Global Langlands Functoriality Conjecture

Global Langlands Functoriality Conjecture

Definitions and mathematical statement

For a number field $F$, ${\mathbb{A}}_F$ is its ring of adeles. A quasisplit connected reductive group has a Borel subgroup over $F$; its $L$-group incorporates its complex dual group and the Galois action. At an unramified place $v$, the Satake parameter $c(\pi_v)$ is a semisimple conjugacy class in the local $L$-group. For an admissible homomorphism $\rho:{}^LG'\to{}^LG$ over the Galois group, the required weak transfer is \[ \forall\pi'\in\operatorname{Aut}(G'),\quad \exists\pi\in\operatorname{Aut}(G),\ \exists S\text{ finite},\quad c(\pi_v)=\rho_v\left(c(\pi'_v)\right)\quad(v\notin S). \] Here $\operatorname{Aut}(G)$ denotes automorphic representations in Arthur's inclusive convention, not automorphisms of $G$. The source convention, including admissibility, is retained; no cuspidality of the transfer or uniqueness is imposed.

Formulation and concept references: [S1].

Short English statement

Does every allowed map of Langlands $L$-groups produce an automorphic transfer matching the local arithmetic data almost everywhere?

Research attempt

Scope and source checkpoint.

The selected assertion is weak global transfer: matching Satake classes outside a finite set, with the inclusive automorphic convention of [S1]. Neither cuspidality nor matching every ramified component is added. Arthur [S1], Section 1, supplies this formulation; [S2] discusses the role of global trace identities. The present argument does not supply those identities or claim a new general transfer theorem. It proves a split-torus subcase directly and constructs an exact obstruction to assembling a transfer merely from individually admissible local parameters.

We use idèle-class characters as the concrete automorphic objects for $\mathrm{GL}_1$. Their conventions are reviewed in [E1], Section 5.8. The distinction between a character of the idèle group and a character of its quotient by the rational points is central to the attempt.

Attempt A: monomial maps between split tori.

Fix a number field $F$ and split tori $T'=\mathbb G_m^r$, $T=\mathbb G_m^s$. Consider the $L$-homomorphism which is the identity on the Galois factor and on dual tori is \[ \rho(z_1,\ldots,z_r)_j=\prod_{i=1}^r z_i^{a_{ji}}, \qquad A=(a_{ji})\in M_{s\times r}(\mathbb Z). \] This fixes a genuine subcase of the catalogue: no additional Galois cocycle in the $L$-homomorphism is being treated. For automorphic characters $\chi_i$ of $F^\times\backslash\mathbb A_F^\times$, define \begin{equation}\label{wsref-0-63-6f-6e-74-37-3a-74-6f-72-75-73} \psi_j=\prod_{i=1}^r\chi_i^{a_{ji}},\qquad \pi=\psi_1\boxtimes\cdots\boxtimes\psi_s. \end{equation} Products and inverses preserve continuity and triviality on $F^\times$. Consequently these are again automorphic characters. In the usual smooth automorphic convention they have the required local smoothness and archimedean character behavior as well. If the source characters are unitary, so are the target characters; unitarity is not needed to replace the inclusive convention of the question.

Let $S$ contain the finitely many places where one of the $\chi_i$ is ramified. At a finite $v\notin S$, choose a uniformizer $\varpi_v$ and use the same reciprocity convention on source and target. Then \[ c(\pi_v)_j=\psi_{j,v}(\varpi_v) =\prod_i\chi_{i,v}(\varpi_v)^{a_{ji}} =\rho\left(c(\pi'_v)\right)_j. \] If the alternative inverse-uniformizer convention is chosen, all the parameters invert and the same identity holds. Thus the required almost-everywhere identity follows without making a local-to-global existence assumption: the global characters were constructed first.

The construction also respects composition. For an integer matrix $B$ describing a second torus map, applying it to (cont7:torus) gives exponents $\sum_jb_{kj}a_{ji}$, exactly the matrix $BA$ describing the composite map. Zero rows give the trivial character; negative entries give inverse characters. Repeated characters and kernels of the torus map cause no problem. This checks several degenerate cases that a claim requiring cuspidal output would mishandle.

Where the same construction stops for general groups.

