Borcherds' thirteen open problems on automorphic forms with singularities on Grassmannians

The singular theta (Howe) correspondence constructed in Borcherds' 'Automorphic forms with singularities on Grassmannians' (Inventiones mathematicae 132, 1998) lifts vector-valued modular forms with poles at cusps for the Weil representation of a lattice $M$ of signature $(2,n)$ to automorphic forms whose exponential $\Psi_M$ is an infinite product (a Borcherds product) with divisor supported on Heegner divisors. The paper's final section posed thirteen open problems on the structure, arithmetic, functoriality, inversion, and scope of this correspondence. Substantial parts are now resolved in restricted settings: the rank-3 classification of reflective Lorentzian lattices (Allcock 2012), the classification of automorphic products of singular weight and their generalized Kac-Moody algebras (Scheithauer 2006; Wang-Williams 2022; and others), converse theorems identifying which meromorphic forms are Borcherds products (Bruinier and others, 2001-2025), regularized lifts over totally real fields (Bruinier 2012), and transfers to unitary groups (Hofmann 2014). The surviving open core is nonempty; the following items locate its main locus, and the list is illustrative rather than exhaustive: necessity of the reflection criterion of Theorem 12.1 and the general classification of reflective Lorentzian lattices, the full classification of generalized Kac-Moody algebras with automorphic denominator functions, Hecke compatibility beyond type $O(2,1)$, the ideal of differential operators annihilating the lift, holomorphic products on genuinely new Hermitian symmetric spaces, best-possible Weyl-vector congruences at the primes 2 and 3, the identification of $b^+=1$ Donaldson invariants with automorphic products, and the construction of Hejhal's pseudo cusp forms by the singular Howe correspondence. All thirteen questions (16.1-16.13) remain required, and the items not named above (16.3, 16.4, 16.10, 16.12, 16.13) also remain open in the generality stated by the source.

Background

Let $M$ be an even lattice, i.e. a free abelian group of finite rank equipped with an integral-valued symmetric bilinear form $(\ ,\ )$ all of whose norms $v^2=(v,v)$ are even; its signature is the signature of the form on $M\otimes\mathbb{R}$. The discriminant group $M'/M$ (with $M'$ the dual lattice of rational vectors having integral inner product with all of $M$) carries a finite quadratic form and hence the Weil representation $\rho_M$ of the metaplectic cover $Mp_2(\mathbb{Z})$ of $SL_2(\mathbb{Z})$; by a modular form of type $\rho_M$ one means a vector-valued holomorphic function on the upper half plane $H$ transforming under $\rho_M$ and allowed to be meromorphic at the cusp (weakly holomorphic). The Grassmannian $G(M)$ of oriented positive-definite 2-planes in $M\otimes\mathbb{R}$ is a Hermitian symmetric domain for the real orthogonal group - the orthogonal analogue of $H$ for $SL_2$ - and automorphic forms of weight $k$ on it are sections of a canonical line bundle. Its rational quadratic divisors, called Heegner divisors, are the analogues of CM points (points of $H$ fixed by imaginary quadratic orders). When $M$ is Lorentzian (signature $(n,1)$), reflections in the hyperplanes $r^\perp$ of vectors with $r^2<0$ generate a subgroup of the integral automorphism group $\mathrm{Aut}(M)$; $M$ is called reflective if this reflection subgroup has finite index in $\mathrm{Aut}(M)$, i.e. if the reflection group is arithmetic. Conway (1983) proved that the even unimodular (i.e. $M'=M$) Lorentzian lattice $II_{1,25}$ has this property, with a Weyl vector of norm 0 (the Weyl vector of a reflection group is the vector $\rho$ in the closed fundamental Weyl chamber - a component of the complement of the reflection hyperplanes - normalized by $(\rho,r)=r^2/2$ for every simple root $r$); Nikulin (1980) proved strong finiteness statements about reflective lattices, and in particular that the number of such lattices is essentially finite in the relevant sense.

The theta (or Howe) correspondence for a reductive dual pair (two mutually centralizing subgroups, here $SL_2$ and the orthogonal group $O_M$) attaches automorphic forms on one member to automorphic forms on the other; the holomorphic case goes back to Oda, Rallis-Schiffmann, and, in the half-integral-weight direction, to the Shimura, Doi-Naganuma, Maass and Gritsenko correspondences. Borcherds (1995) introduced a singular variant for signature $(s+2,2)$ at level 1 (trivial discriminant group): a vector-valued modular form with a pole at the cusp is mapped to a meromorphic automorphic form $\Psi_M$ (regularized theta lift, then exponentiated) whose divisor is a Heegner divisor explicitly computable from the principal part of the input, and which has an explicit infinite product expansion at each 0-dimensional cusp (a Borcherds product). The 1998 paper generalized this to arbitrary even lattices $M$ of signature $(2,b^-)$: the additive lift $F\mapsto\Phi_M$ is defined by a regularized theta integral of the input $F$ (taken almost holomorphic, i.e. holomorphic up to a correction which is a polynomial in $1/\mathrm{Im}(\tau)$) against the Siegel theta kernel, $\Phi_M=\log|\Psi_M|$ for the multiplicative lift, and the divisor of $\Psi_M$ is read off from the coefficients $c_\lambda(m)$ of $F$ for $m<0$. Among the applications obtained: many new generalized Kac-Moody (Borcherds-Kac-Moody) algebras - Lie algebras whose simple root spaces may have dimension $>1$ and whose root system satisfies only weak finiteness axioms, with denominator functions (the formal power series playing the role of $e^{\rho}\prod(1-e^{-\alpha})$ for Kac-Moody algebras) that can now be identified with automorphic products, the fake monster Lie algebra being the model case; a sufficient condition (Theorem 12.1) for a Lorentzian lattice to have a reflection group of finite, or virtually free abelian, index in its automorphism group; extensions of the Shimura and Maass-Gritsenko correspondences to forms with singularities (Theorem 14.3); congruences for theta functions of positive definite lattices; and examples relating the wall-crossing behavior of these forms to Donaldson invariants of 4-manifolds.

