazer. Lehrbuch der Thetafunktionen. Teubner, Leipzig, 1903. Reprinted by Chelsea Publishing Company (New York, 1970).
L. Kronecker. Über die elliptischen Functionen, für welche complexe Multiplication stattfindet. Monatsberichte der Akademie der Wiss. zu Berlin, pages 455–460, 1857. Reprinted in Werke 3, 177–183.
L. Kronecker. Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten. Jl. für die reine u. angew. Math., 53:173–175, 1857. Reprinted in Werke 1, 103–108.
L. Kronecker. Über bilineare Formen. Monatsberichte der Akademie der Wiss. zu Berlin, 1:145–162, 1866. Reprinted in Jl. für die reine u. angew. Math., 68:273–285 and in Werke 1, 145–162.
E. Laguerre. Sur le calcul des systèmes linéaires. Journal École Polytechnique, 62, 1867. Reprinted in Oeuvres 1, 221–267.
S. Lefschetz. On certain numerical invariants of algebraic varieties with application to abelian varieties. Transactions of the American Mathematical Society, 22:327–482, 1921. Reprinted in [413], pp. 41–196.
S. Lefschetz. L’analysis situs et la géométrie algébrique. Gauthier–Villars, Paris, 1924. Reprinted in [413, pp. 283–439].
S. Lefschetz. XV. Transcendental Theory; XVI. Singular Correspondences; XVII. Hyperelliptic Surfaces and Abelian Varieties. In Selected Topics in Algebraic Geometry. Report of the Committee on Rational Transformations, volume 63 of Bulletin of the National Research Council. Washington, 1928. Reprinted in [542, pp. 310–395].
S. Lefschetz. A page of mathematical autobiography. Bulletin of the American Mathematical Society, 74:854–879, 1968. Reprinted in [413, pp.13–38].
B. Riemann. Theorie der Abel’schen Functionen. Jl. für die reine u. angew. Math., 54, 1857. Reprinted in Werke, pp. 88–144.
C. Rosati. Sulle matrici di Riemann. Rendiconti Circolo Mat. Palermo, 53:79–134, 1929.
M. Rosen. Abelian varieties over \(\mathbb{C}\). In G. Cornell and J. Silverman, editors, Arithmetic Geometry, pages 79–101. Springer-Verlag, New York, 1986.
G. Scorza. Intorno alla teoria generale delle matrici di Riemann e ad alcune applicazione. Rendiconti Circolo Mat. Palermo, 41:262–379, 1916.
G. Shimura. Abelian Varieties with Complex Multiplication and Modular Functions. Princeton University Press, 1998.
A. W. Tucker and F. Nebeker. Lefschetz, Solomon. In Dictionary of Scientific Biography, volume 18, pages 534–539. Charles Scribner’s Sons, New York, 1990.
S. G. Vlǎduţ. Kronecker’s Jugendtraum and Modular Functions. Gordon and Breach, 1991.
H. Weber. Ueber die Transformationstheorie der Theta-Functionen, ins Besondere derer von drei Veränderlichen. Annali di matematica, (2) 9:126–166, 1878.
H. Weber. Elliptische Functionen und algebraische Zahlen. Vieweg, Braunschweig, 1891. A second edition was published in 1908 under the same title but as the third volume of the second edition of Weber’s Lehrbuch der Algebra.
K. Weierstrass. Zur Theorie der Abel’schen Functionen. Jl. für die reine u. angew. Math., 47:289–306, 1854. Reprinted in Werke 1, 133–152.
K. Weierstrass. Untersuchungen über die 2r-fach periodischen Functionen von r Veränderlichen. Jl. für die reine u. angew. Math., 89:1–8, 1880. Reprinted in Werke 2, 125–133.
K. Weierstrass. Vorlesungen über die Theorie der abelschen Transcendenten. In G. Hettner and J. Knoblauch, editors, Mathematische Werke von Karl Weierstrass, volume 4. Mayer and Müller, 1902.
E. Wiltheiss. Über die complexe Multiplication hyperelliptischer Functionen zweier Argumente. Math. Ann., 21:385–398, 1883.
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Hawkins, T. (2013). Abelian Functions: Problems of Hermite and Kronecker. In: The Mathematics of Frobenius in Context. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6333-7_10
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