PRH | Essay | 7.17 • From Hadele–Hidele Systems to Toric Frobenioids

From Hadele–Hidele Systems to Toric Frobenioids A Functorial Transfer of the Torus–Graded Valuation Skeleton Aleksandar Perišić August 2025 Abstract We give a clean way to pass from an analytic Hadele–Hidele system (a unitary model of the adelic ax+b action with a torus/Floquet closure of the log–scale) to a purely combinatorial, torus–graded Frobenioid that keeps only degrees, the placement of prime powers on the u –circle, and the circle holonomy. In short: we forget addition and remember the valuation/monodromy layer. We construct a small graded category TF ( O )and a canonical functor T:O 7−→ (TF(O),deg,Lα), functorial under unitary equivalences and degree–preserving intertwiners, compatible with Hecke twists, and—over function fields with circle length L = log q —an equivalence with the Frobenioid of effective divisors. No Fourier/Poisson identity is used; the transfer is extracted from operators and torus holonomy. One–page overview (plain language) • What travels. From the analytic side we keep: (i) the list of prime powers pk (“teeth”); (ii) their degree klog p ; (iii) where each degree sits on the wrapped log–scale u∈R/LZ ; and (iv) the circle’s holonomy eiα (Floquet phase). • What does not travel. The additive/shift operator and any Poisson/theta identities. Those have no functorial home in our target. • The target object. A small graded category TF ( O )whose objects are finite multisets of prime powers and whose morphisms are the degreeand torus–placement–preserving maps between them; it carries a degree functor and an S1local system recording monodromy. • Main theorem in one line. There is a canonical, unitary–invariant functor Tfrom the analytic “tooth/winding” category to TF ; over function fields this identifies TF with the usual Frobenioid of effective divisors. • Why the torus? Wrapping the log–scale ( u∼u + L ) makes monodromy visible. Over number fields this is nontrivial; over K/Fq ( T )with L = log q it collapses to a trivial circle bundle, reflecting the familiar degree. Toy example. Take L = log 10. The object [2 3 ]has degree 3 log 2and sits at angle θ = (3 log 2) mod L on the u –circle. A Hecke twist acts by rephasing the circle local system but leaves degree and θ unchanged. Over a function field with L = log q , θ≡ 0for all prime powers (since log N(p)is a multiple of log q). 1 Quick glossary Hadele/Hidele H= Hamiltonian/Heisenberg/Hilbert. Hadele and Hidele are the generatorlevel flows implementing the ax+baction, not the rings/groups A,A×. Teeth pkSpectral slices indexed by prime powers; degree klog p. Winding operator wRecords laps around the u–circle; W(α) = eiαwhas holonomy eiα. Degree functor Monoid homomorphism deg([pk]) = klog p. Toric Frobenioid The category that keeps only teeth, degree, circle placement, and holonomy. 1 Introduction A recurring theme in arithmetic geometry is that the multiplicative/valuation skeleton is robust (it survives anabelian passage), whereas the additive structure is fragile and often deliberately forgotten. Analytic models of the adelic ax+b action reflect this split: there is a shift (additive) side and a scale/prime (multiplicative) side. Once the scale axis is given a torus (Floquet) closure, the multiplicative side admits a clean, functorial projection to a torus–graded Frobenioid, matching the anabelian skeleton. Concretely, from a Hadele–Hidele system O we build a small graded category TF ( O )—the toric Frobenioid—with prime–tooth objects [ pk ], degree deg ([ pk ]) = klog p , and a circle local system Lα with holonomy eiα inherited from the Floquet closure. The projection is canonical and functorial; in the function field case it becomes an equivalence with the Frobenioid of effective divisors. The point is that TF ( O )can be compared directly to the Frobenioid of effective divisors without making any zetaor theta-type choices on the analytic side. Contributions. We make the following items precise. (C1) Axiomatization of Hadele–Hidele systems (Definition 2.1): additive generator Hade , multiplicative generator Hide = N + Pp ( log p ) Np , commutator [ Hide, Hade ] = iHade , prime counters Np, and torus closure (L, α)with winding operator w. (C2) Construction of the toric Frobenioid TF ( O )with degree