The defect term in Weil's 1935 Riemann-Roch formula for automorphic differential vectors: behavior for multiplier systems with infinite image

In 1935 André Weil proved a Riemann-Roch theorem for matrix rings of functions on a compact Riemann surface, with local integrality conditions prescribed by matrix divisors, and applied it to automorphic forms: for a Fuchsian group of finite signature uniformizing a compact two-dimensional orbifold, he gave a dimension formula for spaces of vector-valued automorphic differentials of integer grade transforming by an arbitrary finite-dimensional complex multiplier representation. The formula is the sum of an explicit Euler-characteristic term, depending only on the signature, the grade, the rank, and the eigenvalue multiplicities of the images of the elliptic generators, and a defect term: the dimension of the complementary-grade space with the contragredient multiplier. The defect is evaluable when the multiplier representation has finite image, where the statement reduces to the Chevalley-Weil theorem. For infinite image the defect's behavior — equivalently, the dimension theory of vector-valued automorphic forms with arbitrary non-unitary multiplier systems in weights below the known high-weight and unitarizable regimes — remains open, already for the modular group. The problem asks for vanishing criteria, an evaluation or sharp structural characterization of the defect term, and the correct infinite-image generalization of the finite-image evaluation.

Background

A compact Riemann surface $X$ of genus $p$ carries, besides its meromorphic functions, meromorphic differentials, and the classical Riemann-Roch theorem relates the two. A divisor $D=\sum_P m_P P$ is a finite formal sum of points with integer coefficients, prescribing allowed zeros and poles; the theorem computes the dimension $\ell(D)$ of the space of meromorphic functions whose poles are bounded by $D$ via

$$\ell(D)=\deg D+1-p+\ell(K-D),$$

where $K$ is a canonical divisor, the divisor of any meromorphic differential. The correction term $\ell(K-D)$ is a defect: it is not determined directly by $D$, but it vanishes once $\deg D>2p-2$, which is what makes the theorem effective in large degree.

Weil's 1935 paper extends this to matrix-valued objects. Instead of single functions one considers columns of $r$ meromorphic functions, and instead of a divisor a matrix divisor $\mathfrak{D}=\prod_\alpha\mathfrak{P}_\alpha^{\theta_\alpha}$: at each of finitely many points $\mathfrak{P}_\alpha$, a nonsingular $r\times r$ matrix $\theta_\alpha$ of local Laurent series prescribes the admissible local behavior. A column $\mathfrak{v}$ is an integral multiple of $\mathfrak{D}^{-1}$ when $\theta_\alpha\mathfrak{v}$ has no pole at $\mathfrak{P}_\alpha$, and a row $d\mathfrak{w}$ of differentials is an integral multiple of $\mathfrak{D}$ when $d\mathfrak{w}\,\theta_\alpha^{-1}$ has no pole there. Writing $N$ and $N'$ for the numbers of linearly independent such columns and rows, and $n$ for the order of $\mathfrak{D}$ (the sum over $\alpha$ of the orders of the determinants $\det\theta_\alpha$), Weil proves

$$N=n-r(p-1)+N',$$

by reduction to the classical theorem through elementary-divisor (Smith normal form) arguments. This is a Riemann-Roch theorem for a matrix ring with prescribed local integrality — in modern terms, for a vector bundle on the curve — and it was later developed algebraically as a Riemann-Roch theorem for orders in rings of valuation vectors of function fields.

