Are Farmer–Pitale–Ryan–Schmidt's tempered balanced analytic $L$-functions automorphic? The Selberg–Langlands $A=Q$ question

Farmer, Pitale, Ryan, and Schmidt proposed a five-axiom definition of a 'tempered balanced analytic $L$-function' that interpolates between Selberg's fully analytic axioms for $L$-functions and Langlands' construction of $L$-functions from cuspidal automorphic representations of $\mathrm{GL}(d)$ over $\mathbb{Q}$. Standard theory of automorphic $L$-functions gives one inclusion, $Q\subset A$: the $L$-function of any tempered balanced unitary cuspidal automorphic representation satisfies the five axioms. The open core is the converse, attributed to Selberg: is every primitive, tempered, balanced, entire analytic $L$-function satisfying the axioms equal to $L(s,\pi)$ for some tempered balanced unitary cuspidal automorphic representation $\pi$ of $\mathrm{GL}(d,\mathbb{A}_\mathbb{Q})$ — in short, is $A=Q$? As of 2026 this is open already at degree 2 with general conductor, and in every degree at least 3; the only case settled without extra hypotheses is degree 2, conductor 1 (Kaczorowski–Perelli, 2022), where the axiomatic objects are exactly level-1 modular $L$-functions.

Background

An $L$-function is a Dirichlet series $L(s)=\sum_{n\ge 1}a_n n^{-s}$ in a complex variable $s=\sigma+it$ that (i) admits an Euler product over the primes, (ii) continues meromorphically to the whole plane, and (iii) satisfies a functional equation relating $s$ and $1-s$. The prototypes are the Riemann zeta function and the Dirichlet $L$-functions of Dirichlet characters; deeper examples come from modular forms and, generally, from automorphic representations. Two general definitions of '$L$-function' dominate the field. Selberg's approach is axiomatic: the Selberg class consists of Dirichlet series with analytic continuation, a functional equation of a specific shape, an Euler product, and a Ramanujan-type bound on coefficients. Langlands' approach is structural: to each cuspidal automorphic representation $\pi$ of the group $\mathrm{GL}(d)$ over the rational adele ring $\mathbb{A}_\mathbb{Q}$ (the adele ring stitches together all completions $\mathbb{Q}_p$ and $\mathbb{R}$ of $\mathbb{Q}$; 'cuspidal automorphic representation' is the representation-theoretic incarnation of a Hecke eigenform, generalizing holomorphic cusp forms to all degrees) one attaches its standard $L$-function $L(s,\pi)=\prod_p L(s,\pi_p)$, an Euler product whose local factors at the primes $p$ are polynomials in $p^{-s}$ of degree $d$. Conjecturally the two pictures describe the same universe of objects, but even making the comparison precise is nontrivial, because the general Ramanujan conjecture (the assertion that all local components of $\pi$ are tempered, i.e. that the Satake parameters — the reciprocal roots $\alpha_{j,p}$ of the local Euler factors — satisfy $|\alpha_{j,p}|=1$ at good primes) is itself open for $d\ge 3$.