functor deg and circle local system Lα—no Fourier/Poisson input. (C3) A canonical functor Tfrom Hadele–Hidele systems to toric Frobenioids (Theorem 5.2), invariant under unitary equivalence and natural for degree–preserving intertwiners. (C4) Compatibility with arithmetic idèle class characters (Hecke twists) as torsor actions on TF (Proposition 6.1). (C5) Over function fields with L = log q , an equivalence between TF ( O )and the Frobenioid of effective divisors (Theorem 7.1). Scope. We do not recast non-abelian anabelian/IUT techniques analytically; we isolate a common torus–graded layer. The additive operator and Poisson/theta identities have no counterpart in our target category and are explicitly left behind. Remark 1.1 (Naming: the “H” in Hadele/Hidele).The prefix Hstands for Hamiltonian/Heisenberg/Hilbert. Our Hadele and Hidele are the operator-level flows on a Hilbert space (generators Hade , Hide ), not the adèles/idèles themselves. Writing H-adele (Hadele) and H-idele (Hidele) flags this viewpoint. 2 Notation We write A for adèles of Q , A× for idèles, and fix a circle length L > 0and Floquet phase α∈R/ 2 πZ . The set Ω = {klog p : pprime, k ∈N} is the prime “tooth” set along the log–scale. 2 Hadele–Hidele systems We abstract just the multiplicative/torus structure we need. Definition 2.1 (Hadele–Hidele system).AHadele–Hidele system Oconsists of data (H, U, Hade, Hide,{Np}p, N, L, α, w, W(α)), where: (H1) H is a complex Hilbert space and U : ( a, v ) ∈A ⋊ A×7→ U ( a, v ) ∈ U ( H )is a unitary representation of the adelic ax+bgroup. (Think: shift by aand scale by vact unitarily.) (H2) Hade and Hide are (essentially self-adjoint on a common core) infinitesimal generators for a7→ U ( a, 1) and u7→ U (0 , vu )with |vu|A = eu .(Think: the shift and dilate Hamiltonians.) (H3) Commutator: [ Hide, Hade ] = i Hade on the core. (Think: the usual dilation–translation relation.) (H4) Prime counters: There are commuting number operators Npwith Hide =N+X p (log p)Np, where N generates the connected norm flow. For each p and k∈N , the tooth klog p∈ Ω is an eigenvalue of Np(the pk–band). (Think: Npcounts how many p’s.) (H5) Torus closure: Fix L > 0and α∈R/ 2 πZ . There is a unitary W ( α ) = eiαw (Floquet holonomy) commuting with Hade, N and all Np , where wis the integer winding operator implementing u∼u+L.(Think: wrap uon a circle and remember the phase.) Two systems are unitarily isomorphic if a unitary intertwiner preserves these structures. Remark 2.2. We retain only the multiplicative/torus layer to be transferred. No Fourier/Poisson identity or theta state is assumed. 3 The torus and winding Let S1 u = R/LZ be the circle obtained by wrapping the log–scale u . The operator wfrom (H(H5)) has spectrum in Z and records winding number; the unitary W ( α )has holonomy eiα along a lap. The pair ( S1 u,Lα )underlies the toric grading, with Lα the S1 –local system of holonomy eiα. 4 The toric Frobenioid We now define the categorical target. Definition 4.1 (Toric Frobenioid).Given O, define a small category TF(O)as follows. • Objects: finite formal sums X = Pcp,k [ pk ]with cp,k ∈N .(Think: multisets of prime powers.) 3 • Morphisms: generated by partial isometries [ pk ] → [ pk ]and inclusions [ pk ] ⊕ [ pk′ ] → [pk+k′], subject to preserving: (M1) degree: deg([pk]) = klog pextends additively; (M2) torus position: θ([pk]) = (klog p) mod L∈S1 u; (M3) winding: the winding attached to klaps equals k(compatible with w). •Degree functor: deg : TF(O)→(R≥0,+). •Local system: a circle local system Lαon S1 u, pulled back to objects via θ. We write (TF(O),deg,Lα)for this toric Frobenioid. Remark 4.2. This is the valuation/degree skeleton with explicit monodromy bookkeeping; it forgets addition. 