{"id":"bc0c58b8-a223-46e7-b587-36952423a262","arxiv_id":"2505.00172","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new Fourier-Mukai equivalence identifies the formal-group circle of an elliptic curve with the affinization of the curve, showing that two definitions of elliptic Hochschild homology agree and degenerate to ordinary and Hodge Hochschild homology.","lead":"This mathematics paper proves that two different ways of defining elliptic Hochschild homology give the same answer, using a new Fourier-Mukai duality for formal groups. The result unifies twisted Hochschild homology theories and suggests a global version modeled on topological modular forms.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main comparison rests on an unpublished Lurie theorem (Theorem 2.8, [Lurb] Prop. 5.1.3) whose symmetric monoidal statement over arbitrary base rings is not proved or publicly verified; if it fails, Corollary 2.9 and Theorem B collapse.","rationale":"The paper's own Proposition 2.5 gives a self-contained argument for formal groups; the only bridge from the formal completion \\hat E to the global elliptic curve E is Lurie's Theorem 2.8. Corollary 2.9 composes the two equivalences and then takes cospectra; any break in Theorem 2.8—either absence of the monoidal structure or failure over discrete bases—means Aeff(E) ≃ S^1_{\\hat E} is not known, and Corollary 2.12 cannot even be stated. The reader's conditional verdict identifies the same point, and I agree. The other candidate concerns (SCR vs CAlg_cn switch, the unreferenced identification of Hom_{QCoh(E)}(O_e,O_e) with Hom_{QCoh(\\hat E)}(O_e,O_e), and the abbreviated cuspidal case in Proposition 2.13) are real but either secondary or fixable; Lemma 2.4 is overgeneralized but only used for finite-rank algebras. Thus the most load-bearing concern is the unpublished input, and a concrete check is to verify or replace Proposition 5.1.3 of [Lurb].","tokens_in":16600,"tokens_out":41861,"duration_ms":442263,"concrete_test":"Obtain the current version of [Lurb] (Lurie, Elliptic Cohomology I) and inspect Proposition 5.1.3: confirm that it states a symmetric monoidal equivalence QCoh(A)^⋆ ≃ QCoh(A^∨)^⊗ for (spectral) abelian varieties over arbitrary E_∞ rings, and that its proof does not require hypotheses failing for discrete rings. If this is not available, provide an independent proof in the discrete case: take the Poincaré line bundle P on E × E (normalized so P|_{E×0} ≅ O_E) and prove that the induced Fourier–Mukai functor Φ_P: QCoh(E) → QCoh(E) sends convolution to tensor product and sends O_e to O_E. This would close the gap in Corollary 2.9.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central comparison HH_{\\hat E}(X) ≃ HH_E(X) (Theorem B / Corollary 2.12) is obtained from Corollary 2.9, which asserts Aeff(E) ≃ S^1_{\\hat E}. The proof of Corollary 2.9 has two inputs: Proposition 2.5 (formal-group Fourier–Mukai, proved in the paper) and Theorem 2.8, quoted verbatim from Proposition 5.1.3 of Lurie's unpublished manuscript [Lurb]. Theorem 2.8 is the only step that connects the elliptic curve E, via the symmetric monoidal equivalence QCoh(E)^⋆ ≃ QCoh(E)^⊗, to the formal completion \\hat E; it is therefore load-bearing for the identification of unit endomorphism algebras Hom_{QCoh(E)^⋆}(O_e,O_e) ≃ C^*(E,O) that yields Aeff(E) ≃ S^1_{\\hat E}. The paper does not supply a proof, a public reference, or a discussion of whether Lurie's statement (formulated for spectral abelian varieties over E_∞ rings) specializes correctly to classical elliptic curves over discrete rings. If Theorem 2.8 is unavailable in the required generality, or if the quoted proposition only gives an equivalence of plain categories without the symmetric monoidal enhancement, then the equivalence of E_∞-algebras in Corollary 2.9 is not established and the main comparison does not follow. This is a verifiable external dependency rather than a detected contradiction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper compares two existing definitions of elliptic Hochschild homology: the formal-group-based twisted Hochschild homology of Moulinos–Robalo–Toën [MRT22] and the mapping-stack construction of Sibilla–Tomasini [ST23]. The main tool is a Fourier–Mukai-type equivalence for one-dimensional formal groups, stated as Theorem A, which identifies the convolution category of quasi-coherent sheaves on a formal group with the ordinary