{"id":"48b05c8e-e1bb-460c-ad9a-1b2625536274","arxiv_id":"2507.15020","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors construct p-adic integration pairings and Q-structures for 1-motives with good reduction, and claim that all linear relations among the resulting p-adic periods at depths 1 and 2 are captured by bilinearity and functoriality.","lead":"This paper builds a p-adic theory of periods for 1-motives with good reduction, constructing an integration pairing and new Q-structures that aim to describe all linear relations among p-adic periods. A reader interested in arithmetic geometry or transcendence will find a p-adic analogue of the Kontsevich-Zagier framework, with conjectures stated at low depth relative to these structures.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2 invokes Proposition 5.1 on u+v without proving that u+v is the p-adic logarithm of a rational point of G^#; the vector part v need not lie in any finite extension of K, so the main structural input to Theorems 4 and 5 is unsupported.","rationale":"The reader's weakest-assumption diagnosis identifies exactly the point where the argument for the p-adic subgroup theorem for 1-motives breaks down. The proof of Theorem 2 needs to apply the p-adic analytic subgroup theorem to an element u + v of Lie(G^#)_Cp, but the theorem only produces algebraic subgroups from logarithms of rational points. The paper does not prove that u + v is such a logarithm: the u-part comes from hp(M,K), while the v-part is an arbitrary element of V(M) ⊗ Cp, and arbitrary Cp-elements cannot be forced into a finite extension of K. This is not a mere technicality: the entire proof of the depth-2 H^ϖ_p- and H^φ_p-period conjectures and the depth-1 hp-period conjecture funnels through Theorem 2, and no alternative argument is offered for the v-component. There is also an additional gap in Theorem 4 where Lemma 5.2 is used to write ω = F^n(γ) for a single γ, although Lemma 5.2 only gives N as a sum of the spaces F^n(X); however, this is secondary because the failure of Theorem 2 already removes the required exact sequence. The reader's verdict of REJECT with moderate confidence is therefore appropriate. I do not see a reason to change the verdict, but the paper contains interesting scaffolding: the p-adic integration pairing of Section 3, the stratified period formalism of Section 4, and the explicit examples in Section 6 are substantive and could form the basis of a revised version if the lifting step in Theorem 2 and the construction of N are supplied.","tokens_in":36030,"tokens_out":7517,"duration_ms":89546,"concrete_test":"Take a split Kummer motive M = [Z → Gm] with G^# ≃ Ga × Gm, and choose x ∈ H^φ_p(M) whose v-component lies in V(M) ⊗ Cp but not in V(M)(K') for any finite K'. Write the explicit equation for γ = (a,b) ∈ Ga(K') × Gm(K') to satisfy log_{G^#}(γ) = u + v, and check whether a solution exists for some finite K'. Since log on Ga is the identity and log on Gm maps K'-points into K', no solution exists when v is outside the algebraic closure. This would show that Proposition 5.1 cannot be applied at the point where Theorem 2 uses it, leaving the proof of Theorem 2 without a substitute input for controlling the v-part.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central structural input is Theorem 2, whose proof in Section 5.2 decomposes x ∈ H^φ_p(M) as x = u + v with u ∈ hp(M,K) and v ∈ V(M) ⊗ Cp, and then applies Proposition 5.1 to u + v ∈ Lie(G^#)_Cp. Proposition 5.1 only applies to tuples u_i that lie in log_G(G(K)_f), i.e., p-adic logarithms of rational points of the algebraic group over a finite extension of K. The proof asserts without argument that u + v has this property. The u-part is a logarithm of a point of the formal p-divisible group or of G, but the lift to G^# is not justified, and the v-part is an arbitrary Cp-linear combination in the vector group V(M); there is no reason for v to lie in V(M)(K') for any finite K'. The subsequent construction of the auxiliary semi-abelian N in the case v ∉ V(M1)_Cp is also under-specified: N = G∨/(K1 ∩ K2), with K1 'a subgroup whose Lie algebra is contained in Ann(v)', is not shown to exist, and the inclusions in diagrams (5.4)–(5.5) are asserted. Since Theorem 2 is the only input that turns a vanishing period into an exact sequence of 1-motives (used in Theorems 4, 5 and Proposition 6.1), this gap is load-bearing. The p-adic analytic subgroup theorem stated as Theorem 3 is itself formulated for γ ∈ G(Q̄)_f with a Q-linear subspace, which is not the standard local formulation; even