Farmer–Pitale–Ryan–Schmidt (2017, published 2019) proposed a bridge: a class of 'tempered balanced analytic $L$-functions' defined by five axioms that are stronger and more structured than Selberg's, yet purely analytic — no automorphic or algebraic object enters. Their axioms, in the normalization where $\Gamma_\mathbb{R}(s)=\pi^{-s/2}\Gamma(s/2)$ and $\Gamma_\mathbb{C}(s)=\Gamma_\mathbb{R}(s)\Gamma_\mathbb{R}(s+1)$ are the standard archimedean factors, are: (Ax 1) $L(s)=\sum a_n n^{-s}$ converges absolutely for $\sigma>1$ and continues meromorphically with only finitely many poles, all on the line $\sigma=1$; (Ax 2) there exist a conductor $N\ge 1$, a degree $d\ge 1$ with $d=d_1+2d_2$, and spectral parameters $\mu_j,\nu_k\in\mathbb{C}$, such that the completed function $\Lambda(s)=N^{s/2}\prod_{j=1}^{d_1}\Gamma_\mathbb{R}(s+\mu_j)\prod_{k=1}^{d_2}\Gamma_\mathbb{C}(s+\nu_k)L(s)$ is bounded in vertical strips away from its poles and satisfies $\Lambda(s)=\varepsilon\,\overline{\Lambda}(1-s)$ with $|\varepsilon|=1$ (the bar denotes Schwartz reflection, $\overline{f}(z)=\overline{f(\overline{z})}$); (Ax 3) $L(s)=\prod_p F_p(p^{-s})^{-1}$ with $F_p$ a polynomial, $F_p(0)=1$, of degree exactly $d$ at the good primes $p\nmid N$ and strictly smaller at the bad primes; (Ax 4) temperedness, i.e. the Selberg bound $\Re(\mu_j)\in\{0,1\}$, $\Re(\nu_k)\in\{\tfrac12,1,\tfrac32,\dots\}$ and the Ramanujan bound $|\alpha_{j,p}|=1$ at good primes with $|\alpha_{j,p}|=p^{-m_j/2}$, $\sum m_j\le d-d_p$, at bad primes, where $d_p$ is the degree of $F_p$; (Ax 5) a central-character axiom: there is a Dirichlet character $\chi$ mod $N$ with $F_p(z)=1-a_pz+\cdots+(-1)^d\chi(p)z^d$ for every $p$, the balance condition $\mathrm{Im}\big(\sum\mu_j+\sum(2\nu_k+1)\big)=0$, and the parity condition $\chi(-1)=(-1)^{\sum\mu_j+\sum(2\nu_k+1)}$. A function satisfying the axioms is primitive if it admits no nontrivial factorization into two analytic $L$-functions (a weak form of multiplicity one, proven conditionally and in strengthened form by the same authors, guarantees such factorizations are essentially unique). Let $A$ denote the class of primitive, tempered, balanced, entire $L$-functions satisfying Axioms 1–5, and let $Q$ denote the class of $L$-functions $L(s,\pi)$ attached to tempered, balanced, unitary cuspidal automorphic representations $\pi$ of $\mathrm{GL}(d,\mathbb{A}_\mathbb{Q})$ — where $\pi$ is called balanced if the restriction of its unitary central character $\omega_\pi=|\cdot|^{iy}\chi$ (with $\chi$ of finite order) to $\mathbb{R}_{>0}$ is trivial, i.e. $y=0$, equivalently if $\omega_\pi$ itself has finite order; by Theorem 3.1(4) of the same paper this implies the analytic balance condition of Ax 5 for $L(s,\pi)$ (unitarity and temperedness are imposed on both sides of the comparison, since temperedness is conjectural in general).

Classical results of Jacquet–Piatetski-Shapiro–Shalika, as formulated by Cogdell, show that $L(s,\pi)$ for such $\pi$ satisfies Axioms 1–2 and, with the temperedness and central-character hypotheses built in, all five axioms: hence $Q\subset A$ is a theorem. The reverse inclusion is the deep open direction. It is exactly the inverse problem that classical converse theorems address only partially: Hecke's and Maass's converse theorems (degree 2, special gamma factors), Weil's (degree 2, requiring functional equations of all twisted series $L(s\otimes\chi)$), and the Cogdell–Piatetski-Shapiro theorem for $\mathrm{GL}(d)$ (requiring functional equations for twists by many automorphic representations of smaller groups) all deduce automorphy from analytic data that is strictly stronger than the single untwisted functional equation and Euler product available from the axioms. The one corner settled without any twisting is degree 2 with conductor 1: Kaczorowski and Perelli, developing the theory of nonlinear twists of the extended Selberg class (the extended class replaces the Euler-product and Ramanujan axioms by weaker hypotheses), proved that every normalized element of degree 2 and conductor 1 is the $L$-function of a level-1 Hecke cusp form, $\zeta(s)^2$, or a level-1 Maass cusp form — automorphic, as expected. Their method is tied to conductor 1, and both they and other authors emphasize that very little is known for higher conductor. The general expectation, recorded by Farmer–Pitale–Ryan–Schmidt in their table of conjectural relationships (where the inclusion $A\subset Q$ is attributed to Selberg), is that $A=Q$; the same paper also records adjacent but distinct open conjectures — the Buzzard–Gee expectation that algebraic type (integral gamma shifts) coincides with arithmetic type (algebraic Dirichlet coefficients after rescaling), the Hasse–Weil, Clozel, and Fontaine–Mazur conjectures linking motives and Galois representations to automorphic forms, and the Ramanujan conjecture itself — which are separate problems and are not part of this question.