tensor category of quasi-coherent sheaves on its Cartier-dual circle. For an elliptic curve E, the completion at the identity is then compared with E itself via Lurie's Fourier–Mukai theory for dual abelian varieties, yielding the central comparison HH_{\\hat E}(X) ≃ HH_E(X) (Corollary 2.12 / Theorem B). The paper also treats nodal and cuspidal cubics, proving directly that their affinizations give the multiplicative and additive formal-group circles, and builds global sheaves of Hochschild-type invariants over moduli stacks of elliptic and cubic curves, with conjectural connections to filtered Hochschild homology and TMF.","tokens_in":16937,"tokens_out":8145,"duration_ms":96232,"significance":"If correct, the paper establishes a genuinely useful bridge between two independent constructions in derived algebraic geometry and gives a clean conceptual explanation of why elliptic Hochschild homology can be computed either from the formal completion or from the full elliptic curve. The comparison is not circular: the definitions being identified come from different sources and are not restatements of one another. The paper is also commendably explicit about its limitations, stating open conjectures for non-smooth cubics and for the filtered circle. Its main technical assets are the formal-group Fourier–Mukai equivalence, the direct push-out arguments for nodal and cuspidal curves, and the construction of global TMF-type Hochschild theories with correct stalk behavior. However, the central comparison depends on substantial external inputs—an unpublished theorem of Lurie and a recent preprint of Torii—and on a derived-geometry dictionary that is asserted rather than proved. These dependencies make the main result conditional as written.","major_comments":[{"comment":"Theorem 2.8 is quoted verbatim from Proposition 5.1.3 of the unpublished manuscript [Lurb], and it is the only input that connects the elliptic curve E to its formal completion. Corollary 2.9 and hence Theorem B collapse if this theorem is not available in the required form. The paper gives no proof, no public reference, and no discussion of how Lurie's statement, formulated for spectral abelian varieties over E-infinity rings, specializes to classical elliptic curves over discrete rings. It also does not verify that the quoted result includes the symmetric monoidal enhancement needed to identify endomorphism algebras as E-infinity algebras. The authors should either supply a proof or an exact public reference for the needed statement, or explicitly reformulate Theorem B as conditional on that external theorem.","section":"§2.2, Theorem 2.8 and Corollary 2.9"},{"comment":"The paper switches from the simplicial-commutative-algebra setting SCR_R used by [MRT22] to the connective-E-infinity-algebra setting CAlg_cn_R with only the remark that the two approaches are 'largely parallel' and that 'in many situations one can freely switch between the two.' This is load-bearing because Theorem B is a comparison with the [MRT22] definition, which is formulated in SCR_R. The manuscript should state precisely which mapping stacks and function algebras are identified under this change of setting, and either prove the identification or cite a theorem that covers the specific sources S^1_{\\hat E} and E considered here.","section":"§2.1 and Corollary 2.12"},{"comment":"The proof of the equivalence QCoh(G) ≃ coMod_{O(G)^*} relies on base change along the atlas a : Spec R → BG^∨ and cites [BZFN10, Proposition 3.10], which requires a to be perfect. The text says this is immediate because a is affine, since it is the atlas of the classifying stack. That implication is not valid as stated: when G^∨ = G_m, the pullback of a along itself is G_m, whose structure sheaf is not a perfect R-module, so BZFN perfectness is not automatic. The authors need to prove the required base-change statement in this setting or replace it with a theorem covering classifying stacks of non-finite flat affine group schemes.","section":"§2.2, proof of Proposition 2.5, step (2)"},{"comment":"The symmetric monoidal enhancement of Proposition 2.5 is established only by invoking [Tor25], a preprint, for the statement that the forgetful functor from comodules is strong symmetric monoidal, and by a sketch that the two tensor products on the intermediate comodule category coincide. Since the monoidal structure is used essentially in Corollary 2.9 to