granting it, the application requires the lifting that is missing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a p-adic integration theory for 1-motives with good reduction, constructs canonical Q-structures H^φ_p, H^ϖ_p, and hp on p-adic realizations from formal p-divisible groups, and introduces a depth-stratified formalism of period conjectures. Its main results (Theorems 4 and 5) claim that the H^ϖ_p- and H^φ_p-period conjectures hold at depth 2 and the hp-period conjecture holds at depth 1, relative to these Q-structures. The proofs rely on a p-adic analytic subgroup theorem for 1-motives (Theorem 2), which is derived from the Bertrand–Fuchs p-adic analytic subgroup theorem. The paper also includes examples and criteria for linear independence of p-adic periods.","tokens_in":36442,"tokens_out":11026,"duration_ms":110943,"significance":"If the central claims were established, the paper would be a valuable contribution to p-adic period theory: Theorem 1 generalizes Colmez–Fontaine integration to 1-motives, the construction of the Q-structures hp, H^φ_p, and H^ϖ_p is original, and the stratified depth formalism provides a new framework for linear relations among p-adic periods. The authors deserve credit for explicit constructions, for the careful statement of limitations (e.g., the unknown relation to André's p-adic periods), and for providing concrete examples. However, the main structural theorem (Theorem 2) and the supporting lemmas currently exhibit load-bearing gaps; without them, the period conjectures at depths 1 and 2 are not established.","major_comments":[{"comment":"The proof decomposes x ∈ H^φ_p(M) as x = u + v with u ∈ hp(M,K) and v ∈ V(M)⊗Cp, and then applies Proposition 5.1 to u+v ∈ Lie(G^♮)_Cp. Proposition 5.1 requires each u_i to lie in log_G(G(Q)_f) (or log_G(G(Q)) for semi-abelian G), i.e., to be the p-adic logarithm of a rational point over a finite extension. The argument only justifies that u is such a logarithm; it does not show that v, an arbitrary Cp-linear combination in the vector group V(M), lies in V(M)(K') for any finite extension K', nor that u+v is the logarithm of a point of G^♮. Consequently, the application of Theorem 3 (the p-adic analytic subgroup theorem) is unjustified, and the existence of H1 ⊆ G^♮ with u+v ∈ Lie(H1)_Cp is unsupported. Since Theorem 2 is the key input to Theorems 4, 5, and Proposition 6.1, this gap is load-bearing.","section":"§5.2, proof of Theorem 2"},{"comment":"In the proof of Theorem 2, for the case v ∉ V(M1)_Cp, the authors define N := G^∨/(K1 ∩ K2), where K1 is 'a subgroup of G^∨ whose Lie algebra is contained in Ann(v), possibly after extending scalars to Cp'. It is not shown that such a subgroup K1 exists as an algebraic subgroup over a finite extension of K; a Lie-algebra condition over Cp does not automatically yield an algebraic subgroup defined over the base field. It is also not demonstrated that the quotient N is a semi-abelian scheme, nor that the inclusions in diagrams (5.4) and (5.5) exist as claimed. The subsequent conclusion that x ∈ H^φ_p(M'_1) and Ann(x) ⊆ T^∨_dR(M'_2)_Cp therefore lacks support. This step is necessary for the full statement of Theorem 2, so the gap is load-bearing.","section":"§5.2, construction of N in the case v ∉ V(M1)_Cp"},{"comment":"The proof of Proposition 5.1 sets V := Ann Ann(u) ⊂ Lie(G)_Cp and then invokes Theorem 3, which is stated for a non-zero Q-linear subspace V ⊆ Lie(G). The proof does not explain how a Cp-subspace is replaced by a Q-subspace to which Theorem 3 applies, nor does it verify that the resulting algebraic subgroup H1 satisfies Ann(u) = coLie(H2)_Cp after using Lemma 5.1. Thus the reformulated subgroup theorem, Proposition 5.1, is not established as stated, independent of the lifting issue in Theorem 2.","section":"§5.1, proof of Proposition 5.1"},{"comment":"Lemma 5.2 claims that for an admissible filtered isocrystal N with nonzero slopes and the stated filtration, the Frobenius translates of X generate N. The proof asserts 'Since N and Y have only positive slopes' without justification; the lemma only assumes nonzero slopes, which allows negative slopes. Moreover, the argument that N/Y contradicts admissibility uses the weak admissibility of N/Y, which is not proved. Since this lemma is used in the proof of Theorem 4 to conclude that ω = F^n(γ) for some γ in coLie(G) ∩ eN, the reduction from a vanishing H^ϖ_p-period to a vanishing H^φ_p-period is incomplete.","section":"§5.3, Lemma 5.2"}],"minor_comments":[{"comment":"The section title 'P-adic peirods of 