The Landen transformation and higher modular correspondences — exact discrete (renormalization-group) symmetries of integrable lattice models such as the square Ising and Baxter models — are algebraic solutions of a Schwarzian differential equation with rational right-hand side. Beyond these classical modular-correspondence series, the same Schwarzian equation admits one-parameter series solutions yn(a,x)=X~(a⋅Q~(x)n)y_n(a,x)=\tilde X(a\cdot \tilde Q(x)^n) built from the inverse nome map X~(q)=1/j(q)\tilde X(q)=1/j(q) (the reciprocal modular jj-function) and the elliptic nome (mirror map) Q~\tilde Q. The series is known to be algebraic at the exhibited 'selected' parameter values (roots of unity and the modular values a=1/1728n−1a=1/1728^{n-1}), globally bounded for every rational parameter value, and generically only differentially algebraic. The open problem is to characterize exactly which parameter values are selected: whether any further algebraic or D-finite specializations exist, whether the globally bounded rational-parameter specializations are ever D-finite or D-D-finite, and whether any D-finite one would be a diagonal of a rational function (an instance of Christol's conjecture), and would therefore in particular reduce to an algebraic function modulo almost every prime.

Background

Integrable lattice models such as the square-lattice Ising model and the Baxter model (the eight-vertex lattice model solved exactly by R. J. Baxter) possess exact discrete symmetries — the Landen transformation k→2k/(1+k)k \to 2\sqrt{k}/(1+k) and its higher analogues, corresponding to τ→Nτ\tau \to N\tau on the ratio τ\tau of the two periods of the underlying elliptic curve — which act as exact renormalization-group transformations on the couplings. These symmetries are encoded in modular equations: algebraic relations between the Hauptmodul — the basic algebraic coordinate generating the functions invariant under the relevant modular group, here taken as x=1728/jx=1728/j, where jj is the modular jj-invariant classifying elliptic curves up to isomorphism — and its transform. The algebraic series solving a modular equation is called a modular correspondence. One route to them is the automorphy of the Gauss hypergeometric function 2F1([1/12,5/12],[1],x)_2F_1([1/12,5/12],[1],x), which uniformizes the modular curve, under pullbacks x→y(x)x \to y(x); eliminating the automorphy prefactor leaves a Schwarzian condition

W(x) − W(y(x))⋅y′(x)2 + {y(x),x} = 0,W(x)\,-\,W(y(x))\cdot y'(x)^2\,+\,\{y(x),x\}\,=\,0,

where {y,x}=y′′′/y′−32(y′′/y′)2\{y,x\}=y'''/y'-\frac{3}{2}(y''/y')^2 is the Schwarzian derivative and, for this uniformization, W(x)=−(32x2−41x+36)/(72x2(x−1)2)W(x)=-(32x^2-41x+36)/(72x^2(x-1)^2) is rational. Solutions of Schwarzian equations are differentially algebraic, and the algebraic modular correspondences are among them.

The source paper found that, beyond the algebraic modular correspondences, this Schwarzian condition admits one-parameter families of series solutions. Let X~(q)=1/j(q)\tilde X(q)=1/j(q), with the integer-coefficient nome expansion q−744q2+356652q3−⋯q-744q^2+356652q^3-\cdots, and let Q~(x)\tilde Q(x) be its compositional inverse — the elliptic nome (the exponential Fourier variable of the jj-invariant, conjugate to the period ratio τ\tau) expressed as a function of the equivalent Hauptmodul 1/j1/j, a mirror map, also with integer coefficients. For every integer n≥1n\ge 1 the series

yn(a,x) = X~(a⋅Q~(x)n)y_n(a,x)\,=\,\tilde X\bigl(a\cdot \tilde Q(x)^n\bigr)

solves the Schwarzian condition in this reciprocal-jj normalization — the same condition with WW replaced by W~(x) = 17282 W(1728x) = −12 (1−1968x+2654208x2)/(x2(1−1728x)2)\tilde W(x)\,=\,1728^2\,W(1728x)\,=\,-\frac{1}{2}\,(1-1968x+2654208x^2)/(x^2(1-1728x)^2) — with the composition law yn(a,ym(b,x))=ymn(abn,x)y_n(a,y_m(b,x))=y_{mn}(ab^n,x) — a structure reminiscent of replicable functions (functions, arising in moonshine theory, whose Fourier coefficients obey Hecke-type replication relations). The base family reads y(a,x)=a x−744 a(a−1)x2+36 a(a−1)(9907a−20845)x3−⋯y(a,x)=a\,x-744\,a(a-1)x^2+36\,a(a-1)(9907a-20845)x^3-\cdots.

