Explicit prime-ideal factorization of the algebraic numbers $\alpha$ satisfying $(f,\Theta_{\mathcal{B}})^{\mathrm{reg}}=\log|\alpha|$ for weight-one modular forms, for arbitrary imaginary quadratic fundamental discriminants

Viazovska proved that for an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-D})$ with $D$ a prime congruent to $3$ modulo $4$, the regularized Petersson product of a weakly holomorphic modular form $f$ of weight one with integral Fourier coefficients and vanishing constant term against the theta series $\Theta_{\mathcal{B}}$ of an ideal class $\mathcal{B}$ equals $\log|\alpha|$ for an algebraic number $\alpha$ in the Hilbert class field $H$ of $K$, and she determined the valuations of $\alpha$ at the primes of $H$ above every rational prime not dividing $D$: they are an explicit sum of Fourier coefficients of $f$ weighted by representation numbers of ideals in a class built from $\mathcal{B}$ for inert primes, and zero for split primes. The valuation at the ramified prime $p=D$ was left conjectural, and the restriction to prime discriminants was stated only as a simplifying hypothesis. Duke and Li independently conjectured the same factorization including the non-split (inert and ramified) primes, and Ehlen (2017) proved that conjecture and extended the full factorization theorem to all odd fundamental discriminants, including composite ones, giving the valuation at every prime ideal including those above the ramified prime. What remains open is the case of even fundamental discriminants $-D\equiv 0\pmod 4$: prove that the regularized product is the logarithm of an algebraic number in the appropriate class field and establish its explicit prime-ideal factorization, in particular at the primes ramified in $K$, including the prime $2$. The obstructions are genus-theoretic (for even $D$ the class number can be even, so the conjugation-fixed prime above a non-split prime need not be unique, which is the linchpin of the class entering the formula) and the missing local arithmetic intersection theory of the relevant CM cycles at the ramified prime $2$, on which the odd-discriminant proof relies. A further open point, already unsettled for odd discriminants, is the conjectured divisibility $r\mid 24\,h_K\,h_H$ of the normalizing integer $r$ (class numbers of $K$ and of $H$); determining $r$ and proving or refuting this divisibility is the third required deliverable, (iii).

Background

Let $K=\mathbb{Q}(\sqrt{-D})$ be an imaginary quadratic field of fundamental discriminant $-D<0$ (the discriminant of a quadratic field is the integer that determines the field up to isomorphism; it is called fundamental when it is the discriminant of the maximal order, the ring of all algebraic integers in the field, which happens exactly when $-D$ is either an odd squarefree integer congruent to $1$ modulo $4$, or $4$ times a squarefree integer congruent to $2$ or $3$ modulo $4$; the even case $-D\equiv 0\pmod 4$ is the one left open below). Let $\mathcal{O}_K$ be the ring of integers, $\mathrm{Cl}(K)$ the ideal class group, $h=|\mathrm{Cl}(K)|$ the class number, and $H$ the Hilbert class field, the maximal unramified abelian extension of $K$; class field theory identifies $\mathrm{Gal}(H/K)$ with $\mathrm{Cl}(K)$ via the Artin map, which sends an ideal class to the Frobenius automorphism of its ideals.