For a general $\rho$, one can still write the desired semisimple conjugacy classes $\rho_v(c(\pi'_v))$. One may also form their local Euler factors, after choosing a representation of the dual target group. Neither operation constructs a vector in an automorphic representation of $G$. In particular, the fact that each local parameter belongs to the correct dual group imposes no visibly global summation identity. The following test makes this distinction exact already in dimension one.

Lemma 7.1: a rigidity test for rational Hecke characters.

Suppose a continuous idèle-class character of $\mathbb Q$ has unramified parameters in $\{1,-1\}$ at all but finitely many primes. Then those parameters agree almost everywhere with a finite-order Dirichlet character.

Proof. The elementary idèle-class description is \[ \mathbb Q^\times\backslash\mathbb A_{\mathbb Q}^\times \simeq\mathbb R_{>0}\times\widehat{\mathbb Z}^{\times}. \] To see the representative explicitly, multiply an idèle by the rational number cancelling each of its finitely many nonzero finite valuations, then use a rational sign to make its real component positive. The finite components are now units, and no nontrivial rational multiplication preserves these normalizations. This also gives the stated product topology.

A continuous complex character of $\mathbb R_{>0}$ is $x\mapsto x^u$ for $u\in\mathbb C$. A continuous character of $\widehat{\mathbb Z}^{\times}$ has finite image: choose an open subgroup mapped into a sufficiently small arc about one in the circle; that arc contains no nontrivial subgroup, so the open subgroup is in the kernel. Here compactness first ensures that the image has modulus one. The finite character therefore factors through $(\mathbb Z/m\mathbb Z)^\times$ for some $m$. Translating to prime parameters, and absorbing inverses into the conventions, gives \begin{equation}\label{wsref-0-63-6f-6e-74-37-3a-68-65-63-6b-65} \alpha_p=\chi(p)p^u\qquad(p\nmid m) \end{equation} for a finite-order Dirichlet character $\chi$.

Since $|\alpha_p|=1$, $\operatorname{Re}u=0$; write $u=it$. If $h$ is the order of $\chi$, then for every prime outside the exceptional set \[ p^{2hit}=1,\qquad ht\log p\in\pi\mathbb Z. \] Choose two distinct such primes $p,q$. If $t\ne0$, both corresponding integers are nonzero, so $\log p/\log q$ is rational. Clearing the denominator would give an equality between a positive power of $p$ and a positive power of $q$, contrary to unique factorization. Thus $t=0$, proving the lemma.$\square$

This proof needs no assertion about primes in arithmetic progressions. Only two distinct primes outside a finite set are required. It also allows the putative global character to have arbitrary finite conductor and an initially unrestricted archimedean component.

Attempt B: diagonalize against all possible global characters.

Enumerate all Dirichlet characters as $\chi_1,\chi_2,\ldots$, retaining a modulus $m_j$ for each. Enumerate all pairs $(j,k)$ of positive integers in a sequence. At the stage assigned to $(j,k)$, choose a previously unused prime $p_{j,k}$ not dividing $m_j$, and assign \[ \alpha_{p_{j,k}}\in\{1,-1\}\setminus\{\chi_j(p_{j,k})\}. \] There is always such a sign. If the character value is neither sign, either sign works. At every finite stage only finitely many primes are excluded, so a new prime is available. Assign $\alpha_p=1$ at any primes not selected by this process.

Proposition 7.2. Every $\alpha_p$ is the parameter of an unramified unitary local character of $\mathbb Q_p^\times$, but the system $(\alpha_p)_p$ is not the almost-everywhere parameter system of any automorphic character of $\mathrm{GL}_1/\mathbb Q$.

Proof. Locally define $\theta_p(x)=\alpha_p^{v_p(x)}$. This is continuous, trivial on units, and unitary. Suppose a global character matched these parameters outside a finite set. Lemma 7.1 would identify its parameters there with one of the enumerated $\chi_j$. But the construction makes them different at every $p_{j,k}$, an infinite set of distinct primes. A finite exceptional set cannot remove all those disagreements. $\square$

One can even form a continuous character of the full idèle group by the product of these local characters and a trivial real component: the product is finite on each idèle because almost all its finite components are units. Its missing property is descent to the quotient by $\mathbb Q^\times$. Changing finitely many local factors and choosing a different archimedean character cannot repair that defect, as Proposition 7.2 proves.