For the geometric questions one needs the following. A smooth oriented 4-manifold has a splitting $H^2(X,\mathbb{R})=H^2_+\oplus H^2_-$ of the second cohomology by the cup-product pairing, with $b^+=\dim H^2_+$; its Donaldson invariants are differential-topological invariants defined from moduli spaces of anti-self-dual $SU(2)$ connections, and for $b^+=1$ they depend on a chamber structure (wall crossing) cut out by the periods of the Riemannian metric; Göttsche (1996) determined the wall-crossing formulas for $b^+=1$, and Moore-Witten (1998) expressed the $b^+=1$ invariants through an integral over the $u$-plane (the modular curve parametrizing the Coulomb branch of the supersymmetric theory). The Koecher boundedness principle is the statement that holomorphic automorphic forms on higher-dimensional arithmetic quotients are automatically bounded at the cusps, which forbids holomorphic forms with poles at cusps for Hilbert modular groups. Hejhal constructed 'pseudo cusp forms': functions on $H$ with logarithmic singularities at imaginary quadratic irrationals which are almost eigenfunctions of the hyperbolic Laplacian, with eigenvalues the imaginary parts of zeros of the Riemann zeta function.

The paper ends with Section 16, 'Open problems', containing thirteen numbered questions (16.1-16.13) which together ask for the structural completion of the singular theta correspondence: which reflective and Lie-theoretic structure it detects, what arithmetic its outputs carry, how it behaves under Hecke operators and differential operators, whether it can be inverted, and how far the construction extends.

Problem Statement

The problem is the set of thirteen open questions posed in Section 16 ('Open problems') of Borcherds, 'Automorphic forms with singularities on Grassmannians' (1998). They share one scientific objective: to complete the structural theory of the singular theta correspondence $F\mapsto\Phi_M$ between modular forms of type $\rho_M$ with controlled singularities at cusps and automorphic forms with Heegner-divisor singularities on the Grassmannian of $M$ - a complete characterization of when the reflective and Lie-theoretic structure it detects exists, what arithmetic and geometry its outputs carry, how it behaves functorially and locally, whether it can be inverted, and how far the construction extends. A resolution of the problem is an answer (proof, classification, or refutation) to each question, in the stated generality; questions settled in restricted settings remain asked here in the generality stated by the source. The thirteen questions are independent of one another and all thirteen belong to the problem.

(16.1) Theorem 12.1 gives a sufficient condition for a Lorentzian lattice to have a reflection group of finite or virtually free abelian index in its automorphism group: the existence of a modular form $F$ of type $\rho_M$ and weight $(1/2-b^-/2,0)$, holomorphic on $H$ and meromorphic at cusps, all of whose Fourier coefficients $c_\lambda(m)$ are real for $m<0$, such that whenever $c_\lambda(\lambda^2/2)\ne 0$ with $\lambda^2<0$, reflection in $\lambda^\perp$ lies in $\mathrm{Aut}(M,F,C)$, the subgroup of $\mathrm{Aut}(M)$ fixing $F$ and the positive cone $C$ of the Lorentzian lattice. Is this condition also necessary? Can it be used to classify the Lorentzian lattices with this property? (Nikulin showed the number of such lattices is essentially finite.)

(16.2) Find all generalized Kac-Moody algebras whose denominator function is an automorphic form; this class includes in particular the cases in which the denominator function is an automorphic form of singular weight (the minimal weight at which a nonzero holomorphic automorphic form on the Grassmannian of signature $(2,b^-)$ can occur, weight $b^-/2-1$ up to sign), and classifying only the singular-weight cases does not by itself resolve the question. These appear to correspond roughly to the cases in which the Lorentzian lattice $M$ has a reflection subgroup with a norm 0 Weyl vector; however there are also many cases in which the generalized Kac-Moody algebra has no real roots and so does not obviously correspond to a reflection group acting on $M$.

(16.3) For lattices of signature $(2,1)$ or $(2,2)$ the construction gives many meromorphic sections of line bundles over modular curves or Hilbert modular surfaces with known zeros and poles (from Theorem 13.3), and similarly on some higher dimensional varieties. Can these be used to give interesting relations between elements of the Picard or Néron-Severi groups represented by the divisors of zeros of these sections? In particular, is it possible to prove the Gross-Zagier theorem along these lines?