The second part of the paper applies this to automorphic forms. Let $\mathfrak{R}$ be a compact Riemann surface of genus $p$ with finitely many marked points $\mathfrak{G}_\lambda$ assigned finite orders $n_\lambda$ (the genus together with these orders is the signature), and let $\mathfrak{G}$ be the corresponding orbifold group: it is generated by the $2p$ standard loops $A_i,B_i$ of the surface together with small loops $C_\lambda$ around the marked points, with relations $C_\lambda^{n_\lambda}=1$ and $\prod_i[A_i,B_i]\prod_\lambda C_\lambda=1$. Let $\mathfrak{A}$ be the covering surface of $\mathfrak{R}$ associated with $\mathfrak{G}$ — when the orbifold is hyperbolic, the upper half-plane, on which $\mathfrak{G}$ acts as a Fuchsian group, that is, a discrete group of Möbius transformations — and let $\mathfrak{M}:\mathfrak{G}\to\mathrm{GL}_r(\mathbb{C})$ be an arbitrary complex matrix representation, the multiplier system. By the Riemann existence theorem there is a single-valued analytic matrix $\theta$ on $\mathfrak{A}$ transforming as $\theta^S=\mathfrak{M}_S\theta$ for $S\in\mathfrak{G}$. A differential vector of grade $f$ is a column of $r$ differentials transforming like $(dz)^f$ under coordinate changes, that is, a column of $f$-fold differentials (negative grades are allowed; for the modular group discussed below, the grade is half the usual modular-form weight); it is automorphic with multiplier $\mathfrak{M}$ if it satisfies the corresponding transformation law under $\mathfrak{G}$, and the objects counted by Weil are those finite, that is holomorphic, everywhere on $\mathfrak{A}$. Applying the matrix Riemann-Roch theorem to the local matrices furnished by $\theta$, Weil obtains the dimension $N$ of this space:

$$N=r(2f-1)(p-1)+\sum_\lambda\sum_\nu N_{\lambda\nu}\Bigl((f-1)\Bigl(1-\frac{1}{n_\lambda}\Bigr)+\Bigl\langle\frac{f-1-\nu}{n_\lambda}\Bigr\rangle\Bigr)+N'.$$

Here $N_{\lambda\nu}$ is the multiplicity of the eigenvalue $e^{2\pi i\nu/n_\lambda}$ of $\mathfrak{M}_{C_\lambda}$ (since $C_\lambda^{n_\lambda}=1$, all eigenvalues are $n_\lambda$-th roots of unity), $\langle x\rangle$ is the fractional part of $x$, and the defect term $N'$ is the dimension of the space of everywhere-finite rows of grade-$(1-f)$ differentials transforming by the contragredient, that is inverse-transpose, system $\mathfrak{w}^S=\mathfrak{w}\,\mathfrak{M}_S^{-1}$. The first two terms depend only on discrete data — signature, grade, rank, eigenvalue multiplicities — while the defect $N'$ is a dual space in the complementary grade $1-f$, exactly parallel to $\ell(K-D)$ in the classical theorem.

When the image $\mathfrak{M}(\mathfrak{G})$ is finite, the defect can be evaluated on a hyperbolic base: $N'=0$ for $f>1$, and for $f=1$ it equals the dimension of the subspace of vectors fixed by all $\mathfrak{M}_S$. In the unramified case with trivial multiplier this recovers the Chevalley-Weil theorem (1934), which describes how the irreducible representations of a finite group of automorphisms of a Riemann surface are distributed among the holomorphic differentials of the covering surface, and which has since been generalized in many directions: to positive characteristic, to equivariant Lefschetz formulas and Galois module structure of cohomology on higher-dimensional varieties, and to orbifold and singular curves.

For multiplier systems with infinite image, no such evaluation is known. In modern language the quotient of $\mathfrak{A}$ by $\mathfrak{G}$ is an orbifold curve $Y$ — a curve with finitely many cone points, that is, points whose neighborhood is modeled on a disk modulo a cyclic rotation — and by the Riemann-Hilbert correspondence, the dictionary between flat connections and their monodromy representations, the multiplier system determines a flat holomorphic vector bundle $\mathcal{E}_\rho$ on $Y$, that is, a bundle carrying a connection of zero curvature; the defect term is $h^0(Y,\mathcal{E}_\rho^*\otimes K_Y^{1-f})$, the dimension of the space of orbifold sections of the dual bundle $\mathcal{E}_\rho^*$ twisted by the power $K_Y^{1-f}$ of the canonical bundle $K_Y$ of $Y$, and the full formula is orbifold Riemann-Roch combined with Serre duality, the standard identification of the correction term in Riemann-Roch with holomorphic sections of a dual bundle. What is missing is the vanishing and evaluation theory for this section space: for finite image, and for unitarizable representations (those conjugate to unitary matrices), positivity arguments force the defect to vanish in high grade, but for a general infinite-image, non-unitarizable multiplier system the holomorphic type of $\mathcal{E}_\rho$ is not controlled by the discrete data above. This is the regime of the modern theory of vector-valued automorphic forms; its most developed case is the modular group $\mathrm{SL}_2(\mathbb{Z})$, where such objects are the vector-valued modular forms — holomorphic functions on the upper half-plane with values in $\mathbb{C}^r$ satisfying the automorphic transformation law with coefficient matrices $\rho(\gamma)$ and holomorphic at the cusp — and where the complementary-grade spaces are accessed through Eichler cohomology, the cohomology of the group with coefficients in polynomials tensored with the representation, introduced in this context by Eichler's 1957 theory of generalized abelian integrals.