Stress test: an Euler product is not an automorphy certificate.

For the same parameters the formal product \[ L_\alpha(s)=\prod_p(1-\alpha_pp^{-s})^{-1} \] is an actual absolutely convergent, nonvanishing holomorphic function in $\operatorname{Re}s>1$. Indeed its logarithm is the absolutely convergent sum \[ \sum_p\sum_{a\ge1}\frac{\alpha_p^a}{a p^{as}}, \] whose absolute value series is bounded on each $\operatorname{Re}s\ge1+\varepsilon$ by a constant times $\sum_{n\ge2}n^{-1-\varepsilon}$. Thus correct degree-one local factors, unit-modulus parameters, multiplicativity, and absolute convergence in a right half-plane can all coexist with failure of global automorphy. No analytic continuation or functional equation is claimed for this product. Those would be substantial extra conditions, not consequences of the displayed convergence calculation.

This example is not a counterexample to functoriality. It has not been obtained by applying an admissible $\rho$ to an automorphic source. What it disproves is the intermediate assertion that arbitrary well-formed local parameters can automatically be assembled into the needed global object. A proof must use the source automorphy in an essential way beyond computing the transformed Euler factors.

Conditional continuation and first unresolved step.

For any fixed source and $\rho$, a successful continuation could construct a nonzero automorphic eigenrepresentation whose spherical Hecke eigenvalues equal the characters determined by $\rho_v(c(\pi'_v))$ outside a finite set. If such a representation is constructed with the catalogue's automorphic convention, the defining Satake correspondence gives the required transfer. What is missing is its global existence, not the formal description of its prospective eigenvalues. Producing compatible finite lists of local eigenvalues does not prove that one automorphic representation realizes all of them; no compactness theorem for that assertion has been established here. In approaches through $L$-functions one would likewise need the specific analytic and twisting hypotheses of an applicable converse theorem, with all its scope restrictions proved.

Verification and outcome.

The torus identity and composition rule are exact character calculations. The obstruction is an infinite diagonal construction with every choice justified; it is not extrapolated from a finite prime sample. The rigidity argument explicitly handles both the finite exceptional set and the archimedean norm twist, two freedoms that are essential to the weak-transfer question.

Outcome: Proved split-torus subcase and a precise failure of local assembly; UNRESOLVED. The general reductive-group transfer requires a global existence argument using the automorphic source. Nothing in this attempt supplies that argument or settles general Langlands functoriality.

Sources

• S1 Classifying automorphic representations, Clay collected-works item76. https://www.claymath.org/library/cw/arthur/pdf/76.pdf. Formulation source.
• S2 Functoriality and the Trace Formula — James Arthur. https://www.math.toronto.edu/arthur/pdf/arthur-func-and-trace-formula.pdf. Further reference; not a proof endorsement.
• S3 Math 590: Open Problems in Number Theory, Spring 2026 — Duke University. https://sites.math.duke.edu/~pierce/documents/Math590.pdf. Further reference; not a proof endorsement.
• E1 Bjorn Poonen, Tate's Thesis, MIT 18.786 course notes, Section 5.8 (idèle-class characters). https://math.mit.edu/~poonen/786/notes.pdf. Primary author notes consulted 27 September 2026. The rational-character rigidity and diagonal obstruction are proved in this attempt.

Global Langlands Functoriality Conjecture

Definitions and mathematical statement

For a number field $F$, ${\mathbb{A}}_F$ is its ring of adeles. A quasisplit connected reductive group has a Borel subgroup over $F$; its $L$-group incorporates its complex dual group and the Galois action. At an unramified place $v$, the Satake parameter $c(\pi_v)$ is a semisimple conjugacy class in the local $L$-group. For an admissible homomorphism $\rho:{}^LG'\to{}^LG$ over the Galois group, the required weak transfer is \[ \forall\pi'\in\operatorname{Aut}(G'),\quad \exists\pi\in\operatorname{Aut}(G),\ \exists S\text{ finite},\quad c(\pi_v)=\rho_v\left(c(\pi'_v)\right)\quad(v\notin S). \] Here $\operatorname{Aut}(G)$ denotes automorphic representations in Arthur's inclusive convention, not automorphisms of $G$. The source convention, including admissibility, is retained; no cuspidality of the transfer or uniqueness is imposed.

