The Riemann–Schottky problem: which principally polarized Abelian varieties (equivalently, which Riemann matrices) arise from Riemann surfaces

The Riemann–Schottky problem asks which principally polarized Abelian varieties (equivalently, which classical Riemann matrices) arise from Riemann surfaces, and is the surviving open component of Hermann Weyl's 1934 Annals paper 'On Generalized Riemann Matrices', which generalized the classical theory of Riemann matrices (complex period matrices carrying a rational principal matrix) by dropping the condition $R^2=-1$ and requiring only total regularity of $S=CR$. The paper left five issues open: the classification in the case where the centrum of the multiplication algebra is a quadratic imaginary extension of a totally real field; the existence of a Riemann matrix with a prescribed involutorial division algebra as its multiplication algebra; a criterion for the principal matrix being symmetric versus skew-symmetric; the question of which abstract Riemann matrices arise from Riemann surfaces; and the role of positive definiteness of $S$. The first, second, third and fifth components were resolved within two years, by A. A. Albert (Ann. of Math. 36 (1935), 886–964) and by Weyl's own factor-set sequel (Ann. of Math. 37 (1936), 709–745). The surviving component — which (classical) Riemann matrices, equivalently which principally polarized Abelian varieties, arise from Riemann surfaces — is the Riemann–Schottky problem: describe the Jacobian locus $J_g=J(M_g)$ inside the moduli space $A_g$ of principally polarized Abelian varieties. It is completely solved for genus at most 4 and open for every genus $g\ge 5$, despite more than a century of work and deep non-explicit characterizations via the KP equation and Kummer trisecants.

Background

Let $C$ be a compact connected Riemann surface of genus $g$. The periods of a basis of holomorphic 1-forms over a basis of $H_1(C,\mathbb{Z})$ form a $g\times 2g$ complex matrix, the period matrix; Riemann's bilinear relations say that, in a symplectic basis, it can be written $\Omega=(D\;\tau)$ with $\tau$ in the Siegel upper half space $\mathbb{H}_g$ (symmetric complex matrices with positive definite imaginary part). The complex torus $J(C)=\mathbb{C}^g/(\mathbb{Z}^g+\tau\mathbb{Z}^g)$ is the Jacobian of $C$, a principally polarized Abelian variety (ppav) — a polarized torus whose polarization is given by a unimodular integral alternating form; the moduli space of ppavs is the quotient $A_g=\mathbb{H}_g/\mathrm{Sp}(2g,\mathbb{Z})$. By Torelli's theorem the map $J:M_g\to A_g$ from the moduli space of curves is injective, and its image $J_g$, the Jacobian locus, is a proper $3g-3$-dimensional subvariety of the $g(g+1)/2$-dimensional $A_g$ for $g\ge 4$.

In the algebraic literature of the 1920s–30s (Rosati, Scorza, Lefschetz, Albert, Weyl), a (classical) Riemann matrix of genus $p$ is a $p\times 2p$ complex matrix $\omega$ for which there is a nonsingular rational skew-symmetric $2p\times 2p$ matrix $C$, a principal matrix, such that $\omega C^{-1}\omega^{T}=0$ and $\omega(iC^{-1})\overline{\omega}^{T}$ is positive definite Hermitian (this is the definition in Albert's 1965 normal-form paper; it is equivalent, via the bilinear relations, to a ppav). Two such matrices are identified if they differ by $\omega\mapsto\alpha\omega A$ with $\alpha$ complex nonsingular and $A$ rational nonsingular. The algebra of rational matrices $A$ satisfying $M\omega=\omega A$ for some complex $M$ is the multiplication algebra, i.e. the endomorphism algebra of the ppav, a finite-dimensional semisimple algebra over $\mathbb{Q}$ carrying the Rosati involution $A\mapsto C^{-1}A^{T}C$ induced by the polarization.

Weyl's 1934 Annals paper 'On Generalized Riemann Matrices' recast this theory: a real matrix $R=C^{-1}S$ is a generalized Riemann matrix when $C$ is a nonsingular rational symmetric or skew-symmetric matrix and $S$ is a real symmetric matrix that is totally regular (nondegenerate on every rational subspace); the classical case is the subclass with $R^2=-1$ and $S$ positive definite. Weyl showed that Poincaré reducibility, Schur's lemma and the Rosati involution survive without $R^2=-1$, so that the multiplication algebra of a pure (indecomposable) generalized Riemann matrix is a division algebra with an anti-automorphic involution (an involutorial division algebra), and he raised the classification, existence, parity and realizability questions. These were settled by A. A. Albert's 'Involutorial Simple Algebras and Real Riemann Matrices' (1935) — which classified the involutorial simple algebras, identified the exact total-positivity condition on the associated factor system (a 2-cocycle of a field extension encoding the Brauer class of the algebra), and proved existence for every admissible division algebra — and by Weyl's sequel 'Generalized Riemann Matrices and Factor Sets' (1936), which gave the complete necessary and sufficient conditions, including an exceptions table governing the symmetric/skew parity of $C$.