Problem Statement

Let $\mathfrak{R}$ be a Riemann surface of genus $p$ with finitely many marked points, either compact with all orders $n_\lambda$ finite (Weil's own setting) or of finite hyperbolic area with additional parabolic marked points (cusps), and of negative orbifold Euler characteristic $2-2p-\sum_\lambda(1-1/n_\lambda)-s<0$, where $s$ is the number of cusps (so $s=0$ in the compact case); let $\mathfrak{G}$ be the associated Fuchsian group of finite signature — the modular group $\mathrm{SL}_2(\mathbb{Z})$, of signature $(0;2,3,\infty)$, being the guiding non-compact example — acting as a Fuchsian group on its covering surface $\mathfrak{A}$ (the upper half-plane), let $\mathfrak{M}:\mathfrak{G}\to\mathrm{GL}_r(\mathbb{C})$ be a multiplier representation, and let $f$ be an integer grade. Weil's formula expresses the dimension of the space of everywhere-finite grade-$f$ automorphic differential vectors with multiplier $\mathfrak{M}$ on $\mathfrak{A}$ as an explicit Euler-characteristic term plus the defect term

$$\sigma=\dim\{\text{everywhere-finite grade-}(1-f)\text{ differential rows }\mathfrak{w}\text{ with }\mathfrak{w}^S=\mathfrak{w}\,\mathfrak{M}_S^{-1}\}.$$

When the image $\mathfrak{M}(\mathfrak{G})$ is finite, Weil's formula reduces to the Chevalley-Weil theorem and the defect is evaluable: $\sigma=0$ for $f>1$, and $\sigma$ equals the dimension of the fixed subspace $\{v:\mathfrak{M}_Sv=v\}$ for $f=1$. The problem is to determine the behavior of $\sigma$ when the image of $\mathfrak{M}$ is infinite.

1. Vanishing. Give sharp criteria on $(\mathfrak{G},\mathfrak{M},f)$ under which $\sigma=0$ — in particular, an analogue of the finite-image vanishing $\sigma=0$ for $f>1$ valid for infinite image, extending the known high-grade and unitarizable cases.

2. Evaluation and interpretation. When $\sigma\neq 0$, compute or characterize it in terms of the representation $\mathfrak{M}$ and the signature — or else prove precise limits on such characterizations: determine whether $\sigma$ is a function of the discrete data entering the Euler-characteristic term, and if not, exhibit explicit multiplier systems with identical discrete data but different $\sigma$.

3. Generalization. Formulate and prove the correct extension of Weil's finite-image evaluation to infinite image in the modern formulation: determine $h^0(Y,\mathcal{E}_\rho^{*}\otimes K_Y^{1-f})$ for the flat orbifold vector bundle $\mathcal{E}_\rho$ attached to an arbitrary multiplier system $\rho$ on the orbifold curve $Y=\mathfrak{A}/\mathfrak{G}$, in all weights, beyond the regimes where it is already known to vanish or to be computable from the Euler characteristic.

Equivalently: complete Weil's identity into a genuine dimension formula for vector-valued automorphic forms with arbitrary multiplier systems, by solving — or sharply delimiting — the defect term.

References

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