To each ideal class $\mathcal{B}\in\mathrm{Cl}(K)$ one attaches the theta series $\Theta_{\mathcal{B}}(\tau)=\sum q^{N(\mathfrak{a})}$, summing over the integral ideals $\mathfrak{a}\subset\mathcal{O}_K$ in the class $\mathcal{B}$, where $q=e^{2\pi i\tau}$ for $\tau$ in the upper half-plane and $N(\mathfrak{a})$ is the absolute norm. It is a holomorphic modular form of weight one (a holomorphic function on the upper half-plane transforming with the factor $(c\tau+d)^{-1}$ under the fractional-linear changes of variable coming from a discrete subgroup of $\mathrm{SL}_2(\mathbb{R})$) for the congruence subgroup $\Gamma_0(D)$ (the subgroup of $\mathrm{SL}_2(\mathbb{Z})$ of matrices whose lower-left entry is divisible by $D$), with the quadratic character attached to $K$ (the Dirichlet character modulo $D$ encoding the splitting of primes in $K$); the canonical isomorphism of Bruinier and Bundschuh between scalar modular forms of this type and vector-valued modular forms identifies it with the vector-valued theta series used in the source papers, a form for the Weil representation $\rho$ of $\mathrm{SL}_2(\mathbb{Z})$ attached to the lattice $\mathcal{O}_K$ (the Weil representation is the finite-dimensional representation determined by the discriminant group of the lattice, the quotient $\mathcal{O}_K'/\mathcal{O}_K$ of the dual lattice $\mathcal{O}_K'$, with its induced quadratic form; vector-valued means the form takes values in that space, with the group acting through $\rho$; for $D$ prime this module is $\mathbb{Z}/D\mathbb{Z}$, while for even $D$ it has order $D$ but is not cyclic). A weakly holomorphic modular form of weight one is holomorphic on the upper half-plane but allowed poles at the cusps (boundary points of the modular curve); if its constant term vanishes it is called a weakly holomorphic cusp form, and the space of such forms for $\rho$ is denoted $S_1^!(\rho)$, with Fourier expansion $f=\sum_{t\gg-\infty}\sum_{\nu}c_{\nu}(t)\,e^{2\pi i t\tau/D}\,\mathbf{e}_{\nu}$, where $t$ ranges over the integers bounded below, $\nu$ over the discriminant group of the lattice, and $\mathbf{e}_{\nu}$ over the standard basis vectors.

The Petersson product of two weight-one forms is the invariant integral $\int\langle f(\tau),\overline{\Theta_{\mathcal{B}}(\tau)}\rangle\,y^{-1}dx\,dy$ over the quotient of the upper half-plane by $\mathrm{SL}_2(\mathbb{Z})$, where $\langle\cdot,\cdot\rangle$ is the standard Hermitian pairing on the representation space; it diverges for non-cuspidal forms, so Viazovska regularizes it as the limit, as $T\to\infty$, of the same integral over the fundamental domain of $\mathrm{SL}_2(\mathbb{Z})$ truncated at height $T$, the truncation restoring convergence: the regularized Petersson product $(f,\Theta_{\mathcal{B}})^{\mathrm{reg}}$. For the cusp form $g_\chi=\sum_{\mathcal{C}}\chi(\mathcal{C})\Theta_{\mathcal{C}}$ attached to a character $\chi$ of the class group, the Rankin-Selberg method (an unfolding technique expressing such integrals through $L$-functions) expresses $(g_\chi,\Theta_{\mathcal{B}})$ through Artin $L$-values (the $L$-functions attached to representations of Galois groups), and Stark's theorem on units then exhibits it as a logarithm of an algebraic unit in an abelian extension of $K$. Motivated by the Gross-Zagier conjecture on the algebraicity of CM values (values at points with complex multiplication, that is, points parameterizing elliptic curves whose endomorphism ring is $\mathcal{O}_K$) of automorphic Green functions (the analogues, on modular curves and their higher-rank relatives, of the Green's functions of differential geometry: Laplace eigenfunctions with prescribed logarithmic singularities), Viazovska, and independently Duke and Li (coming from the theory of mock modular forms, the holomorphic parts of harmonic weak Maass forms, which are modular forms harmonic for the hyperbolic Laplacian but allowed exponential growth at the cusps), extended this to weakly holomorphic $f$ in the cases settled below: if the $c_{\nu}(t)$ are integers with vanishing constant term, then $(f,\Theta_{\mathcal{B}})^{\mathrm{reg}}=\log|\alpha|$ for an algebraic number $\alpha$ in the Hilbert class field $H$ of $K$.

