Michel–Venkatesh's questions from “The subconvexity problem for GL2”: subconvexity-inspired fast computation of central L-values, and the asymptotic count of cusp forms by analytic conductor

The source survey poses two questions. (1) The approximate functional equation computes a central L-value $L(\pi,1/2)$ in time $\mathrm{C}(\pi)^{1/2+\varepsilon}$, where $\mathrm{C}$ is the analytic conductor; can the ideas entering the proofs of subconvexity be reinterpreted as faster, “subconvex-time” algorithms with exponent below $1/2$? (2) Let $N(X)$ count cuspidal representations of $\mathrm{GL}_n(\mathbb{Q})$ with analytic conductor at most $X$; determine the asymptotic behavior of $N(X)$, in particular whether $\log N(X)/\log X\to n+1$. Question (2) is answered affirmatively by Brumley–Milićević for $n\le 2$ without restriction and for $n\ge 3$ within the archimedean spherical spectrum — pure power growth $|\mathfrak{F}(Q)|\sim\mathcal{C}(\mathfrak{F})Q^{n+1}$ with no logarithmic factors — leaving open only the case of non-spherical archimedean components for $n\ge 3$; no follow-up removing this restriction has appeared. Question (1) remains open in the general conductor aspect: aspect-specific subconvex-time algorithms due to Hiary (Riemann zeta; Dirichlet $L$-functions at smooth moduli), Vishe ($\mathrm{GL}_2$ spectral aspect; twists by smooth moduli), and amortized regimes (Odlyzko–Schönhage, Booker–Then) settle only special aspects, and no uniform $C^{1/2-\delta}$ algorithm for $\mathrm{GL}_1/\mathrm{GL}_2$ central values is known. The open core is the algorithmic question, together with the residual counting case.

Background

An automorphic representation of $\mathrm{GL}_n$ over $\mathbb{Q}$ is an irreducible constituent of the space of complex-valued functions on the adèle group $\mathrm{GL}_n(\mathbb{A}_\mathbb{Q})$ (the restricted product of $\mathrm{GL}_n(\mathbb{R})$ with the compact $p$-adic groups $\mathrm{GL}_n(\mathbb{Z}_p)$) transforming under the left action of $\mathrm{GL}_n(\mathbb{Q})$; it is cuspidal when the functions are orthogonal to the images of all proper parabolic subgroups. For $n=1$ the cuspidal representations are Dirichlet characters (multiplicative characters of the invertible residue classes modulo an integer, necessarily primitive when viewed as automorphic forms), and for $n=2$ they include classical holomorphic cusp forms and Maass cusp forms (non-holomorphic eigenfunctions of the hyperbolic Laplacian) on the upper half-plane. To each unitary cuspidal representation $\pi$ is attached an $L$-function

$$L(\pi,s)=\prod_v L_v(\pi,s),$$

an Euler product over the places $v$ of $\mathbb{Q}$ that converges in a right half-plane, extends to an entire function, and satisfies a functional equation relating $s$ and $1-s$. For unitary $\pi$ the functional equation is symmetric about the central point $s=1/2$, and $L(\pi,1/2)$ is the central value.

The complexity of $\pi$ is measured by the analytic conductor $\mathrm{C}(\pi)$, a positive real number packaging the finite ramification (the product of the local conductors $q(\pi_v)$ over the finite primes) and the archimedean spectral parameters (archimedean = at the real place; roughly, a polynomial in the eigenvalues of the Lie algebra of $\mathrm{GL}_n(\mathbb{R})$): for $\zeta(1/2+it)$ it is comparable to $1+|t|$, and for a Dirichlet character of modulus $q$ with trivial archimedean data it is $q$. The functional equation combined with the Phragmén–Lindelöf principle gives the convexity bound $|L(\pi,1/2)|\ll \mathrm{C}(\pi)^{1/4+\varepsilon}$, and the subconvexity problem asks for a bound $\mathrm{C}(\pi)^{1/4-\delta}$ for some fixed $\delta>0$, uniformly in the aspect in which $\pi$ varies (spectral parameter, conductor, or character twists). The source monograph-length survey of Michel–Venkatesh established subconvexity for all $L$-functions of $\mathrm{GL}_1$ and $\mathrm{GL}_2$ over number fields, uniformly in all aspects, and organized the open questions of the area.

Computationally, the standard tool is the approximate functional equation: a representation of $L(\pi,1/2)$ as a smoothly truncated sum over roughly $\mathrm{C}(\pi)^{1/2}$ Dirichlet coefficients (Hecke eigenvalues) with explicit test functions. Since each term is computable in polylogarithmic time given the coefficients, a single central value can be computed to any fixed precision in about $\mathrm{C}(\pi)^{1/2+\varepsilon}$ arithmetic operations. The exponent $1/2$ is thus the algorithmic twin of the analytic convexity exponent $1/4$, and one asks whether the extra cancellation exploited in subconvexity proofs — which replace the soft convexity argument by genuine structure of the family — can be reinterpreted as an algorithm running in time $\mathrm{C}^{1/2-\delta}$. There was a precedent: for $\zeta(1/2+it)$, Riemann–Siegel-type methods already achieve essentially $t^{1/2}$, and refined summation technologies — the amortized method of Odlyzko–Schönhage for many evaluation points, and Hiary's algorithms with exponents $4/13$ and, in variants, $1/3$ — show that the barrier can be beaten in special settings. Throughout, run time is measured either in unit-cost arithmetic operations on numbers of $O(\log\mathrm{C})$ bits, or in bit complexity, with a demand of $p$ bits of precision contributing polynomially in $p$.