1-motives' contains a typo; it should read 'P-adic periods of 1-motives'.","section":"§4 header"},{"comment":"The citation 'Hubber and Wüstholz' should be 'Huber and Wüstholz'.","section":"Example 4.1"},{"comment":"The phrase 'and and providing' contains a duplicated 'and'.","section":"§1, paragraph 7"},{"comment":"The notation 'M n' in the exact sequence 0 → M1 → M n → M2 → 0 is ambiguous; it should be written M^{⊕ n} to indicate the direct sum of n copies of M.","section":"Theorem 2 statement"},{"comment":"The symbol K is used both for the number field and for a p-adic completion in the same subsection, which is confusing; a distinct notation such as K_v would help.","section":"§4.2"}],"recommendation":"reject","confidential_remarks":"The paper contains substantial constructive ideas, particularly Theorem 1 and the Q-structure formalism. However, the central theorem (Theorem 2) has a fundamental gap in the application of the p-adic analytic subgroup theorem, and the proof of Lemma 5.2 is also incomplete. These issues are load-bearing for the main period conjectures. I recommend rejection; the authors may wish to resubmit if they can supply a correct lifting argument or appropriate additional hypotheses."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth serious refereeing, but the central structural theorem (Theorem 2) has a genuine gap that the headline results inherit.\n\nWhat is actually new and good: Theorem 1 constructs a p-adic integration pairing for 1-motives with good reduction, extending Colmez's abelian-variety theory; the construction is detailed and plausible. The Q-structures H^φ_p and H^ϖ_p are carefully defined via pullbacks along Fontaine's map and the integration map, and the stratified formalism for period relations at depths 1 and 2 is a useful framework. The paper credits the relevant prior work, including Huber–Wüstholz and Andreatta–Barbieri-Viale.\n\nThe soft spot is where your report puts it. In the proof of Theorem 2 (Section 5.2), x ∈ H^φ_p(M) is decomposed as u + v with u ∈ hp(M,K) and v ∈ V(M) ⊗ Cp. Proposition 5.1 is then applied to u + v ∈ Lie(G^#)_Cp. But Proposition 5.1 requires the input to lie in log_G(G(Qbar)_f) or log_G(G(Qbar)) for semi-abelian G. The u part has that property by construction; the v part is an arbitrary Cp-linear combination in the vector group and there is no justification that u+v is the logarithm of a rational point of G^#. The auxiliary semi-abelian N in the v ∉ V(M1) case is also under-specified: N = G^∨/(K1 ∩ K2) with K1 'a subgroup whose Lie algebra is contained in Ann(v)'—no proof of existence is given, and the inclusions in diagrams (5.4)–(5.5) are asserted. This is not a minor technicality; Theorem 2 is the only input that turns a vanishing period into an exact sequence of 1-motives, and Theorems 4, 5, and Proposition 6.1 all depend on it.\n\nIs the paper salvageable? Possibly. The integration construction and the period formalism are independent contributions that may survive with a repaired subgroup theorem. But as written, the main results are unsupported.\n\nWho should read this: specialists in p-adic Hodge theory will want to study the pairing and the Q-structures; they should not yet rely on the depth-1/depth-2 conjecture proofs. It deserves a serious referee rather than a desk rejection, because the scaffold is substantive, but the referee should be asked to focus on whether the lifting step in Theorem 2 can be justified.","headline":"Ambitious p-adic period framework with a real integration pairing, but the central subgroup theorem has an unproved lifting step that the main theorems inherit.","tokens_in":36982,"tokens_out":3552,"would_cite":false,"duration_ms":37121,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F30","11S80","14C15","14C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes p-adic period conjectures for 1-motives at depths 1 and 2, relative to canonical Q-structures, by showing all linear relations among these p-adic periods are induced by bilinearity and functoriality.","keywords":["p-adic periods","1-motives","p-adic integration","crystalline cohomology","period conjectures","p-adic analytic subgroup theorem","Q-structures","B_dR period ring"],"falsifier":"Take a Kummer 1-motive $M = [\\mathbb{Z}\\to\\mathbb{G}_m]$ over a number field with good reduction at $p$, attached to a non-root-of-unity $g$, and compute which elements of $h_p(M) + V(M)C_p \\subseteq \\mathrm{Lie}(G^{\\natural})_{C_p}$ lie in the image of the logarithm map on $G^{\\natural}(K')$ for finite extensions $K'$; exhibiting one element that