The relevant classes of series: a series is D-finite (holonomic) if it satisfies a linear ODE with polynomial coefficients; D-D-finite if it satisfies a linear ODE with D-finite function coefficients; differentially algebraic if it satisfies a polynomial nonlinear ODE; and globally bounded if it has rational coefficients with only finitely many primes in the denominators, so that a rescaling of xx yields integer coefficients.

What is known about the parameter dependence: (i) for aa an NN-th root of unity, y(a,x)y(a,x) is algebraic — it is a solution series of the modular equation ΓN2(x,y)=0\Gamma_{N^2}(x,y)=0 associated with q→qN2q\to q^{N^2}; (ii) for a=1/1728n−1a=1/1728^{n-1}, yn(a,x)y_n(a,x) is the algebraic modular-correspondence series associated with q→qnq\to q^n (and root-of-unity multiples of these values give further algebraic correspondence series); (iii) generically in aa the series is only differentially algebraic, and the order of the linear differential operator annihilating the algebraic specializations at aN=1a^N=1 grows with NN, so no linear ODE of bounded order with coefficients polynomial in xx and aa annihilates the whole family; (iv) for every rational a=P/Qa=P/Q the series is globally bounded — for a=3a=3 it begins 3x−4464x2+1917216x3−⋯3x-4464x^2+1917216x^3-\cdots and reduces, modulo the prime 2, to an algebraic series.

The arithmetic stake is sharpened by Christol's conjecture on diagonals: diagonals of rational functions of several variables are always globally bounded and D-finite, and the conjecture — proved in several hypergeometric families but open in general — predicts the converse for globally bounded D-finite series; a necessary condition is reduction to an algebraic function modulo (almost) every prime. Deciding D-finiteness of the globally bounded specializations and their possible diagonality are thus two sides of one question. In related truncated (polynomial) examples in the same paper, the nome-like series is D-finite but its order-one annihilating operator fails global nilpotence (nonzero pp-curvatures at a positive-density set of primes; here pp-curvature measures the reduction of the operator modulo a prime, and global nilpotence of all pp-curvatures is an arithmeticity criterion), so these arithmetic diagnostics must be applied with care.

Problem Statement

Determine the differential-algebraic nature of the one-parameter series yn(a,x)=X~(a⋅Q~(x)n)y_n(a,x)=\tilde X(a\cdot \tilde Q(x)^n) — solutions of the Schwarzian condition W~(x)−W~(y(x))y′(x)2+{y(x),x}=0\tilde W(x)-\tilde W(y(x))y'(x)^2+\{y(x),x\}=0 with W~(x)=−12 (1−1968x+2654208x2)/(x2(1−1728x)2)\tilde W(x)=-\frac{1}{2}\,(1-1968x+2654208x^2)/(x^2(1-1728x)^2), where X~(q)=1/j(q)\tilde X(q)=1/j(q) and Q~\tilde Q is its compositional inverse (the elliptic nome) — as a function of the parameter aa. Concretely:

  1. Characterize all parameter values aa for which yn(a,x)y_n(a,x) is an algebraic function of xx. The exhibited selected values are the NN-th roots of unity (base family n=1n=1) and, for every n≥1n\ge 1, the modular values a=1/1728n−1a=1/1728^{n-1} together with all their root-of-unity multiples ω/1728n−1\omega/1728^{n-1}, where ωM=1\omega^M=1 is arbitrary and unconstrained relative to nn. Are there any others?

  2. Determine whether there exists a parameter value — in particular a rational value — for which yn(a,x)y_n(a,x) is D-finite but not algebraic. The source expects that the globally bounded integer- and rational-parameter specializations are never D-finite; prove or refute this.

  3. Decide whether the globally bounded rational-parameter specializations are D-D-finite. If any of them is D-finite, determine whether it is a diagonal of a rational function — an instance of Christol's conjecture — and in particular whether it reduces to an algebraic function modulo almost every prime (a necessary condition for diagonality, whose sufficiency is precisely Christol's conjecture).

The common objective is a complete classification of the parameter values at which the differentially algebraic family drops to algebraic, D-finite or D-D-finite series, together with the arithmetic characterization (global boundedness, reduction modulo primes, diagonal status) of its globally bounded members.

References

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