{"id":"c81160b2-84bd-44b6-b843-71ae4d31dd55","arxiv_id":"1908.08678","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors construct canonical isogeny characters for Drinfeld-Stuhler modules and use them to give explicit examples of Drinfeld-Stuhler curves that violate the Hasse principle.","lead":"This paper develops a theory of canonical isogeny characters for Drinfeld-Stuhler modules, which are function field analogues of abelian surfaces with quaternionic multiplication, and uses it to prove explicit criteria for the absence of rational points on the associated modular varieties. It then produces explicit families of Drinfeld-Stuhler curves over F_q(T) that violate the Hasse principle, meaning they have points over every completion but none over the global field.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Hasse-principle criteria rest on Proposition 3.3's degree bound q^d−1; the proof's normality/abelian step for the good-reduction extension is unjustified and needs an independent check.","rationale":"The paper's main theorems are conditional on the canonical isogeny character having controlled ramification, specifically q^d−1. The reader already identified Proposition 3.3 as the weakest assumption, and I agree. Our pass found a precise unproved step inside Proposition 3.3: the extension cut out by ΓθN is asserted to be abelian/totally tame and its Galois group is injected into Aut_k(φbar). Since ΓθN is not obviously normal, this is a real gap rather than just a missing citation. The remainder of the paper—Lemma 6.5, Proposition 4.8, and the congruence arguments in Section 6—uses exactly this bound. The examples in Section 7 are computational and not shipped, but that is secondary; if Proposition 3.3 is fixed, the computational claims are checkable. If the bound fails only mildly, the statements may still be true, but the proof would need a different ramification argument. The verdict remains CONDITIONAL because the concern is specific and testable, not a demonstrated counterexample.","tokens_in":28359,"tokens_out":27360,"duration_ms":302936,"concrete_test":"Carry out an independent verification of Proposition 3.3: (i) compute whether H=ΓθN is normal in G_K for a local Galois group with a nontrivial tame inertia action; if not, (ii) prove directly that the natural map I_K/N → Aut_k(φbar) is well-defined and injective using Gardeyn's criterion [7] and the Taguchi–Tamagawa Tate conjecture [32], replacing the unproved 'Gal(L/K)' step; and (iii) for d=2, q=3, construct a Drinfeld-Stuhler module over F_3((T)) via the Morita reduction in Theorem 3.2 and compute the minimal totally tamely ramified extension over which it has good reduction, checking that its degree divides q^2−1=8. If the injection cannot be supplied or the degree bound fails, the congruence arguments in Theorems 6.10 and 6.13 collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 3.3 is load-bearing for Theorems 6.6, 6.10, 6.13 and Proposition 4.8, because it bounds the ramification of the canonical isogeny character by q^d−1. The proof of the degree bound has a gap: after setting N = ker(ρ|I_K) and H = ΓθN, the paper says the field L cut out by H is totally tamely ramified and then treats L/K as an abelian extension, using Gal(L/K) → Aut_k(φbar) to bound [L:K]. But H = ΓθN need not be normal in G_K: conjugating θ by an element of inertia can change it by an element of I_K not contained in N. Hence L/K need not be Galois, and 'Gal(L/K)' and the fixed-field argument are not justified. What is needed is an injection of the inertia quotient I_K/N into Aut_k(φbar); the text does not prove such a map is well-defined. Without this, the minimal good-reduction extension could be larger than q^d−1, and the surjectivity in Lemma 6.5 and the ramification bounds in Prop. 4.8 would not follow. This is the weak point on which the explicit Hasse-principle examples depend.