lies in no such image would falsify the unproved lifting assertion in Section 5.2, invalidate Theorem 2, and collapse Theorem 4 and Theorem 5.","tokens_in":35840,"feed_emoji":"🧮","tokens_out":11031,"duration_ms":102762,"temperature":0.7,"pith_summary":"This paper develops a $p$-adic analogue of the classical theory of periods for 1-motives, the mixed motives built from a lattice mapping into a semi-abelian variety. It constructs a $p$-adic integration pairing that generalizes the Colmez–Fontaine–Messing theory from abelian varieties to all 1-motives with good reduction, and it uses this pairing to build canonical $\\mathbb{Q}$-structures on the $p$-adic realizations. The central result is that, relative to these structures, all linear relations among the corresponding $p$-adic periods are induced by bilinearity and functoriality: this is proved at depth 1 for the $h_p$-periods and at depth 2 for the $H^\\varphi_p$- and $H^\\varpi_p$-periods. Because these motives include Kummer extensions, tori, and abelian varieties, the theorem gives a $p$-adic counterpart to the Kontsevich–Zagier conjecture for 1-motives over $\\bar{\\mathbb{Q}}$. The paper matters because it converts a transcendence-style conjecture about $p$-adic numbers into a provable structural statement about exact sequences of 1-motives.","feed_headline":"P-adic period conjectures for 1-motives proven at depths 1 and 2","feed_subtitle":"A new p-adic integration pairing shows all linear relations among these periods reduce to bilinearity and functoriality.","key_machinery":"The load-bearing objects are the $p$-adic integration pairing $\\int^{\\varpi}\\colon T_p(M)\\times T^\\vee_{\\mathrm{dR}}(M)\\to B_2$, with $B_2 = B^+_{\\mathrm{dR}}/t^2B^+_{\\mathrm{dR}}$, which is bilinear, perfect, Galois-equivariant, and compatible with the Hodge filtration (Theorem 1), and Fontaine's map $\\varphi_M\\colon T_p(M)\\to \\mathrm{Lie}(G)\\otimes C_p(1)$ that it refines. On top of these sit the canonical $\\mathbb{Q}$-structures $h_p(M)$, $H^\\varphi_p(M)$, and $H^\\varpi_p(M)$, built from rational-point logarithms and their pullbacks along $\\varphi_M$ and $\\varpi_M$. The structural engine is the $p$-adic subgroup theorem for 1-motives (Theorem 2), which uses the $p$-adic analytic subgroup theorem of Bertrand and Fuchs to attach to every period class $x\\in H^\\varphi_p(M)$ a short exact sequence $0\\to M_1\\to M^n\\to M_2\\to 0$ with $x\\in H^\\varphi_p(M_1)$ and $\\mathrm{Ann}(x)\\subseteq T^\\vee_{\\mathrm{dR}}(M_2)$. The depth-2 conclusion for $H^\\varpi_p$ additionally uses Lemma 5.2, which shows that the degree-one Hodge component of an admissible filtered isocrystal with nonzero slopes generates the whole isocrystal under Frobenius, allowing the differential form in a vanishing period to be transported to $\\mathrm{coLie}(G)$.","core_discovery":"The paper's central claim is that the $p$-adic period conjectures for 1-motives over number fields with good reduction at $p$ hold at depth 2 for the period spaces built from the integration pairing $\\varpi_M$ and from Fontaine's map $\\varphi_M$, and at depth 1 for the period space built from rational-point logarithms $h_p(M)$. Concretely, Theorem 4 and Theorem 5 assert that every linear relation among $H^\\varpi_p$-periods or $H^\\varphi_p$-periods is a formal consequence of bilinearity and functoriality applied through a short exact sequence of 1-motives, and the same holds for $h_p$-periods already at depth 1. These results are established relative to the canonical $\\mathbb{Q}$-structures $H^\\varphi_p(M)$, $H^\\varpi_p(M)$, and $h_p(M)$ constructed in Section 4.2, which the authors obtain by pulling back rational points of the formal $p$-divisible group along the integration and Fontaine maps. The proof routes through a $p$-adic analytic subgroup theorem for 1-motives that controls the annihilator of any period class by the de Rham realization of a subquotient motive, mirroring Wüstholz's classical theorem.","pith_inferences":["The depth-stratified formalism suggests a hierarchy the paper does not develop: relations among periods attached to $k$-fold extensions of 1-motives should be governed by depth-$k$ versions of the subgroup theorem, with depths 1 and 2 as the first two rungs.","Because the canonical $\\mathbb{Q}$-structures come from rational points of the formal $p$-divisible group rather than from algebraic classes in crystalline cohomology, the same template could define period conjectures for other mixed motives that admit a $p$-adic integration map.","A testable consequence of the depth-1 