compare E-infinity algebras of endomorphisms, this dependency is load-bearing. The paper should either include a self-contained proof of the monoidality of the composite equivalence, or clearly list the precise statement from [Tor25] that is being assumed.","section":"§2.2, monoidality in Proposition 2.5 and Corollary 2.7"}],"minor_comments":[{"comment":"The sentence 'when G is either ˆGa, ˆGa or ˆE' contains a typo: the second group should presumably be ˆGm.","section":"§1.1"},{"comment":"The proof asserts t0(Map_{dSt_R}(E,X)) ≃ t0(X) without justification; a one-line argument or a reference would be helpful.","section":"Corollary 2.12"},{"comment":"The notation T[-1]T for the shifted tangent bundle is not defined; it should be introduced explicitly.","section":"Proposition 2.13"},{"comment":"The notation HHTMF_*, HHTmf_*, and HHtmf_* is used both for sheaves over the relevant moduli stack and for their global sections; this is confusing and should be disambiguated.","section":"§3"},{"comment":"The paper relies on [Lurb], [Tor25], and [FPT], all of which are unpublished or in preparation; it would be helpful to indicate in the text which results are needed from each and to mark these dependencies clearly in the introduction.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The reliance on Lurie's unpublished [Lurb, Proposition 5.1.3] is the central risk of this paper. The authors should be encouraged to either provide a detailed proof of the specialized statement or to publish the comparison as a conditional theorem. The other main gap, the SCR_R versus CAlg_cn_R dictionary, may be standard to experts but is still load-bearing for the claimed comparison to [MRT22]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's central claim is that the MRT22 formal-group definition of elliptic Hochschild homology agrees with the ST23 mapping-stack definition. That is a real cross-check between two independent constructions, and I believe it is the first time the two are shown to match. The genuine novelty is Proposition 2.5: a symmetric monoidal equivalence QCoh(G)^star ≃ QCoh(S^1_G)^tensor for any one-dimensional formal group G, proved via comodules over the Hopf algebra of distributions. This is a new result, close in spirit to prior characteristic-zero statements but proven here over a general discrete base ring. The paper also uses it to identify Aeff(E) with S^1_hatE (Cor 2.9) and then to get the comparison, and it sets up global TMF-Hochschild homology over moduli stacks with honest, clearly labeled conjectures. The writing is clear and the limitations are stated.\n\nThe soft spots are real but not fatal. The load-bearing input is Lurie's Theorem 2.8, quoted from an unpublished manuscript [Lurb, Prop 5.1.3], giving a symmetric monoidal Fourier–Mukai equivalence for elliptic curves over E_∞ rings. The paper does not prove it, does not give a public reference, and does not discuss whether the symmetric monoidal enhancement survives the specialization to discrete rings. If that enhancement fails, Corollary 2.9 and the main comparison collapse. This is an external dependency, not a contradiction, and in this area relying on [Lurb] is common currency—but because everything rests on it, the paper should either prove the needed statement or state it as a clearly marked assumption. The monoidality argument in Proposition 2.5 is sketched and leans on another recent preprint [Tor25]; the interchangeability of SCR_R and CAlg_cn_R settings is asserted rather than proved; and the cuspidal part of Proposition 2.13 is handled by a quick pushout argument that leaves a gap (the pullback square for the cusp is not fully justified). None of these are fatal, and the paper would benefit from a careful referee asking for details.\n\nThe citation pattern is fine: the two definitions come from different sets of authors, and the paper's reliance on [ST23] is for the comparison target, not as a proof input. There is no circularity. This is a high-quality preprint that deserves a serious referee. I would send it to peer review with a request to clarify the dependence on [Lurb] and [Tor25], and to fill the small gaps. For anyone working on derived algebraic geometry, Hochschild homology, or TMF, this is worth reading and likely worth citing.","headline":"A genuinely new comparison between two definitions of elliptic Hochschild homology, held together by an unpublished Lurie theorem that the