The main theorem of Viazovska's paper determines the prime-ideal factorization of $\alpha$. Under the simplifying hypothesis that $D$ is a prime congruent to $3$ modulo $4$ (so, by genus theory, the structure theory of the $2$-torsion of the class group, $h$ is odd), she proved $\alpha\in H$ and: for a prime ideal $\mathfrak{P}$ of $\mathcal{O}_H$ above a rational prime $p$ with Kronecker symbol $\left(\frac{p}{D}\right)=1$ (the quadratic residue symbol, extended to composite 'denominators'; $p$ splits in $K$), $\mathrm{ord}_{\mathfrak{P}}(\alpha)=0$; while for $\left(\frac{p}{D}\right)=-1$ ($p$ inert in $K$), $\mathrm{ord}_{\mathfrak{P}}(\alpha)=\sum_{t<0}\sum_{\nu}c_{\nu}(t)\,r_{\mathcal{BA}^{2}}(-t/p)\,\mathrm{ord}_{p}(t),$ where $\mathrm{ord}_{\mathfrak{P}}$ is the normalized discrete valuation, $r_{\mathcal{C}}(n)$ counts integral ideals of $\mathcal{O}_K$ of norm $n$ in the class $\mathcal{C}$, and $\mathcal{A}=\mathcal{A}(\mathfrak{P})$ is the ideal class attached to $\mathfrak{P}$ as follows: complex conjugation acts on the primes of $H$ above $p$; since $h$ is odd, exactly one of them, say $\mathfrak{P}_1$, is fixed by conjugation, and $\mathcal{A}$ is the class with $\mathfrak{P}^{\sigma_{\mathcal{A}}}=\mathfrak{P}_1$, where $\sigma_{\mathcal{A}}\in\mathrm{Gal}(H/K)$ is the automorphism corresponding to $\mathcal{A}$ under the Artin isomorphism. Schofer had earlier proved the averaged statement (summing over all classes $\mathcal{B}$ yields the logarithm of a rational number with an explicit factorization); Viazovska's theorem refines it class by class. Her proof identifies $(f,\Theta_{\mathcal{B}})^{\mathrm{reg}}$ with the logarithm of the value at a CM point (a point of the modular curve parameterizing an elliptic curve with complex multiplication by $K$) of a Borcherds product, a meromorphic modular function given by an infinite product built from $f$ whose divisor is supported on Heegner divisors (divisors of CM points), and computes the valuations through the local height pairings of Heegner divisors on the arithmetic modular curve $X_0(D)$ (the modular curve viewed over the integers; a local height pairing is a rational-valued bilinear pairing between divisors, defined place by place on such an arithmetic surface) computed by Gross and Zagier.

Two restrictions were left open in the source. First, the ramified prime: for $p=D$, the only prime ramified in $K$, the valuation of $\alpha$ was conjectured but not proved (Viazovska's Remark 3, with a numerical example for $D=23$ as supporting evidence). Second, the hypothesis that $D$ is prime was introduced only 'for simplicity', the general-discriminant case being left open. Both were subsequently resolved for odd discriminants: Duke and Li conjectured the factorization for all non-split primes from numerical experiments, and Ehlen proved their conjecture and extended the theorem to all odd fundamental discriminants, including composite ones, proving $\mathrm{ord}_{\mathfrak{P}}(\alpha)=2r\,Z(m)_{\mathfrak{P}}$ for all prime ideals $\mathfrak{P}\subset\mathcal{O}_{H}$ with completely explicit multiplicities, where $r$ is a normalizing integer and $Z(m)$ is an arithmetic special cycle (a divisor on an arithmetic model of a Shimura curve, a quotient of the upper half-plane by an arithmetic group playing the role of a modular curve, cut out by special endomorphisms of CM elliptic curves, in the sense of Kudla-Rapoport-Yang), $Z(m)_{\mathfrak{P}}$ denoting the multiplicity of its component at $\mathfrak{P}$. The case left open by the literature is that of even fundamental discriminants $-D\equiv 0\pmod 4$ (for instance $D=4,8,20,24,40$), where the prime $2$ is ramified in $K$, the class number can be even, and the $2$-adic intersection theory of the relevant CM cycles is not available.

Problem Statement

Let $-D<0$ be a fundamental discriminant and $K=\mathbb{Q}(\sqrt{-D})$ the imaginary quadratic field of ring of integers $\mathcal{O}_K$, class group $\mathrm{Cl}(K)$, and Hilbert class field $H$. Let $\rho$ be the Weil representation of $\mathrm{SL}_2(\mathbb{Z})$ attached to the lattice $\mathcal{O}_K$, let $S_1^!(\rho)$ be the space of weakly holomorphic cusp forms of weight one for $\rho$, with Fourier expansion $f=\sum_{t\gg-\infty}\sum_{\nu}c_{\nu}(t)\,e^{2\pi i t\tau/D}\,\mathbf{e}_{\nu}$ (the index $t$ ranging over the integers bounded below and $\nu$ over the discriminant group of the lattice), and let $\Theta_{\mathcal{B}}$ be the theta series of an ideal class $\mathcal{B}\in\mathrm{Cl}(K)$. The problem is to remove the parity restriction from the known factorization theorems: the case of $D$ prime (Viazovska, with the ramified prime completed by Ehlen) and, more generally, of $-D$ any odd fundamental discriminant (Ehlen) are settled, and the open case is that of even fundamental discriminants $-D\equiv 0\pmod 4$ (for instance $-4,-8,-20,-24,-40$), for which the following must be established.

