Integrality of singular values of the generalized lambda function $\Lambda_k=W_{[k,2,1]}$ in the excluded regime $\delta=\gcd(k,N)>1$ with $3\mid\delta$ or $N/\delta$ a prime power

The generalized lambda function $\Lambda_k=W_{[k,2,1]}$ is the ratio of differences of Weierstrass $\wp$-values $\big(\wp(k/N;\tau)-\wp(1/N;\tau)\big)/\big(\wp(2/N;\tau)-\wp(1/N;\tau)\big)$, a modular function for the congruence subgroup $\Gamma_1(N)$ introduced in Lang's framework. Ishii (2011 preprint arXiv:1110.4429, generalized in Ramanujan J. 33 (2014)) proved that $\Lambda_k$ together with the modular invariant $j$ generates the modular function field of $\Gamma_1(N)$, so that CM values of $\Lambda_k$ generate ray class fields, and proved that, writing $\delta=\gcd(k,N)$, whenever $\delta=1$, or $\delta>1$ with $3\nmid\delta$ and $N/\delta$ not a prime power, every $\mathrm{SL}_2(\mathbb{Z})$-transform of $\Lambda_k$ has $q$-expansion over $\mathbb{Z}[\zeta_N]$ and every CM value $\Lambda_k(\alpha)$ is an algebraic integer. In the complementary regime, $\delta>1$ with $3\mid\delta$ or $N/\delta$ a prime power, the status of both the $q$-expansion integrality and the algebraic integrality of the CM values is open.

Background

A central theme of explicit class field theory for imaginary quadratic fields is to produce, in the spirit of Hilbert's twelfth problem, concrete modular functions whose special values generate the abelian extensions of such a field $K$. If $\alpha$ lies in the complex upper half plane $\mathbb{H}$ and $\mathbb{Q}(\alpha)=K$ is imaginary quadratic, $\alpha$ is called an imaginary quadratic point or CM point; the classical singular moduli $j(\alpha)$ generate the Hilbert class field (the maximal unramified abelian extension) of $K$. Refinements with level structure generate ray class fields, the abelian extensions classified by class field theory with a prescribed modulus, and are sought as class invariants: generators with small height that are algebraic integers, so that their minimal polynomials over $\mathbb{Z}$ (class polynomials) have integer coefficients.

Fix an integer $N\ge 6$ and let $\Gamma_1(N)=\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{SL}_2(\mathbb{Z})\ :\ a\equiv 1,\ c\equiv 0 \pmod{N}\right\},$ a congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$; denote by $\mathcal{A}_1(N)$ its field of modular functions (meromorphic functions on $\mathbb{H}$ invariant under $\Gamma_1(N)$ with compatible behavior at the cusps). Write $L_\tau=\mathbb{Z}+\mathbb{Z}\tau$ for the lattice generated by $1$ and $\tau$, and $\wp(z;L)$ for the Weierstrass $\wp$-function of a lattice $L$. In his book Elliptic Functions, Lang introduced the generalized lambda functions: for a triple $a=[a_1,a_2,a_3]$ of distinct integers with $0<a_i\le N/2$, $W_a(\tau)=\frac{\wp(a_1/N;L_\tau)-\wp(a_3/N;L_\tau)}{\wp(a_2/N;L_\tau)-\wp(a_3/N;L_\tau)},$ a family generalizing the classical elliptic modular lambda function (the squared elliptic modulus parametrizing elliptic curves with a basis of $2$-torsion, itself the corresponding ratio of $\wp$-differences at the half-periods). Lang proposed investigating the special values of these functions at imaginary quadratic points, in particular whether they generate ray class fields. The function studied in Ishii's preprint arXiv:1110.4429 is $\Lambda_k:=W_{[k,2,1]}$ (note that the related display $\big(\wp(k\tau/N;\tau)-\wp(\tau/N;\tau)\big)/\big(\wp(2\tau/N;\tau)-\wp(\tau/N;\tau)\big)$, which appears in Corollary 3.3 of the preprint and Corollary 4.6 of the published version, equals $\Lambda_k\circ S$ for $S=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\in\mathrm{SL}_2(\mathbb{Z})$ by the transformation of $\wp$ under the homothety $L_{-1/\tau}=(-1/\tau)L_\tau$, not $\Lambda_k$ itself); the surrounding theory of modular units is relevant here: a modular unit is a modular function with no zeros or poles on $\mathbb{H}$, and for congruence subgroups such units are generated by Siegel functions, explicit $q$-products built from the Dedekind eta function.