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a theory of canonical isogeny characters for Drinfeld-Stuhler modules over F_q(T), analogous to Jordan's theory for abelian surfaces with quaternionic multiplication. It proves that a Drinfeld-Stuhler module over a local field acquires good reduction over a totally tamely ramified extension of degree dividing q^d-1 (Proposition 3.3), uses this to control the ramification of the canonical isogeny characters (Proposition 4.8), and then derives three global criteria (Theorems 6.6, 6.10, 6.13) for the absence of K-rational points on Drinfeld-Stuhler varieties. In the quaternionic dimension d=2 case, the criteria are combined with the local results of [23] to produce explicit pairs (X_D,K) for which X_D(K)=∅ while X_D(K_v)≠∅ for every completion K_v, thereby giving explicit violations of the Hasse principle (Examples 7.7, 7.8, 7.10).","tokens_in":41,"tokens_out":19078,"duration_ms":259996,"significance":"If the local input in Proposition 3.3 is made fully rigorous, the paper represents a substantial contribution: it extends the canonical isogeny character method to Drinfeld-Stuhler modules in arbitrary dimension d, not just quaternion algebras and curves, and it provides effective criteria for non-existence of rational points together with explicit counterexamples to the Hasse principle over function fields. The explicitness of the criteria is a real strength: the finite sets P'(y,s) and P(y,s) in Theorems 6.10 and 6.13 are computable, the arithmetic conditions in the examples are concrete, and the Magma verifications, while not shipped as code, are described in enough detail to be reproduced. The paper also engages carefully with the field-of-moduli-versus-field-of-definition issue for coarse moduli schemes, using the earlier results of [25] and [23]. The main weakness is that Proposition 3.3, which underpins Propositions 4.8 and the two class-group and congruence criteria, is proved by an argument that contains a genuine gap in its final degree bound.","major_comments":[{"comment":"The subgroup H = Γ_θ N of G_K is not shown to be normal, so the extension L/K cut out by H need not be Galois; nevertheless the proof then speaks of 'Gal(L/K)' and uses the fixed field of the subgroup generated by a single element g of that so-called Galois group. This invalidates the argument bounding [L:K] by q^d-1 as written. The gap is load-bearing because Proposition 3.3 is used to bound the ramification of the canonical isogeny characters in Proposition 4.8, to prove the surjectivity in Lemma 6.5, and to justify the reduction arguments in Theorems 6.10 and 6.13. A natural repair is to work directly with the finite quotient I_K/N and its faithful action on T_p(φ̅), applying the Taguchi–Tamagawa isogeny theorem to bound the degree without asserting that the minimal good-reduction extension is abelian; the authors need to supply such a complete argument.","section":"Section 3, proof of Proposition 3.3"}],"minor_comments":[{"comment":"The name 'Drinfeld-Shutler modules' in the first sentence of the introduction should be 'Drinfeld-Stuhler modules'.","section":"Section 1, first paragraph"},{"comment":"In condition (ii) of the definition, 'identity matirx' should be 'identity matrix'.","section":"Section 2, definition of Drinfeld-Stuhler module"},{"comment":"The assumptions write 'D ⊗ K ≅ M_d(K)' and 'D ⊗ K ≅ M_2(K)' without a base; these should read 'D ⊗_F K ≅ M_d(K)' and 'D ⊗_F K ≅ M_2(K)' respectively.","section":"Theorems 6.10 and 6.13"},{"comment":"W(y) is said to consist of elements π of F, but condition (2) [F(π):F]=d is impossible for π∈F; the definition should specify elements of a fixed algebraic closure of F.","section":"Definition 6.9"},{"comment":"The formula for n' in the definition of D'(y,s) appears garbled in the text; it should be displayed consistently with the quantity n in Proposition 6.8 when s=deg(p), namely n' = d(q^{sd}-1)/((q^s-1)(q^d-1)).","section":"Section 6.3, Definition 6.9"},{"comment":"The word 'claracter' appears twice in the text around the definition of ε and should be corrected to 'character'.","section":"Section 6.3"}],"recommendation":"major_revision","confidential_remarks":"The paper's central contribution is the canonical isogeny character theory and the effective Hasse-principle criteria; the reliance on [23] and [25] is appropriate and clearly acknowledged. The gap in Proposition 3.3 is serious but, in my assessment, repairable: one can replace the non-normal subgroup Γ_θ N by the finite quotient I_K/N and use the faithful action on the special-fiber Tate module to bound the degree without asserting that the minimal extension is abelian. I would not reject on this basis, but the authors must rewrite the proof of Proposition 3.3 and check that the subsequent arguments (Propositions 4.8, Lemma 6.5, Theorems 6.10 and 6.13) are then fully justified. The paper would also benefit from a proofreading pass to correct the numerous typographical errors."