(i) Algebraicity. For every $f\in S_1^!(\rho)$ with integral Fourier coefficients and vanishing constant term, and every ideal class $\mathcal{B}$, prove that $(f,\Theta_{\mathcal{B}})^{\mathrm{reg}}=\log|\alpha|$ for an algebraic number $\alpha$ in the Hilbert class field $H$ of $K$. This is an independent partial milestone, resolved by the algebraicity together with the identification of $H$ as the field of definition, and is subsumed by a complete (ii), which additionally supplies the explicit factorization; the genus-theory-compatible reformulation of (ii) serves only to index the classes $\mathcal{C}(\mathfrak{P})$.

(ii) Explicit factorization. Determine the prime-ideal factorization of $\alpha$: $\mathrm{ord}_{\mathfrak{P}}(\alpha)=0$ for the primes $\mathfrak{P}$ above the split primes $p$ ($\left(\frac{-D}{p}\right)=1$); and for the primes $\mathfrak{P}$ above the non-split primes $p$ (the inert primes and, in particular, the ramified primes $p\mid D$, including $p=2$), an explicit formula for $\mathrm{ord}_{\mathfrak{P}}(\alpha)$ as a finite sum over the negative indices $t$ of the Fourier coefficients $c_{\nu}(t)$ weighted by the numbers $r_{\mathcal{C}(\mathfrak{P})}(n)$ of integral ideals of a specified class $\mathcal{C}(\mathfrak{P})$ (built from $\mathcal{B}$ and the Artin class attached to $\mathfrak{P}$) and a specified norm, analogous to Viazovska's proved inert formula and to her conjectural ramified formula (formula (9) of her paper, stated in Remark 3 there); the precise normalizations (the power of $p$ entering the norm and the class in which ideals are counted) are part of what must be determined. Since for even $-D$ the class number can be even, so that the conjugation-fixed prime above a non-split $p$ need not be unique and the class $\mathcal{A}(\mathfrak{P})$ of the odd-discriminant formula is not directly available, the classes $\mathcal{C}(\mathfrak{P})$ are to be indexed compatibly with genus theory, while what is to be determined is the individual $\mathfrak{P}$-valuations of $\alpha$ in $H$ itself.

(iii) Normalization. In the normalization $\mathrm{ord}_{\mathfrak{P}}(\alpha)=2r\,Z(m)_{\mathfrak{P}}$ of the Duke-Li-Ehlen framework, determine the normalizing integer $r$ and prove or refute the conjectured divisibility $r\mid 24\,h_K\,h_{H}$; either outcome — a proof or a disproof of this divisibility — resolves this deliverable.

The common objective of (i)-(iii) is a single factorization theorem, uniform over all imaginary quadratic fields, for the algebraic numbers measuring regularized Petersson products of weight-one modular forms; the already-settled odd-discriminant cases are the specialization against which the even-discriminant result must be consistent.