authors need to make explicit.","tokens_in":17474,"tokens_out":1898,"would_cite":true,"duration_ms":20101,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F08","14L05","55N34"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Fourier–Mukai duality for formal groups unifies elliptic Hochschild homology","keywords":["Fourier–Mukai duality","formal groups","elliptic Hochschild homology","Cartier duality","twisted Hochschild homology","mapping stacks","topological modular forms","moduli of elliptic curves"],"falsifier":"Take a specific elliptic curve over a non-field ring such as the integers, compute the $E_\\infty$-algebra of endomorphisms of the skyscraper sheaf at the identity in the convolution category, and compute the $E_\\infty$-algebra of global functions on the elliptic curve; if these are not equivalent, then $\\operatorname{Aff}(E) \\simeq S^1_{\\widehat{E}}$ is false and the two Hochschild definitions differ.","tokens_in":16377,"feed_emoji":"🔁","tokens_out":11429,"duration_ms":107834,"temperature":0.7,"pith_summary":"The paper is trying to show that two independent definitions of elliptic Hochschild homology are in fact one theory: a formal-group construction based on Cartier duality and a mapping-stack construction that takes functions on maps from an elliptic curve. The engine is a new Fourier–Mukai duality for formal groups, comparing convolution of sheaves on a one-dimensional formal group with ordinary sheaves on an associated twisted circle. When the formal group is the completion of an elliptic curve, the duality identifies the twisted circle with the affinization of the curve, which forces the two Hochschild definitions to agree. This matters because it brings ordinary, Hodge, and elliptic Hochschild homology into a single formal-group mechanism and yields global universal versions over moduli stacks of elliptic and cubic curves.","feed_headline":"Formal-group Fourier–Mukai duality unifies Hochschild theories","feed_subtitle":"Two independent definitions of elliptic Hochschild homology coincide; nodal and cuspidal limits recover ordinary and Hodge versions.","key_machinery":"The load-bearing object is the twisted circle $S^1_G := B G^\\vee$, the classifying stack of the Cartier dual of a one-dimensional formal group $G$. The key identity is the symmetric monoidal equivalence $\\mathrm{QCoh}(G)^\\star \\simeq \\mathrm{QCoh}(S^1_G)^\\otimes$: convolution along the group law on $G$ becomes ordinary tensor product on the circle. The proof's crucial computation is $\\mathrm{QCoh}(G) \\simeq \\mathrm{coMod}_{\\mathcal{O}(G)^*}$, identifying quasi-coherent sheaves with comodules over the Hopf algebra of distributions, and then identifying that same comodule category with sheaves on $B G^\\vee$; monoidality is established by showing both tensor structures reduce to tensor product of comodules over $R$. The skyscraper sheaf $\\mathcal{O}_e$ at the identity, as the unit for convolution, supplies the endomorphism algebra whose cospectrum recovers the circle.","core_discovery":"Central claim: for every one-dimensional abelian formal group $G$ over a discrete commutative ring, the category of quasi-coherent sheaves on $G$ with convolution product is symmetric monoidally equivalent to the category of quasi-coherent sheaves on the twisted circle $S^1_G := B G^\\vee$ with ordinary tensor product. The proof passes through comodules over the distribution Hopf algebra $\\mathcal{O}(G)^*$ and identifies the same comodule category with sheaves on the classifying stack of the Cartier dual $G^\\vee$. When $G = \\widehat{E}$ is the completion of an elliptic curve, this equivalence composes with the Fourier–Mukai duality for the elliptic curve itself to give $\\operatorname{Aff}(E) \\simeq S^1_{\\widehat{E}}$. Because maps out of an affinization agree with maps out of the original curve for affine targets, the two definitions of elliptic Hochschild homology—functions on maps from the formal-group circle, and functions on maps from the elliptic curve—coincide for affine schemes. The same pattern identifies the affinizations of nodal and cuspidal cubics with the multiplicative and additive formal-group circles, so ordinary Hochschild and Hodge Hochschild homology appear as degenerate cubic cases.","pith_inferences":["Because the equivalence turns the affinization of an elliptic curve into a classifying stack, computations of elliptic Hochschild homology in mixed