5 The transfer functor We first isolate a small ∗–category extracted from O. Definition 5.1 (Tooth/winding ∗ -category).Let C( O )be the strict ∗ –category whose objects are finite orthogonal sums of spectral subspaces of the Np indexed by teeth [ pk ], and whose morphisms are bounded operators generated by partial isometries that commute with N and w and preserve tooth labels. Theorem 5.2 (Functorial transfer).There is a canonical functor T:C(O)−→ TF(O) that on objects sends a spectral subspace for tooth [ pk ]to the symbol [ pk ], and on morphisms sends any degree/torus–preserving partial isometry to the corresponding generator in TF . The functor is: (F1) well-defined (independent of representatives); (F2) functorial in O(unitary isomorphisms induce isomorphisms of toric Frobenioids); (F3) essentially surjective on objects and full on degree–preserving morphisms; (F4) compatible with the degree functor and the pullback of the circle local system Lα. Proof sketch. (F1) Spectral subspaces for Np decompose H into sums labeled by [ pk ]; sending these to symbols is canonical. (F2) Unitary intertwiners preserve spectra of Np , N , and w. (F3) Any finite sum of symbols is realized by spectral subspaces; fullness holds because any combinatorial degree/torus–preserving map is implemented by a partial isometry supported on the slices. (F4) Compatibility with deg follows from Hide = N + P ( log p ) Np ; compatibility with Lαuses commutation with W(α) = eiαw. Remark 5.3 (What is not transferred).No use is made of Hade beyond the commutator axiom; Poisson/theta identities and analytic readouts are intentionally absent from the target. The functor Tis a projection onto the torus–graded valuation layer. 4 6 Arithmetic twists as torsor actions Arithmetic idèle class characters act naturally on the toric Frobenioid by phase torsors. Proposition 6.1 (Hecke compatibility).Let ω : CQ = A×/Q×→S1 be a unitary idèle class character. Then ω defines an autoequivalence of TF ( O )that fixes objects and tensors the local system along S1 uby a constant phase, preserving deg and θ. This is functorial: (O, ω)7−→ (TF(O),deg,Lα⊗ω) canonically as graded categories with local system. Idea. A Hecke character restricts to phases on prime powers compatible with multiplicativity. Since TF forgets addition and keeps only degree and torus placement, the effect is to rephase the circle local system by a constant S1twist, leaving deg and θunchanged. 7 Function field case: an equivalence The function field setting affords an exact identification. Theorem 7.1 (Equivalence over function fields).Let K/Fq ( T )be a global function field and choose L= log q. Then for any prime ideal pand k≥1, θ([pk]) ≡0 (mod L),deg([pk]) = kdeg(p) log q. The functor Tinduces an equivalence TF(O)≃Diveff (K), where Diveff ( K )is the Frobenioid of effective divisors with deg ([ pk ]) = kdeg ( p ) log q and trivial circle local system. Proof sketch. Since N( p ) = qdeg p , we have log N( p ) = deg p·log q , so ( klog N( p )) mod log q≡ 0. Thus torus placement is constant, and the remaining grading is the familiar integer degree (scaled by log q). Objects and morphisms then match those of Diveff (K). 8 Functoriality and stability Proposition 8.1 (Unitary invariance).If O and O′ are unitarily isomorphic Hadele–Hidele systems, then TF(O)∼ =TF(O′)canonically as graded categories with local system. Sketch. A unitary intertwiner preserves spectra of Np , N , and w, hence preserves deg , θ , and the circle local system. Proposition 8.2 (Stability under degree–preserving intertwiners).Let T : O → O′ be a bounded intertwiner commuting with N , all Np , and W ( α ). Then Tmaps T to a morphism of toric Frobenioids preserving deg and Lα. Sketch. Such T respects the tooth decomposition and torus holonomy, hence descends to the combinatorial data of TF. 5 9 Limitations and outlook The projection Tis intentionally blind to addition: it cannot see Hade beyond its commutator with Hide, and it uses no Poisson/theta identity. Two directions remain: • Tannakian lift. Enhance TF to a neutral Tannakian category whose fundamental group recovers (at least) the abelianized Galois group via class field theory; arithmetic twists act as tensor autoequivalences. • Non-abelian layer. Extending beyond valuations would need categorical structures encoding tempered fundamental groups; our analytic data alone are insufficient. • Expanding flows. One may adjoin further one–parameter flows with prescribed commutators with Hade , Hide and W ( α ). When such flows preserve N , Np and w, they pass through T; otherwise they remain analytic decorations that the projection forgets. 