Problem Statement

Let $L(s)$ be a primitive, tempered, balanced, entire analytic $L$-function in the sense of Farmer–Pitale–Ryan–Schmidt, i.e. a Dirichlet series satisfying: (Ax 1) absolute convergence for $\sigma>1$ and meromorphic continuation to $\mathbb{C}$ with at most finitely many poles, all on $\sigma=1$ (primitive entire members have none); (Ax 2) for some conductor $N$, degree $d=d_1+2d_2$, and spectral parameters $\mu_j,\nu_k$, the completed function $\Lambda(s)=N^{s/2}\prod_{j=1}^{d_1}\Gamma_\mathbb{R}(s+\mu_j)\prod_{k=1}^{d_2}\Gamma_\mathbb{C}(s+\nu_k)L(s)$ is bounded in vertical strips away from its poles and satisfies $\Lambda(s)=\varepsilon\,\overline{\Lambda}(1-s)$ with $|\varepsilon|=1$; (Ax 3) an Euler product $L(s)=\prod_p F_p(p^{-s})^{-1}$, absolutely convergent for $\sigma>1$, with $F_p$ a polynomial with $F_p(0)=1$ of degree $d$ for $p\nmid N$ and $<d$ for $p\mid N$; (Ax 4) temperedness: $\Re(\mu_j)\in\{0,1\}$, $\Re(\nu_k)\in\{\tfrac12,1,\tfrac32,\dots\}$, and for the factorization $F_p(z)=\prod_j(1-\alpha_{j,p}z)$ one has $|\alpha_{j,p}|=1$ at good primes and $|\alpha_{j,p}|=p^{-m_j/2}$ with $\sum m_j\le d-d_p$ at bad primes; (Ax 5) a central character: a Dirichlet character $\chi$ mod $N$ with $F_p(z)=1-a_pz+\cdots+(-1)^d\chi(p)z^d$ for all $p$, $\mathrm{Im}\big(\sum\mu_j+\sum(2\nu_k+1)\big)=0$, and $\chi(-1)=(-1)^{\sum\mu_j+\sum(2\nu_k+1)}$. Let $A$ be the class of such functions, and let $Q$ be the class of standard $L$-functions $L(s,\pi)$ of tempered, balanced, unitary cuspidal automorphic representations $\pi$ of $\mathrm{GL}(d,\mathbb{A}_\mathbb{Q})$ (over all $d$; $\pi$ is balanced if its unitary central character $\omega_\pi=|\cdot|^{iy}\chi$ satisfies $y=0$, equivalently if $\omega_\pi$ has finite order). It is known that $Q\subset A$. The problem is the converse inclusion, conjectured by Selberg in the axiomatic setting: prove that every $L\in A$ equals $L(s,\pi)$ for some such $\pi$ of matching degree and conductor — equivalently, prove $A=Q$ — for all degrees $d$ and conductors $N$; or disprove it by exhibiting an explicit member of $A$ together with a rigorous proof that no tempered balanced unitary cuspidal automorphic representation of any $\mathrm{GL}(d,\mathbb{A}_\mathbb{Q})$ has this standard $L$-function.

References

1. David W. Farmer, Ameya Pitale, Nathan C. Ryan, Ralf Schmidt, 'Analytic L-functions: Definitions, theorems, and connections', Bulletin of the American Mathematical Society 56 (2019), no. 2, 261–280. DOI: 10.1090/bull/1646. arXiv:1711.10375, https://arxiv.org/abs/1711.10375
2. J. Kaczorowski, A. Perelli, 'Classification of L-functions of degree 2 and conductor 1', Advances in Mathematics 408 (2022), 108569. DOI: 10.1016/j.aim.2022.108569. arXiv:2009.12329, https://arxiv.org/abs/2009.12329
3. Jerzy Kaczorowski, Alberto Perelli, 'L-functions of degree 2 and conductor 1: underlying ideas and a generalisation', Functiones et Approximatio Commentarii Mathematici 71 (2024), no. 1. DOI: 10.7169/facm/2091. arXiv:2503.02492, https://arxiv.org/abs/2503.02492
4. Andrew R. Booker, Michael Farmer, Min Lee, 'An extension of Venkatesh's converse theorem to the Selberg class', Forum of Mathematics, Sigma 11 (2023), e26. DOI: 10.1017/fms.2023.22. arXiv:2207.00451, https://arxiv.org/abs/2207.00451
5. David W. Farmer, Sally Koutsoliotas, Stefan Lemurell, David P. Roberts, 'The Landscape of L-functions: degree 3 and conductor 1', arXiv:2303.00079 (2023). DOI: 10.48550/arXiv.2303.00079, https://arxiv.org/abs/2303.00079
6. David W. Farmer, Ameya Pitale, Nathan C. Ryan, Ralf Schmidt, 'Multiplicity one for analytic L-functions and applications', Mathematical Proceedings of the Cambridge Philosophical Society 181 (2026), no. 1, 885–911. DOI: 10.1017/S0305004126102011. arXiv:1305.3972 (there titled 'Multiplicity one for $L$-functions and applications'), https://arxiv.org/abs/1305.3972
7. Jerzy Kaczorowski, Alberto Perelli, 'A weak converse theorem for degree 2 L-functions with conductor 1', Publications of the Research Institute for Mathematical Sciences 53 (2017), 337–347. DOI: 10.4171/prims/53-2-5, https://doi.org/10.4171/prims/53-2-5
8. Michael Farmer, 'A converse theorem for degree 2 elements of the Selberg class with restricted gamma factor', Acta Arithmetica 205 (2022), no. 1, 41–52. DOI: 10.4064/aa211001-27-6, https://doi.org/10.4064/aa211001-27-6