The surviving question — which Riemann matrices arise from Riemann surfaces — is the Riemann–Schottky problem, studied by Riemann in unpublished notes and first attacked by Schottky (1888) and Schottky–Jung (1909). In its standard modern form: describe the image $J_g:=J(M_g)\subset A_g$; in period-matrix language, characterize exactly the period matrices of curves among all ppavs. Known: $J_g$ equals the locus $A_g^{\mathrm{ind}}$ of indecomposable ppavs (those not expressible as a nontrivial product) for $g\le 3$, and for $g=4$ it is cut out by a single explicit weight-8 Siegel modular form (Schottky's relation, proved rigorously by Igusa and Freitag, as recounted in Grushevsky's survey). For $g\ge 5$ no explicit description is known. Deep characterizations exist but quantify over unknown parameters and hence yield neither defining equations nor an effective membership test: by Novikov's conjecture, proved by Shiota (Theorem 7.6 in Grushevsky's survey), an indecomposable ppav is a Jacobian iff its second-order theta functions satisfy the Kadomtsev–Petviashvili (KP) equation from soliton theory; by Krichever's theorem (Theorem 7.7 in the survey; cf. Krichever–Shiota), iff its Kummer variety — the quotient by the involution $x\mapsto -x$, embedded by second-order theta constants — admits a trisecant line. Partial ('weak') solutions, i.e. explicit loci having $J_g$ as one irreducible component, are known in several approaches (theta-constant and Prym-variety methods, the Andreotti–Mayer loci of ppavs with singular theta divisors), and the problem remains a focus of current research in computational, tropical and integrable-systems directions.

Problem Statement

The Riemann–Schottky problem: describe, explicitly and exactly, the Jacobian locus $J_g=J(M_g)\subset A_g$ — the image of the moduli space of smooth genus-$g$ curves under the Torelli map inside the moduli space of principally polarized Abelian varieties. Equivalently, in the period-matrix language of the source question: characterize exactly which Riemann matrices (i.e. which principally polarized Abelian varieties, or in Weyl's original phrasing, which abstract Riemann matrices with $R^2=-1$ and positive definite $S$ up to Weyl equivalence) arise from a compact Riemann surface. The problem is posed for all genera $g\ge 2$; it is completely solved for $g\le 4$, and the open content is for every $g\ge 5$, the first open case being $g=5$. The description must be exact — the described locus is $J_g$ itself, not a locus having $J_g$ as only one irreducible component — and explicit, involving no existential quantification over unknown parameters and determining the locus by defining equations in the period-matrix coordinates of $\mathbb{H}_g$ or by an effective membership test.

References

1. Hermann Weyl, 'On Generalized Riemann Matrices', Annals of Mathematics, Second Series 35, no. 4 (1934), 714–729. DOI: 10.2307/1968487, https://doi.org/10.2307/1968487. (Source paper of the problem.)
2. Hermann Weyl, 'Generalized Riemann Matrices and Factor Sets', Annals of Mathematics, Second Series 37, no. 3 (1936), 709–745. DOI: 10.2307/1968485, https://doi.org/10.2307/1968485. (Weyl's sequel resolving the classification, parity and positivity components.)
3. A. Adrian Albert, 'Involutorial Simple Algebras and Real Riemann Matrices', Annals of Mathematics, Second Series 36, no. 4 (1935), 886–964. DOI: 10.2307/1968595, https://doi.org/10.2307/1968595. (Existence theorem and sign conditions.)
4. A. A. Albert, 'A Normal Form for Riemann Matrices', Canadian Journal of Mathematics 17 (1965), 1025–1029. DOI: 10.4153/cjm-1965-097-0, https://doi.org/10.4153/cjm-1965-097-0. (States the classical definition of Riemann matrices and the isomorphism $\omega\mapsto\alpha\omega A$.)
5. Nathan Jacobson, 'Abraham Adrian Albert 1905–1972', Bulletin of the American Mathematical Society 80, no. 6 (1974), 1075–1100. DOI: 10.1090/S0002-9904-1974-13622-0, https://doi.org/10.1090/S0002-9904-1974-13622-0. (Memorial article with a bibliography of Albert's works.)
6. Samuel Grushevsky, 'The Schottky Problem', survey lecture notes (2010). arXiv:1009.0369, https://arxiv.org/abs/1009.0369. (Covers the genus $\le 4$ state of knowledge, Open Problem 1 (explicit solution for $g=5$ completely open), Theorem 7.6 (Shiota) and Theorem 7.7 (Krichever), and the account of weak solutions.)
7. Samuel Grushevsky, Yuancheng Xie, 'Integrable Systems Approach to the Schottky Problem and Related Questions', lecture notes (2025, revised 2026). arXiv:2504.20243, https://arxiv.org/abs/2504.20243. (Includes Krichever's proof in the flex-line case of the Schottky problem.)
8. I. Krichever, T. Shiota, 'Soliton Equations and the Riemann–Schottky Problem' (2011). arXiv:1111.0164, https://arxiv.org/abs/1111.0164. (States the trisecant characterization: an indecomposable ppav is the Jacobian of a curve iff its Kummer variety has a trisecant line.)
9. Daniele Agostini, Lynn Chua, 'On the Schottky Problem for Genus Five Jacobians with a Vanishing Theta Null' (2019; published version 2021). arXiv:1905.09366, https://arxiv.org/abs/1905.09366; DOI: 10.2422/2036-2145.201909_013, https://doi.org/10.2422/2036-2145.201909_013. (Example of current partial progress: a solution of the weak Schottky problem in genus 5 with a vanishing theta null.)
10. E. Brandon de Leon, J. Frauendiener, C. Klein, 'Computational Approach to the Schottky Problem' (2023). arXiv:2303.15249, https://arxiv.org/abs/2303.15249. (Numerical test based on Fay's trisecant identity, presented as a characterization equivalent to being a Jacobian, as shown by Krichever.)