10 Connections worth a casual look The following links seem naturally comparable to the torus–graded valuation layer ( deg , placement modL, holonomy): • Dynamical zeta (Ruelle/Selberg). Primitive periodic orbits ↔ prime powers; transfer operators and twisted Euler products. • Ihara zeta and graphs. Primitive cycles ↔ prime powers; families of weighted graphs with cycle lengths approximating klog p(mod L). • Noncommutative geometry/KMS (Bost–Connes). Time evolution from the multiplicative generator on a semigroup C∗ -algebra; KMS phases controlled by torus–graded data. • Thermodynamic formalism. Weights on teeth as a potential; topological pressure vs. log Euler products. • Tropical/Berkovich/Arakelov skeletons. Package as a metrized graph plus an S1 local system; check functoriality under pullbacks. • SYZ–style mirror toys. The u –circle as base, prime teeth as singular fibers; Landau– Ginzburg monodromy. • Renormalization Hopf algebras. Primes as primitives and flows as characters; compare associated gradeds with the toric Frobenioid. 11 A micro example: the prime 3 We close with a concrete three-step example showing how a single prime power travels from the analytic Hadele–Hidele side to the toric Frobenioid, and how it looks in the function field case. Step 1: analytic side (a single tooth) Fix a Hadele–Hidele system Owith circle length L > 0. For the prime p= 3, let H3k⊂ H be a nonzero spectral subspace on which: 6 •N3acts by the eigenvalue k; •Npacts by 0for all p= 3; •Nacts with some spectrum contained in an interval Ik⊂R. On H3kthe multiplicative generator restricts to Hide H3k=N H3k+klog 3 ·id, so the tooth [3 k ]has degree klog 3in the sense of our construction. Wrapping the log–scale u on S1 u=R/LZ, this tooth sits at circle angle θk:= (klog 3) mod L∈S1 u, and the winding operator wrecords how many laps around S1 u the flow has made. The Floquet unitary W(α) = eiαwacts on H3kby a phase determined by the winding number. Step 2: image in the toric Frobenioid Applying the functor Tof Theorem 5.2, the spectral subspace H3kis sent to the object [3k]∈TF(O). On this object we retain exactly: •the degree deg([3k]) = klog 3; •the torus position θ([3k]) = θk= (klog 3) mod L∈S1 u; •the fiber of the circle local system Lαat θk, with holonomy eiα around the circle. Any partial isometry V:H3k−→ H3k that commutes with N and wand preserves the tooth label [3 k ]is mapped by Tto a morphism T(V) : [3k]−→ [3k] in TF ( O ), i.e. to an endomorphism of the corresponding object that respects degree and torus placement. If L= log 10, for example, then deg([32]) = 2 log 3, θ([32]) = (2 log 3) mod log 10, so the tooth [32]is a point on the u–circle at that angle, carrying the local system Lα. 7 Step 3: function field shadow Now specialize to the function field situation of Theorem 7.1. Take a global function field K/Fq ( T )with q = 3 and choose L = log q = log 3. For any prime ideal p⊂ OK of degree deg p , we have log N(p) = deg p·log 3, so θ([pk]) = (klog N(p)) mod log 3 ≡0 (mod log 3). In particular, the circle coordinate of every tooth collapses to 0, and the only remaining data are the integer degrees kdeg(p). Under the equivalence TF(O)≃Diveff (K) from Theorem 7.1, our analytic tooth [3 k ]is seen only through its degree: it corresponds to an effective divisor of the appropriate degree (up to the usual identification of primes with places of K ). The toric refinement (the angle θk and the holonomy of Lα ) is invisible in the function field Frobenioid, which matches the usual picture: over function fields, the valuation skeleton is purely discrete. This tiny example illustrates the general philosophy: On the analytic side, a tooth such as [3 k ]lives on a torus with a nontrivial local system; the functor Tkeeps exactly the degree, the circle position, and the holonomy. In the function field limit with L = log q , the circle collapses and only the discrete Frobenioid of effective divisors survives. Acknowledgments Discussions around the additive/multiplicative split and torus closures inspired this formalization; any remaining imprecision is the author’s. References [1] J. T. Tate, Fourier Analysis in Number Fields and Hecke’s Zeta-Functions, Ph.D. thesis, Princeton (1950); reprinted in Cassels–Fröhlich, Algebraic Number Theory, Academic Press (1967). [2] A. Weil, Basic Number Theory, Second Edition, Springer (1974). [3] S. Mochizuki, The Geometry of Frobenioids, Publ. RIMS Kyoto Univ. 37 (2001), 163–223. [4] S. Mochizuki, Topics in Absolute Anabelian Geometry, Publ. RIMS Kyoto Univ. 40 (2004), 1–40. 8