Put $\zeta=e^{2\pi i/N}$ and $q=e^{2\pi i\tau/N}$. The standard route from $q$-expansion arithmetic to integrality of special values is the modular equation argument: if $f$ is a modular function for $\Gamma_1(N)$ with no poles on $\mathbb{H}$ and every transform $f\circ A$ ($A\in\mathrm{SL}_2(\mathbb{Z})$) has Laurent expansion $f\circ A\in\mathbb{Z}[\zeta]((q))$ (a formal Laurent series in $q$ with coefficients in the ring of cyclotomic integers $\mathbb{Z}[\zeta]$), then the product $\Phi(X,j)=\prod_{A}(X-f\circ A)$ over coset representatives of $\mathrm{SL}_2(\mathbb{Z})/\Gamma_1(N)\{\pm I\}$ is a monic polynomial with coefficients in $\mathbb{Z}[\zeta][j]$; Galois invariance gives $\Phi\in\mathbb{Z}[j][X]$, so $f$ is integral over $\mathbb{Z}[j]$, and since $j(\alpha)$ is an algebraic integer at every CM point $\alpha$ (a classical fact of complex multiplication), $f(\alpha)$ is an algebraic integer.

The lineage of results is as follows. Ishida and Ishii built the function-field machinery of $\mathcal{A}_1(N)$; Hong and Koo obtained ray class field generators from singular values of $\Gamma_1(N)$-modular functions; Ishii and Kobayashi proved that $W_a$ is integral over $\mathbb{Z}[j]$ (with algebraic-integer CM values) under a strong coprimality condition on the triple, essentially $a_1a_2a_3(a_1\pm a_3)(a_2\pm a_3)$ prime to $N$. Ishii's preprint arXiv:1110.4429 and its generalization in Ramanujan J. 33 (2014) 121-130 (arXiv:1110.6489) prove that $\Lambda_{k,\ell}=W_{[k,\ell,1]}$ generates $\mathcal{A}_1(N)$ over $\mathbb{C}(j)$ for all $1<\ell\ne k<N/2$, and that consequently $\Lambda_k(\alpha)$ together with $\zeta$ generates the ray class field of $K=\mathbb{Q}(\alpha)$ modulo $N$ over its Hilbert class field whenever $\mathbb{Z}[\alpha]$ is the maximal order (via the Gee-Stevenhagen form of Shimura reciprocity, which computes the Galois action on such special values). For the integrality statements, with $\delta=\gcd(k,N)$, the hypothesis is: (i) $\delta=1$, or (ii) $\delta>1$, $(\delta,3)=1$ and $N/\delta$ is not a power of a prime. Under it, $\Lambda_k\circ A\in\mathbb{Z}[\zeta]((q))$ for every $A\in\mathrm{SL}_2(\mathbb{Z})$, $\Lambda_k$ is integral over $\mathbb{Z}[j]$, and all CM values of all transforms are algebraic integers. Where the hypothesis bites: the leading coefficient of $\Lambda_k\circ A$ is a product of ratios of the shape $(1-\zeta^a)/(1-\zeta^b)$, whose integrality is governed by the cyclotomic-unit lemma that $1-\zeta^k$ is a unit of $\mathbb{Z}[\zeta]$ exactly when $N/\gcd(k,N)$ is not a prime power (used in the branch $c\equiv 0\pmod N$ of the transform computation), while the branch ${2c}={c}$ (here $\{x\}$ denotes the integer with $0\le\{x\}\le N/2$ such that $x\equiv\pm\{x\}\pmod N$, as defined in Section 2 of the preprint), possible only when $3\mid N$, introduces a denominator $1-\zeta^{3d}$ and thereby the condition $(\delta,3)=1$.