Formulation and concept references: [S1].

Short English statement

Does every allowed map of Langlands $L$-groups produce an automorphic transfer matching the local arithmetic data almost everywhere?

Research attempt

Current frontier.

Research check: 27 September 2026. Arthur's Principle 1.1 uses the inclusive automorphic convention, not only the discrete or cuspidal spectrum. Its reference to Langlands's 1979 paper is substantive: Lemma 1 and Proposition 2 identify global constituents of parabolic induction from cuspidal data as automorphic representations. These statements were inspected before using that convention here. [S1], [E1]. Established functoriality includes major classical-group transfers and, for regular algebraic non-CM cuspidal representations of $\operatorname{GL}_2({\mathbb{A}}_{{\mathbb{Q}}})$, all symmetric powers by Newton--Thorne, Theorem A. The latter is not a theorem for all number fields and all admissible maps. Arthur's trace-formula discussion and the 2026 Duke course outline do not supply such a general theorem. [S1], [S2], [S3], [E2].

Attack route.

A trace-formula comparison needs a global spectral identity isolating transferred data. A converse-theorem approach needs the appropriate analytic properties of sufficiently many twisted $L$-functions. A third route embeds the target dual group in a general linear group, constructs the composite transfer, and descends it. We first check the construction in a case where automorphy can actually be supplied, then stress-test the descent route. None of these approaches may assume an already constructed universal Langlands group.

Attempt: monomial transfers and an induced subcase.

Let the source be the split torus $T'=(\mathbf G_m)^r$ over an arbitrary number field $F$. Its automorphic representations are tuples of idele-class quasicharacters $\chi_1,\ldots,\chi_r$. First take a split torus target $T=(\mathbf G_m)^s$ and a dual-torus map given by an integer matrix $B=(b_{ji})$: \[ \rho(z_1,\ldots,z_r)_j=\prod_{i=1}^r z_i^{b_{ji}}. \tag{7.1} \] Here the $L$-map acts as the identity on the Galois factor and has no additional Galois cocycle. Define $\psi_j=\prod_i\chi_i^{b_{ji}}$. Negative exponents cause no problem because quasicharacters are nonvanishing. Each $\psi_j$ is trivial on $F^\times$, is continuous, and has the same automorphic smoothness and moderate-growth properties as a finite product of such characters. Outside the union of their ramification sets, \[ \psi_{j,v}(\varpi_v)=\prod_i\chi_{i,v}(\varpi_v)^{b_{ji}}, \tag{7.2} \] which is precisely the desired equality of torus Satake parameters. This is an explicit construction in a solved subcase, not a new general transfer theorem.

Now take target $\operatorname{GL}_s$ and let the restriction of $\rho$ to the complex dual torus be any algebraic $s$-dimensional representation, still with the trivial cocycle convention. A representation of a complex torus decomposes into weight spaces, so, after conjugation, its diagonal weights are $7.1$. Put the resulting $\psi_j$ on the diagonal Levi of the upper-triangular Borel and form normalized parabolic induction. At an unramified finite place select its spherical irreducible constituent; at the finitely many remaining places select an irreducible constituent. The restricted tensor product is automorphic by Langlands's Lemma 1 and Proposition 2. Its unramified Satake class is \[ \left[\operatorname{diag} \left(\psi_{1,v}(\varpi_v),\ldots,\psi_{s,v}(\varpi_v)\right)\right]. \tag{7.3} \] Thus it has the required weak transfer. [E1]. The normalization is essential: normalized induction uses the square root of the Borel modulus. Switching induction conventions absorbs this factor in the inducing torus character; the product formula makes that modulus character trivial on the rational torus. One must not instead insert unwanted powers of the residue cardinality into $7.3$. The construction is allowed to be noncuspidal or a subquotient, exactly as the recorded target permits. Requiring a cuspidal lift would make this argument inappropriate.

Attempt: composition is valid, but descent is additional.

Suppose weak transfer is known for admissible maps ${}^LG_0\xrightarrow{\alpha}{}^LG_1\xrightarrow{\beta}{}^LG_2$. Given $\pi_0$, choose a transfer $\pi_1$, then a transfer $\pi_2$. Outside the union of their two finite exceptional sets, \[ c(\pi_{2,v})=\beta_v(c(\pi_{1,v})) =\beta_v\alpha_v(c(\pi_{0,v})). \tag{7.4} \] This proves transfer for the composite without any uniqueness assumption. For torus maps it agrees with multiplication of the integer exponent matrices. An attempted factorization of a general admissible map into known cases would therefore be legitimate, but no such factorization is produced here.