characteristic might be reducible to finite group-quotient presentations, where both sides are otherwise hard to compute.","If the paper's conjectural Fourier–Mukai duality for arbitrary cubic curves holds, the same mechanism would imply that the formal completion of every cubic curve determines its affinization, making the universal tmf theory genuinely universal.","The formal-group circle suggests a spectral lift: over $E_\\infty$-ring spectra, the same comparison should identify elliptic Hochschild homology with functions on a spectral mapping stack, tying the affine statement to topological modular forms."],"forward_implications":["For any one-dimensional formal group, convolution sheaf theory is represented by a circle stack, so Fourier–Mukai-style dualities are not special to elliptic curves.","The equivalence of the formal-group and mapping-stack definitions of elliptic Hochschild homology for affine schemes means results proven on either side transfer directly.","Nodal and cuspidal degenerations of the same global theory recover ordinary Hochschild and Hodge Hochschild homology, placing all three theories in one family.","The sheafified theories over the moduli stacks of elliptic and cubic curves have stalks $\\widehat{E}$-Hochschild at elliptic fibers, $\\widehat{\\mathbb{G}}_m$-Hochschild at nodal fibers, and $\\widehat{\\mathbb{G}}_a$-Hochschild at cuspidal fibers, so they interpolate between the three theories."],"supporting_citations":[{"why":"Defines G-Hochschild homology as functions on mapping stacks out of the twisted circle $S^1_G$; this is the construction Theorem B compares.","marker":"[MRT22]"},{"why":"Supplies the general-base Cartier duality and the twisted-circle formalism used throughout the paper.","marker":"[Mou24]"},{"why":"Defines elliptic Hochschild homology as global sections of the almost-constant mapping stack from an elliptic curve; this is the theory identified with the formal-group version.","marker":"[ST23]"},{"why":"Quoted source of the Fourier–Mukai equivalence for dual abelian varieties over arbitrary ring spectra, used to obtain the affinization identification for elliptic curves.","marker":"[Lurb]"},{"why":"Provides the perfect-stack base-change and integral-transform foundations used in the monoidal equivalences and in computing fibres.","marker":"[BZFN10]"},{"why":"Establishes the symmetric monoidal structure on comodules needed to prove that the formal-group equivalence is monoidal.","marker":"[Tor25]"}],"fun_headline_variants":["New duality unifies elliptic Hochschild homologies","One duality to rule all Hochschild theories","Elliptic Hochschild: two definitions, one proof","Formal-group duality matches two elliptic Hochschild homologies","Fourier–Mukai duality settles elliptic Hochschild homology"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central comparison rests on an unpublished, quoted Fourier–Mukai equivalence for dual abelian varieties over arbitrary ring spectra; if that equivalence fails in the stated generality, the identification between the affinization of an elliptic curve and its formal-group circle—and therefore the equality of the two Hochschild theories—does not follow.","fun_headline_variants_meta":{"raw":{"variants":["New duality unifies elliptic Hochschild homologies","One duality to rule all Hochschild theories","Elliptic Hochschild: two definitions, one proof","Formal-group duality matches two elliptic Hochschild homologies","Fourier–Mukai duality settles elliptic Hochschild homology"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001515,"raw_usage":{"total_tokens":6081,"prompt_tokens":965,"completion_tokens":5116,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":581,"completion_tokens_details":{"reasoning_tokens":5039}},"tokens_in":581,"tokens_out":5116,"duration_ms":36567,"temperature":1.0,"reasoning_tokens":5039,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:49:24.463660+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a specific elliptic curve over a non-field ring such as the integers, compute the $E_\\infty$-algebra of endomorphisms of the skyscraper sheaf at the identity in the convolution category, and compute the $E_\\infty$-algebra of global functions on the elliptic curve; if these are not equivalent, then $\\operatorname{Aff}(E) \\simeq S^1_{\\widehat{E}}$ is false and the two Hochschild definitions differ.","supporting_citations":[],"review_version":1}