The second question concerns the size of the family of all such objects. Let $\mathfrak{F}$ denote the universal family of unitary cuspidal representations of $\mathrm{GL}_n(\mathbb{A}_\mathbb{Q})$, taken up to twisting by the unitary characters $|\det|^{it}$ (without this quotient — or without fixing the central character, as in the source's finiteness corollary — every unitary twist orbit would contribute a continuum of members), and let $N(X)$ be the number of its members with analytic conductor at most $X$. Michel–Venkatesh proved that $N(X)$ is bounded by a power of $X$ and asked for the correct growth rate, suggesting $\log N(X)/\log X\to n+1$; the answer controls, for example, the number of twists required in converse-theorem characterizations of automorphic representations. A sanity check at $n=1$: the number of Dirichlet characters of conductor at most $X$ is asymptotic to $(18/\pi^4)X^2$, consistent with the exponent $n+1=2$. Brumley–Milićević resolved the question in the main cases: denoting by $\mathfrak{F}(Q)$ the members of conductor at most $Q$, they proved the Weyl–Schanuel law

$$|\mathfrak{F}(Q)|\sim\mathcal{C}(\mathfrak{F})\,Q^{n+1},$$

pure power growth with no logarithmic factors, for $n\le 2$ unconditionally and for $n\ge 3$ restricted to the archimedean spherical spectrum, i.e. to representations $\pi$ whose archimedean component $\pi_\infty$ contains a nonzero vector fixed by the maximal compact subgroup $\mathrm{O}_n(\mathbb{R})$ (the analogue of weight zero at infinity). The restriction is technical, stemming from the explicit spectral inversion of archimedean test functions in the Arthur trace formula (a noncommutative generalization of the Selberg trace formula, relating the spectrum of $\mathrm{GL}_n$ to orbital integrals), and the authors state their belief that it can be removed in a subsequent work. Their method rests on an Effective Limit Multiplicity property for the trace formula and yields as by-products uniform Weyl laws with logarithmic savings in the level and quantitative bounds on the non-tempered spectrum (representations whose local components fail the expected Ramanujan-type size bounds).

Problem Statement

The source poses two questions; they are logically independent but originate from the same survey, and the problem is completed only when each surviving core below is resolved.

(A) Fast computation of central L-values — open; the primary surviving core. By the approximate functional equation, $L(\pi,1/2)$ for a unitary cuspidal representation $\pi$ of $\mathrm{GL}_n(\mathbb{A}_\mathbb{Q})$, $n\le 2$, is computable to any prescribed precision in time $\mathrm{C}(\pi)^{1/2+\varepsilon}$, where $\mathrm{C}$ is the analytic conductor. Does there exist an algorithm computing $L(\pi,1/2)$ in time $\mathrm{C}(\pi)^{1/2-\delta+o(1)}$ for some fixed $\delta>0$ — in the source's phrasing, an algorithmic interpretation of the ideas entering the proofs of subconvexity, though an answer by any method resolves the mathematical target? The following aspects are settled and form the boundary of the open core: $\zeta(1/2+it)$ (Hiary: exponent $4/13$, with variants at $2/5$ and $1/3$, improving the classical $1/2$ of Riemann–Siegel, $3/8$ of Schönhage and $1/3$ of Heath-Brown) and Dirichlet $L$-functions at power-full (e.g. smooth) moduli (Hiary: $q^{1/3+\varepsilon}$); central values of $\mathrm{GL}_2$ cusp forms in the spectral aspect (Vishe: $T^{7/8+o(1)}$); twists $L(1/2,f\times\chi_q)$ by Dirichlet characters of smooth or highly composite modulus $q$ (Vishe: $q^{5/6+o(1)}$); and amortized regimes for many evaluation points (Odlyzko–Schönhage; Booker–Then). The open core is the general conductor aspect — for example Dirichlet $L$-functions at prime or arbitrary moduli, $\mathrm{GL}_2$ central values varying in the level aspect, or ideally a uniform statement over all aspects — and, beyond that, the extension to $n\ge 3$, where even the analytic subconvexity bounds remain unknown. A proof that no such algorithm exists in a stated computational model would equally resolve this part.

(B) The asymptotic count of the universal family — largely resolved; residual core. Let $N(X)$ be the number of cuspidal representations of $\mathrm{GL}_n(\mathbb{Q})$ with analytic conductor at most $X$, counted up to unitary twist — that is, the universal family $\mathfrak{F}$ of the Background. Determine the asymptotic behavior of $N(X)$, in particular whether $\log N(X)/\log X\to n+1$. Brumley–Milićević prove $|\mathfrak{F}(Q)|\sim\mathcal{C}(\mathfrak{F})Q^{n+1}$ for $n\le 2$ without restriction and for $n\ge 3$ under the archimedean spherical hypothesis, which answers the source question affirmatively in those cases. The surviving core is the case $n\ge 3$ with arbitrary — in particular non-spherical — archimedean component: establishing the Effective Limit Multiplicity property, or the Weyl–Schanuel law directly, for non-spherical inducing data, thereby removing the restriction that Brumley–Milićević identify as purely technical but do not remove; or a disproof of the $n+1$ prediction for the unrestricted family.

Completion boundary: the problem is answered when (A) is resolved by a proven subconvex-time algorithm for a conductor aspect outside the settled list above (the uniform all-aspects statement over $\mathrm{GL}_1/\mathrm{GL}_2$ being the natural full target), or by a proven impossibility result in a stated computational model; and (B) is resolved by the asymptotic for the unrestricted $\mathrm{GL}_n$ universal family, $n\ge 3$, or by a refutation. Partial progress on either core alone does not complete the problem, but is meaningful.

References

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