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper is worth taking seriously. It does two genuinely new things: it develops canonical isogeny characters for Drinfeld-Stuhler modules in arbitrary dimension d, and it uses them to produce the first explicit counterexamples to the Hasse principle for Drinfeld-Stuhler curves. The examples in Section 7 are concrete, with explicit polynomials p, q, K, and the criteria in Theorems 6.6, 6.10, and 6.13 are real number-theoretic conditions rather than tautologies. This is a solid step forward for the function-field analogue of the Shimura curve story.\n\nThe biggest soft spot is Proposition 3.3, which is load-bearing for the ramification bounds in Section 4 and hence for the main theorems. The stress-test note has a point: the proof claims that L/K is abelian and then reasons with Gal(L/K), but the subgroup H = Γθ N need not be normal in G_K. So as written, the degree bound [L:K] ≤ q^d − 1 is not justified. I cannot call this a fatal flaw, because there is an easy repair: N is normal in G_K, and the inertia quotient I_K/N injects into Aut_k(φbar) through reduction. That gives exactly the same bound. The authors should rewrite the last paragraph of the proof along those lines and stop talking about Gal(L/K) when they mean the inertia quotient. Until that is done, the proof of Proposition 3.3 is incomplete.\n\nThe other concerns are more minor. The paper leans heavily on prior results from [23] and [25], and on Magma computations that are not shipped. The finite sets P and P′ are explicit enough that the computations can in principle be reproduced, but the authors should say exactly what was computed and make the verification reproducible, especially because the counterexamples depend on those checks. The congruence arguments in Sections 5–6 look coherent to me, and I did not find hidden circularity.\n\nOverall: this is an important paper with credible, checkable results. It deserves a serious referee, but the referee should be asked to verify Proposition 3.3 carefully and to get the authors to clean up the proof and document the Magma computations. I would bring it to a reading group, and I would cite it if I worked in this area.","headline":"Real, checkable Hasse-principle counterexamples for Drinfeld-Stuhler curves and a genuinely new isogeny character theory; Proposition 3.3 is written too loosely and needs repair, but the underlying bound looks salvageable.","tokens_in":29160,"tokens_out":10839,"would_cite":true,"duration_ms":117194,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G09","11R52"],"pacs":[],"model":"deepseek-v4-flash","headline":"Drinfeld-Stuhler curves over $F_q(T)$ can fail the Hasse principle, and explicit examples exist.","keywords":["Drinfeld-Stuhler modules","D-elliptic sheaves","canonical isogeny characters","Hasse principle","Drinfeld modular curves","function field arithmetic","quaternion algebras","rational points"],"falsifier":"Construct a single Drinfeld-Stuhler module over a local field of characteristic $p$ whose good reduction is first achieved over a totally tamely ramified extension of degree not dividing $q^d-1$. Such an example would invalidate Proposition 3.3 and break the ramification control used in the main theorems.","tokens_in":28129,"feed_emoji":"🧮","tokens_out":9625,"duration_ms":91723,"temperature":0.7,"pith_summary":"The paper sets out to prove that Drinfeld-Stuhler modular varieties — function-field analogues of Shimura curves attached to quaternion algebras — can have no rational points over prescribed extensions of $F_q(T)$, and to make this failure explicit enough to produce counterexamples to the Hasse principle. The tool is a canonical isogeny character attached to the $p$-torsion of a Drinfeld-Stuhler module at a ramified prime $p$; this character controls both the Frobenius polynomial of the reduction and the ray class group of the field. The main theorems list explicit finite conditions under which $X_D(K)=\\emptyset$, and