(16.4) Everything in the paper is done for quadratic forms over the integers. Extend the construction to quadratic forms over the rings of integers of algebraic number fields and function fields. The major obstacle is the Koecher boundedness principle: holomorphic automorphic forms with poles at cusps are rare in higher dimensions and do not exist for $SL_2$ over any totally real field other than $\mathbb{Q}$, so it is unclear how to extend the results to Hilbert modular varieties; it may still be possible to do something with $SL_2$ over the integers of a real quadratic field or a quaternion algebra over $\mathbb{Q}$, where the symmetric space is hyperbolic space of dimension 3 or 5.

(16.5) Describe how the correspondence behaves under the action of Hecke operators. In the holomorphic cases, correspondences such as the Shimura correspondence take eigenforms to eigenforms, but for forms with singularities this statement is usually vacuous since eigenforms do not exist in general. A possible replacement: a homomorphism from the Hecke algebra of $O_{b^+,b^-}$ to that of $SL_2$ compatible with the correspondences of Theorems 13.3 and 14.3, where for $O_{2,b^-}$ the Hecke operators should act multiplicatively rather than additively on the meromorphic infinite products.

(16.6) What local properties do the functions $\Phi_M$ have? Are they eigenfunctions of the Laplacian at their nonsingular points? More generally, are their restrictions to nonsingular values killed by an ideal of finite index in the center of the universal enveloping algebra of $O_M(\mathbb{R})$ - in other words, do they satisfy the same local conditions as automorphic forms except at their singular points?

(16.7) When $b^+=1$, the wall-crossing formula and the fact that $\Phi_M$ is piecewise polynomial are remarkably similar to statements about Donaldson invariants of 4-manifolds with $H^2$ equal to $I_{1,b^-}$ (the odd unimodular lattice of signature $(1,b^-)$). When do Donaldson polynomials for some 4-manifolds have the same wall-crossing formulas as some functions $\Phi_M$? When they do, the piecewise polynomial Donaldson invariants split as the sum of a polynomial invariant and an automorphic form $\Phi_M$ with singularities, giving polynomial invariants for 4-manifolds with $b^+=1$ that do not depend on a choice of Weyl chamber (the source verifies this for piecewise linear Donaldson invariants when $b^-\le 9$).

(16.8) Investigate the functions on other Hermitian symmetric spaces. Holomorphic infinite products are obtained only on symmetric spaces with hermitian symmetric subspaces of complex dimension one less - the spaces of $O_{2,b^-}(\mathbb{R})$ and $U(1,n)$, the unit balls embedding in those of $O(2,2n)$. On other symmetric spaces, such as Siegel upper half planes of genus $g>2$ (the Hermitian symmetric spaces of the symplectic groups $Sp_{2g}(\mathbb{R})$), one does not get holomorphic automorphic forms as infinite products, but one does get real analytic automorphic forms with singularities along Siegel upper half planes of genus $g-1$, by embedding the Siegel upper half plane in the Grassmannian $G(\mathbb{R}^{2g,2g})$ and restricting. What can be done with these? Is it possible to find holomorphic sections of vector bundles on Siegel upper half planes with known zeros?

(16.9) What congruence conditions (especially at the primes 2 and 3 dividing $|M'/M|$) does the Weyl vector satisfy when $F$ has integral coefficients? What are the best possible congruences satisfied by lattices - that is, what is the lattice generated by the theta functions of lattices of some fixed genus (two integral lattices lie in the same genus when they are isomorphic over the $p$-adic integers for every prime $p$ and over $\mathbb{R}$)?

(16.10) Can one reverse the correspondence from modular forms to automorphic forms with singularities, and reconstruct modular forms from automorphic forms with singularities? In particular, when are there isomorphisms between spaces?

(16.11) Hejhal constructed 'pseudo cusp forms': functions on the upper half plane with logarithmic singularities at imaginary quadratic irrationals which are almost eigenfunctions of the Laplacian with eigenvalues given by the imaginary parts of zeros of the Riemann zeta function. Can these pseudo cusp forms be constructed using the singular Howe correspondence?

(16.12) Generalize the correspondence of Theorem 14.3 to vector-valued forms by using polynomials $p$ with $m^- > 0$ (in the notation of that theorem, $p$ is a homogeneous polynomial on $M\otimes\mathbb{R}$ of degree $m^+$ in the positive and $m^-$ in the negative directions, and the case $m^-=0$ is what Theorem 14.3 treats).

(16.13) What happens if Theorem 7.1 (which computes the Fourier expansion of $\Phi_M$) is applied to functions which are not almost holomorphic? For example, Freitag asked whether there are analogues of Maass wave forms (non-holomorphic eigenfunctions of the hyperbolic Laplacian on $H$, square-integrable at the cusps) with singularities at cusps, in which case these could be inserted into Theorem 7.1; Hejhal's trace-formula volumes provide many examples of such functions.

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