Evidence from a sibling setting points to genuine arithmetic sensitivity of exactly the excluded regimes: for the principal-congruence analogue $\Lambda(\tau;Q_1,Q_2)$ (the same ratio of $\wp$-differences at points $Q_1,Q_2,Q_1+Q_2$ with $\{Q_1,Q_2\}$ a basis of $(\mathbb{Z}/N)^2$), Ishii proved that $C_N\Lambda$ is integral over $\mathbb{Z}[j]$ with $C_N=1$ precisely when $N$ is not a prime power ($C_N=p^2$ for $p=2,3$, $C_N=p$ for $p>3$ otherwise), sharpened this to $(1-\zeta)^3\Lambda(\alpha)$ and $(1-\zeta)^3/\Lambda(\alpha)$ being algebraic integers at all CM points $\alpha$, proved $\Lambda(\alpha)$ is a unit when $N$ is not a prime power, and exhibited explicit CM values that are not algebraic integers when $N$ is a prime power (norm $3^{-3}$ at $N=3$, norm $1/2$ at $N=4$). Those results, however, are proved for a different group and a different function and do not transfer to $\Lambda_k=W_{[k,2,1]}$.

Problem Statement

Let $N\ge 6$, let $k$ be an integer with $2<k<N/2$, put $\delta=\gcd(k,N)$, and let $\Lambda_k(\tau)=W_{[k,2,1]}(\tau)=\frac{\wp(k/N;L_\tau)-\wp(1/N;L_\tau)}{\wp(2/N;L_\tau)-\wp(1/N;L_\tau)},$ where $L_\tau=\mathbb{Z}+\mathbb{Z}\tau$ is the lattice generated by $1$ and $\tau$, $\wp(z;L)$ denotes the Weierstrass $\wp$-function of a lattice $L$, $\zeta=e^{2\pi i/N}$, $q=e^{2\pi i\tau/N}$, $\mathbb{Z}[\zeta]((q))$ is the ring of formal Laurent series in $q$ with coefficients in the cyclotomic integer ring $\mathbb{Z}[\zeta]$, $j$ denotes the classical elliptic modular invariant, and an imaginary quadratic point is an $\alpha$ in the complex upper half plane $\mathbb{H}$ such that $\mathbb{Q}(\alpha)$ is an imaginary quadratic field. It is known that if $\delta=1$, or if $\delta>1$ with $(\delta,3)=1$ and $N/\delta$ not a power of a prime, then $\Lambda_k\circ A\in\mathbb{Z}[\zeta]((q))$ for every $A\in\mathrm{SL}_2(\mathbb{Z})$, $\Lambda_k$ is integral over $\mathbb{Z}[j]$, and $\Lambda_k(\alpha)$ is an algebraic integer for every imaginary quadratic point $\alpha$. Determine the complementary regime, namely $\delta>1$ with $3\mid\delta$ or $N/\delta$ a power of a prime:

(a) for which such pairs $(k,N)$ does $\Lambda_k\circ A\in\mathbb{Z}[\zeta]((q))$ hold for every $A\in\mathrm{SL}_2(\mathbb{Z})$;

(b) for which such pairs $(k,N)$ is $\Lambda_k$ integral over $\mathbb{Z}[j]$, and in particular $\Lambda_k(\alpha)$ an algebraic integer at every imaginary quadratic point $\alpha$.

The two prongs are linked parts of one objective, completing the integrality picture of the family: (a) implies (b) through the modular equation argument, while failure of (a) at some transform does not by itself decide (b), since denominators may cancel in the Galois-invariant product $\prod_A(X-\Lambda_k\circ A)$. A complete answer either proves that (a) and (b) hold throughout the regime, or establishes exactly which pairs $(k,N)$ fail, with explicit witnesses for the failing pairs (an imaginary quadratic point $\alpha$ with $\Lambda_k(\alpha)$ not an algebraic integer, or a transform $A$ with $\Lambda_k\circ A\notin\mathbb{Z}[\zeta]((q))$) together with the corresponding statements for the remaining pairs.

References

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