for $d=2$ they combine with local-point criteria to exhibit specific pairs $(X_D,K)$ where the curve has points everywhere locally but no global point.","feed_headline":"Explicit Drinfeld-Stuhler curves violate Hasse principle","feed_subtitle":"Canonical isogeny characters yield checkable criteria for empty rational points and explicit counterexamples.","key_machinery":"The carrying object is the canonical subgroup $C_{\\varphi,p}$ — the unique one-dimensional $F_p^{(d)}$-submodule of $\\varphi[p]$ stable under the order $O_D$ — together with its Galois character $\\rho_{\\varphi,p}$. The paper shows that the norm of $\\rho_{\\varphi,p}$ is the determinant character and that its $(q-1)$-power agrees with the Carlitz module character; local ramification estimates from Proposition 3.3 then bound where $\\rho_{\\varphi,p}^{q^d-1}$ can ramify. These identities convert the existence of a Drinfeld-Stuhler module over $K$ into a congruence on Frobenius polynomials (Proposition 6.8) and a surjectivity statement for ray class groups (Lemma 6.5).","core_discovery":"The paper's central claim is that non-existence of rational points on $X_D(K)$ can be forced by the canonical isogeny character. For a central division algebra $D$ of dimension $d^2$ over $F=F_q(T)$ with a prime $p$ of invariant $1/d$, any Drinfeld-Stuhler module $\\varphi$ over a degree-$d$ splitting field $K$ has a canonical subgroup $C_{\\varphi,p}\\cong F_p^{(d)}$ in $\\varphi[p]$; the Galois action defines the character $\\rho_{\\varphi,p}$. The paper proves that $\\rho_{\\varphi,p}$ satisfies a norm/determinant identity with the Carlitz module and that local ramification is controlled by a tameness bound of degree dividing $q^d-1$. It follows that if $y$ is a totally ramified prime outside $\\operatorname{Ram}(D)$ and $p\\notin P'(y,\\deg p)$ — or $p\\notin P(y,\\deg p)$ when $d=2$ — then $X_D(K)=\\emptyset$. Theorems 6.6, 6.10, and 6.13 are the formal statements; Section 7 turns them into explicit quaternion-algebra examples over $F_3$ and $F_5$ violating the Hasse principle.","pith_inferences":["One can view the finite set $P(y,\\deg p)$ as a precomputable obstruction set: for fixed $q,d,y$, the same computer search that produced the examples can be rerun for all primes of a given degree, yielding a census of Hasse-principle violations.","If the embedding condition discussed in Remark 7.12 holds for the $d=2$ examples, then the Brauer-Manin obstruction is the only obstruction, so these are explicit function-field cases where failure of the Hasse principle is explained by the Brauer-Manin set rather than by local solubility alone.","The congruence in Proposition 6.8 constrains Frobenius elements at the totally ramified prime, so the same canonical character should also obstruct weak approximation or constrain $X_D(K)$ when it is nonempty; the paper does not pursue this."],"forward_implications":["Theorems 6.6, 6.10, and 6.13 turn a rational-point question into a finite computation: checking membership in $P'(y,\\deg p)$ or $P(y,\\deg p)$ and checking a ray-class-group surjectivity condition.","For $d=2$, Examples 7.7, 7.8, and 7.10 provide explicit quaternion algebras and quadratic fields over $F_3$ and $F_5$ such that $X_D(K)=\\emptyset$ while $X_D(K_v)\\neq\\emptyset$ for every place $v$.","The same canonical-isogeny-character criteria work for arbitrary $d\\geq 2$; higher-dimensional Hasse-principle violations are blocked only by the absence of an analogous local-point theory for higher-dimensional Drinfeld-Stuhler varieties.","The obstruction is encoded in a congruence on the reduction's Frobenius polynomial, so checking the criteria does not require searching for rational points directly."],"supporting_citations":[{"why":"Supplies the template theory of canonical isogeny characters for abelian surfaces with quaternionic multiplication that this paper transplants to Drinfeld-Stuhler modules.","marker":"[13]"},{"why":"Constructs the Drinfeld-Stuhler modular varieties and contains the implicit potentially-good-reduction statement that Section 3 reproves.","marker":"[17]"},{"why":"Provides the Drinfeld-Stuhler module foundations: Tate modules, Morita equivalence, automorphism groups, and the field-of-moduli versus field-of-definition theorem.","marker":"[25]"},{"why":"Supplies