References

1. Maryna Viazovska, "Petersson inner products of weight-one modular forms", Journal für die reine und angewandte Mathematik (Crelle's Journal) 749 (2019), 133-159. DOI: 10.1515/crelle-2016-0042, https://doi.org/10.1515/crelle-2016-0042 (author's preprint arXiv:1211.4715, https://arxiv.org/abs/1211.4715).
2. Maryna Viazovska, "CM Values of Higher Green's Functions", preprint arXiv:1110.4654, https://arxiv.org/abs/1110.4654.
3. W. Duke and Y. Li, "Harmonic Maass forms of weight 1", Duke Mathematical Journal 164 (2015), no. 1, 39-113. DOI: 10.1215/00127094-2838436, https://doi.org/10.1215/00127094-2838436.
4. Stephan Ehlen, "CM values of regularized theta lifts and harmonic weak Maaß forms of weight 1", Duke Mathematical Journal 166 (2017), no. 13, 2447-2519. DOI: 10.1215/00127094-2017-0005, https://doi.org/10.1215/00127094-2017-0005 (errata: DOI 10.1215/00127094-2017-0046; preprint arXiv:1701.03662, https://arxiv.org/abs/1701.03662).
5. Stephan Ehlen, "Singular moduli of higher level and special cycles", Research in the Mathematical Sciences 2 (2015), no. 1, Article 16. DOI: 10.1186/s40687-015-0030-0, https://doi.org/10.1186/s40687-015-0030-0 (preprint arXiv:1505.02711, https://arxiv.org/abs/1505.02711).
6. David R. Dorman, "Special values of the elliptic modular function and factorization formulae", Journal für die reine und angewandte Mathematik (Crelle's Journal) 383 (1988), 207-220. DOI: 10.1515/crll.1988.383.207, https://doi.org/10.1515/crll.1988.383.207.
7. Benedict H. Gross and Don B. Zagier, "Heegner points and derivatives of L-series", Inventiones Mathematicae 84 (1986), no. 2, 225-320. DOI: 10.1007/BF01388809, https://doi.org/10.1007/BF01388809.
8. Jarad Schofer, "Borcherds forms and generalizations of singular moduli", Journal für die reine und angewandte Mathematik (Crelle's Journal) 629 (2009), 1-36. DOI: 10.1515/crelle.2009.025, https://doi.org/10.1515/crelle.2009.025.
9. Jan Hendrik Bruinier and Michael Bundschuh, "On Borcherds Products Associated with Lattices of Prime Discriminant", The Ramanujan Journal 7 (2003), no. 1-3, 49-61. DOI: 10.1023/A:1026222507219, https://doi.org/10.1023/A:1026222507219.
10. Stephen S. Kudla, Michael Rapoport, and Tonghai Yang, "On the derivative of an Eisenstein series of weight one", International Mathematics Research Notices 1999, no. 7, 347-385. DOI: 10.1155/S1073792899000185, https://doi.org/10.1155/S1073792899000185.
11. Jan Hendrik Bruinier, Stephan Ehlen, and Tonghai Yang, "CM values of higher automorphic Green functions for orthogonal groups", Inventiones Mathematicae 225 (2021), no. 3, 693-785. DOI: 10.1007/s00222-021-01038-0, https://doi.org/10.1007/s00222-021-01038-0.
12. Yingkun Li, "Algebraicity of higher Green functions at a CM point", Inventiones Mathematicae 234 (2023), no. 1, 375-418. DOI: 10.1007/s00222-023-01205-5, https://doi.org/10.1007/s00222-023-01205-5 (preprint arXiv:2106.13653, https://arxiv.org/abs/2106.13653).
13. Yingkun Li and Markus Schwagenscheidt, "Mock modular forms with integral Fourier coefficients", Advances in Mathematics 399 (2022), 108264. DOI: 10.1016/j.aim.2022.108264, https://doi.org/10.1016/j.aim.2022.108264 (preprint arXiv:2101.05583, https://arxiv.org/abs/2101.05583).
14. Stephan Ehlen, Yingkun Li, and Markus Schwagenscheidt, "Harmonic Maass forms associated with CM newforms", Journal für die reine und angewandte Mathematik (Crelle's Journal), published online 11 June 2024. DOI: 10.1515/crelle-2024-0036, https://doi.org/10.1515/crelle-2024-0036 (preprint arXiv:2210.07341, https://arxiv.org/abs/2210.07341).
15. Sebastián Herrero, Özlem Imamoglu, Anna-Maria von Pippich, and Markus Schwagenscheidt, "Special values of Green's functions on hyperbolic 3-space", preprint arXiv:2405.01219, https://arxiv.org/abs/2405.01219.
16. Wei-Lun Tsai and Dongxi Ye, "On the Petersson norm $\langle\theta_\psi,\theta_\psi\rangle$", preprint arXiv:2608.01583, https://arxiv.org/abs/2608.01583.