result is that for a Kummer motive with non-root-of-unity $g$ and good reduction at $p$, the $h_p$-periods should admit no nontrivial $\\mathbb{Q}$-linear relations; numerically computing $p$-adic logarithms of powers of $g$ and testing their linear independence would probe Theorem 5 directly."],"forward_implications":["All linear relations among $h_p$-periods of 1-motives with good reduction are induced by bilinearity and functoriality (Corollary 5.2), so the $p$-adic analogue of the Kontsevich–Zagier conjecture holds at depth 1 for this class.","Every zero $H^\\varpi_p$-period or $H^\\varphi_p$-period is explained by a short exact sequence of 1-motives: the period vector lies in the sub-motive and the form in the quotient, so vanishing periods receive a motivic classification (Theorem 6 for the Picard 1-motives of varieties).","Periods of motives whose generated categories admit no nontrivial morphisms are linearly independent; for instance, a nonzero $H^\\varpi_p$-period of an abelian variety is linearly independent over the coefficient field from one of $\\mathbb{G}_m$ (Corollary 6.1).","The depth-2 statement is optimal: for Kummer motives, the depth-1 formal period space differs from the depth-2 space, showing that genuine relations beyond bilinearity and functoriality exist and are exactly captured at depth 2 (Example 6.3)."],"supporting_citations":[{"why":"Colmez's p-adic periods of abelian varieties provide the pairing that Theorem 1 generalizes to 1-motives; the proof of perfectness and Hodge compatibility for the semi-abelian case invokes Colmez's Theorem 5.2 directly.","marker":"[Col92]"},{"why":"Huber–Wüstholz's linear-relations theorem for 1-periods over Q is the classical statement the paper recovers at depth 1 in its framework and uses as the model for the stratified formalism.","marker":"[HW22]"},{"why":"Wüstholz's analytic subgroup theorem is the classical tool whose p-adic analogue (Theorem 2) is the paper's main structural input.","marker":"[Wus89]"},{"why":"Fuchs–Pham's p-adic analytic subgroup theorem (with Bertrand's earlier version) is cited as the p-adic result that Proposition 5.1 adapts to the 1-motive setting.","marker":"[FP15]"},{"why":"Andreatta–Barbieri-Viale's crystalline realizations of 1-motives supply the crystalline realization and the crystalline–de Rham comparison used for the isocrystal part of the argument.","marker":"[ABV05]"},{"why":"Tate's p-divisible groups theorem supplies the Hodge–Tate weights and vanishing of Galois cohomology used in Proposition 2.2 and Lemma 3.1.","marker":"[Tat67]"}],"fun_headline_variants":["P-adic period conjectures for 1-motives: depths 1 and 2 proven","Depth-2 p-adic period conjecture for 1-motives resolved","P-adic integration settles period conjectures for 1-motives","All linear relations among 1-motive p-adic periods identified"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes without proof that every $p$-adic period class can be split into a logarithm of a rational point of a finite extension plus a vector-group part, and that the combined element still lies in the range of the logarithm map; if this lifting fails for any 1-motive, the subgroup theorem, and with it the depth-1 and depth-2 period conjectures, loses its main structural support.","fun_headline_variants_meta":{"raw":{"variants":["P-adic period conjectures for 1-motives: depths 1 and 2 proven","Depth-2 p-adic period conjecture for 1-motives resolved","P-adic integration settles period conjectures for 1-motives","All linear relations among 1-motive p-adic periods identified"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000275,"raw_usage":{"total_tokens":1740,"prompt_tokens":1141,"completion_tokens":599,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":757,"completion_tokens_details":{"reasoning_tokens":517}},"tokens_in":757,"tokens_out":599,"duration_ms":6217,"temperature":1.0,"reasoning_tokens":517,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:43:58.103376+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a Kummer 1-motive $M = [\\mathbb{Z}\\to\\mathbb{G}_m]$ over a number field with good reduction at $p$, attached to a non-root-of-unity $g$, and compute which elements of $h_p(M) + V(M)C_p \\subseteq \\mathrm{Lie}(G^{\\natural})_{C_p}$ lie in the image of the logarithm map on $G^{\\natural}(K')$ for finite extensions $K'$; exhibiting one element that lies in no such image would falsify the unproved lifting assertion in Section 5.2, invalidate Theorem 2, and collapse Theorem 4 and Theorem 5.","supporting_citations":[],"review_version":1}