Conversely, suppose an admissible map $r:{}^LG\to{}^L\operatorname{GL}_N$ has been chosen and a transfer for $r\rho$ constructed. Obtaining a representation $\Pi$ whose Satake classes equal $r\rho(c(\pi'_v))$ is not yet an automorphic representation of $G$. The missing assertion must descend this global $\Pi$ to an automorphic representation of $G$ with the prescribed classes $\rho(c(\pi'_v))$, not merely produce some classes with the same images. It includes both a global existence issue and a conjugacy issue.

The conjugacy issue already occurs for a faithful embedding of a torus: \[ r:{\mathbb{C}}^\times\hookrightarrow\operatorname{GL}_2({\mathbb{C}}),\quad r(t)=\begin{pmatrix}t&0\\0&t^{-1}\end{pmatrix},\qquad P=\begin{pmatrix}0&1\\1&0\end{pmatrix}. \tag{7.5} \] We have $Pr(t)P^{-1}=r(t^{-1})$. For $t=2$, the two source elements are not conjugate in the abelian source, although their images are conjugate in the target. Thus faithfulness of $r$ does not imply injectivity on semisimple conjugacy classes. Even perfectly matching all characteristic polynomials after this embedding may lose required local information. This does not obstruct the explicit torus construction above; it refutes only a shortcut that tries to reconstruct that construction from the larger group's conjugacy classes alone.

Stress test.

The induced subcase uses an established automorphy theorem; merely forming a restricted tensor product of arbitrary local representations would not supply it. The general descent argument has no replacement for that theorem. Strong multiplicity-one reasoning, where available, can compare representations already known to exist; it cannot create the missing automorphic representation of $G$. Likewise, equality of formal Euler products is not an automorphy theorem.

The tensor product $7.3$ need not lie in the cuspidal or discrete $L^2$ spectrum. This is not a defect for the source's inclusive formulation, but it would be a defect for a silently strengthened version. The proof has addressed only split torus sources and the specified untwisted maps; nonsplit groups, nontrivial Galois cocycles and nonabelian source representations are not covered. No ramified local prescription, uniqueness, or cuspidality is being claimed.

Outcome.

Outcome: Exploratory approach with unresolved gap. The calculations recover an explicit family of valid weak transfers and prove the composition lemma. The proposed general-linear descent route has been narrowed to a missing global automorphy statement with prescribed local classes. The faithful-embedding counterexample shows that an image-only descent assertion is insufficient. Neither this local correction nor the solved torus subcase resolves the full functoriality assertion.

Sources

• S1 Classifying automorphic representations, Clay collected-works item76. https://www.claymath.org/library/cw/arthur/pdf/76.pdf. Catalogue formulation source.
• S2 Functoriality and the Trace Formula — James Arthur. https://www.math.toronto.edu/arthur/pdf/arthur-func-and-trace-formula.pdf. Catalogue research/status reference; not a proof endorsement.
• S3 Math 590: Open Problems in Number Theory, Spring 2026 — Duke University. https://sites.math.duke.edu/~pierce/documents/Math590.pdf. Catalogue research/status reference; not a proof endorsement.
• E1 Robert P. Langlands, On the notion of an automorphic representation, Proceedings of Symposia in Pure Mathematics 33, Part 1 $1979$, 203--207, Lemma 1 and Proposition 2. https://publications.ias.edu/sites/default/files/On-the-notion-of-an-automorphic-representation-rpl.pdf. Added research source. Primary source incorporated by Arthur's inclusive automorphic convention; induction and irreducible constituents. Consulted 27 September 2026.
• E2 James Newton and Jack A. Thorne, Symmetric power functoriality for holomorphic modular forms, II, arXiv:2009.07180v2 (27 September 2021), Theorem A and Appendix A. https://arxiv.org/html/2009.07180v2. Added research source. Symmetric powers over Q in the specified holomorphic/regular-algebraic cases, not general functoriality. Consulted 27 September 2026.