the local-point criteria for Drinfeld-Stuhler curves used in Section 7 to verify Hasse-principle violations.","marker":"[23]"},{"why":"Supplies the Galois criterion for good reduction of tau-sheaves used to finish the proof of Theorem 3.2.","marker":"[7]"},{"why":"Supplies the statement that Drinfeld modules acquire good reduction over tamely ramified extensions, used for the $q^d-1$ degree bound in Proposition 3.3.","marker":"[33]"},{"why":"Supplies the Tate-isogeny-conjecture input used to bound automorphism groups in the proof of Proposition 3.3.","marker":"[32]"},{"why":"Supplies Anderson's determinant and duality constructions for t-motives used to identify the norm of the canonical character with the determinant character.","marker":"[1]"}],"fun_headline_variants":["Canonical isogeny characters yield Hasse principle counterexamples","New criteria for empty rational points on Drinfeld-Stuhler curves","Hasse principle failures via canonical isogeny characters","Explicit Hasse principle counterexamples from Drinfeld-Stuhler"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire argument rests on the claim that every Drinfeld-Stuhler module over a local field becomes well-behaved (good reduction) over a totally tamely ramified extension of degree dividing $q^d-1$; if that degree bound is false, the main emptiness criteria need not follow.","fun_headline_variants_meta":{"raw":{"variants":["Canonical isogeny characters yield Hasse principle counterexamples","New criteria for empty rational points on Drinfeld-Stuhler curves","Hasse principle failures via canonical isogeny characters","Explicit Hasse principle counterexamples from Drinfeld-Stuhler"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000703,"raw_usage":{"total_tokens":3142,"prompt_tokens":888,"completion_tokens":2254,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":504,"completion_tokens_details":{"reasoning_tokens":2180}},"tokens_in":504,"tokens_out":2254,"duration_ms":17139,"temperature":1.0,"reasoning_tokens":2180,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:32:23.738894+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a single Drinfeld-Stuhler module over a local field of characteristic $p$ whose good reduction is first achieved over a totally tamely ramified extension of degree not dividing $q^d-1$. Such an example would invalidate Proposition 3.3 and break the ramification control used in the main theorems.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the template theory of canonical isogeny characters for abelian surfaces with quaternionic multiplication that this paper transplants to Drinfeld-Stuhler modules."},{"cited_title":"D-elliptic sheaves and the Langlands correspondence","cited_arxiv_id":null,"evidence_quote":"Constructs the Drinfeld-Stuhler modular varieties and contains the implicit potentially-good-reduction statement that Section 3 reproves."},{"cited_title":"Drinfeld-Stuhler modules","cited_arxiv_id":null,"evidence_quote":"Provides the Drinfeld-Stuhler module foundations: Tate modules, Morita equivalence, automorphism groups, and the field-of-moduli versus field-of-definition theorem."},{"cited_title":"Local Diophantine properties of modular curves of D-elliptic sheaves","cited_arxiv_id":null,"evidence_quote":"Supplies the local-point criteria for Drinfeld-Stuhler curves used in Section 7 to verify Hasse-principle violations."},{"cited_title":"A Galois criterion for good reduction of τ-sheaves","cited_arxiv_id":null,"evidence_quote":"Supplies the Galois criterion for good reduction of tau-sheaves used to finish the proof of Theorem 3.2."},{"cited_title":"Good reduction of elliptic modules","cited_arxiv_id":null,"evidence_quote":"Supplies the statement that Drinfeld modules acquire good reduction over tamely ramified extensions, used for the $q^d-1$ degree bound in Proposition 3.3."},{"cited_title":"The Tate conjecture for t-motives","cited_arxiv_id":null,"evidence_quote":"Supplies the Tate-isogeny-conjecture input used to bound automorphism groups in the proof of Proposition 3.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Anderson's determinant and duality constructions for t-motives used to identify the norm of the canonical character with the determinant character."}],"review_version":1}