{"id":"16e8377a-e6fd-4b2c-a505-5f034f1cdfbe","arxiv_id":"2412.14313","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For prime power level p^r, all ℓ-primary rational torsion of the cuspidal generalized Drinfeld Jacobian vanishes unless ℓ divides q(q^2-1).","lead":"For Drinfeld modular curves of prime power level, the author proves that the generalized Jacobian with cuspidal modulus has no rational torsion of prime order outside a small explicit set of primes. The result mirrors the classical Yamazaki-Yang theorem and, under a standard conjecture, gives the full torsion group.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Determinant nonvanishing over Q is not enough to prove δ-injectivity on the finite group C(p^r); Proposition 5.9 only gives det ≡ ±1 mod |p|, not a unit modulo the group exponent.","rationale":"The reader's conditional verdict focuses on the transfer of Lemma 3.1 and the asserted exact sequence (4). That is a legitimate concern about external input and notation, but the more direct logical gap is in Section 5: the determinant computation is used to conclude injectivity of a homomorphism from a finite abelian group into (Q/Z)^r, yet the argument only proves that the determinant is a nonzero integer of the special form ±1+|p|f(|p|). Nonvanishing over Q does not imply that the reduction modulo the primes dividing the order of C(p^r) is invertible; a cyclic group of order N mapping to Q/Z via multiplication by an integer a is injective if and only if gcd(a,N)=1, not merely if a≠0. Since the orders in Theorem 2.8 contain factors M(p) and N(p) whose prime divisors need not divide q(q^2−1), the unconditional part of Theorem 1.3 requires control modulo exactly those primes, and the Conjecture C part requires control modulo all primes dividing the group exponent. The paper's Corollary 5.10 does not provide that control. I am not asserting the theorem is false; the Hessenberg structure may well imply a stronger congruence modulo (|p|^2−1), as the small cases r≤6 suggest. But the written proof is incomplete at a load-bearing step. The concrete test proposed would determine whether the gap is merely expository or reflects a real failure of the determinant criterion. The paper otherwise contains a coherent strategy, cites relevant prior work (Ho, Wei-Yamazaki, Gekeler), and gives a plausible reduction to the determinant computation, so I do not move to REJECT; the appropriate verdict remains CONDITIONAL pending either a strengthened determinant unit proof or an explicit citation showing the ℓ-primary part follows from Wei-Yamazaki [24].","tokens_in":25781,"tokens_out":37572,"duration_ms":326634,"concrete_test":"Use Theorem 5.8 to compute det(M_h^δ) for the matrix defined in Claim 5.5 in the case r=7, q=2, deg p=3 (so |p|=8, M(p)=21, N(p)=7), and reduce the determinant modulo 7. If det≡0 mod 7, then δ|C(p^7) has a nontrivial kernel on the 7-primary part, directly contradicting Theorem 1.3. If det is a unit mod 7, repeat the computation for all primes ℓ dividing M(p)N(p) for a range of pairs (q,r) with deg p>1; if a prime ℓ∤q(q^2−1) is ever found dividing the determinant, the central claim fails. If no such prime is found, the paper still needs to replace Corollary 5.10 by an explicit proof that the determinant is a unit modulo the exponent of C(p^r), since the current argument only establishes det≠0.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 5.3 conflates nonsingularity over Q with injectivity of a homomorphism from a finite group to (Q/Z)^r. After the factorisation in Equation (7), δ factors as φ∘δbar with φ injective, so injectivity of δ|C(n) is equivalent to injectivity of δbar: C(n)→Λ⊗Q/Z≅(Q/Z)^r. For a map from a finite abelian group to (Q/Z)^r, injectivity requires the representing integer matrix to be a unit modulo the exponent of C(n): for every prime ℓ dividing |C(n)|, the reduction of the matrix modulo ℓ must have full rank. Proposition 5.9 and Corollary 5.10 only prove det(M_h^δ)=±1+|p|f(|p|) and hence det≠0 as an integer. This shows the determinant is a unit modulo |p|, but it says nothing about primes ℓ dividing M(p)=(|p|^2-1)/(q^2-1) or N(p), which can occur without dividing q(q^2−1). For example, q=2, deg p=3 gives |p|=8, M(p)=21, so ℓ=7 divides |C(p^r)| but 7∤q(q^2−1)=6; the unconditional statement of Theorem 1.3 explicitly covers such primes. The congruence det≡±1 mod |p| gives no control modulo 7. Thus Theorem 5.3, and with it Theorem 3.7 and Theorem 1.3, does not follow from the determinant computation as written. A stronger congruence, for example det≡±1 mod (|p|^2−1), would suffice, but it is neither stated nor proved in the paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the rational torsion of the generalised Drinfeld modular Jacobian J_0(p^r)_m for a cuspidal modulus m, where p is a monic irreducible element of A=F_q[T]. The main result, Theorem 1.3, claims that for every odd prime ell not dividing q(q^2-1), the ell-primary part J_0(p^r)_m(K)[ell^infinity] is trivial, and that, assuming Conjecture C, the full rational torsion is isomorphic to r copies of Z/(q-1)Z. The proof reduces the problem, using Ho's description of C(p^r), Gekeler's residue-field computations, and a transfer argument from Yamazaki-Yang, to the injectivity of a boundary map delta restricted to C(p^r). The final section represents delta|C(p^r) by an r x r integer matrix M_delta and attempts to prove injectivity by showing that det(M_delta) is nonzero.","tokens_in":26129,"tokens_out":14820,"duration_ms":140489,"significance":"If the main theorem is correct, it gives a clean function-field analogue of Yamazaki-Yang's classical result: for prime-power Drinfeld level, rational torsion of the cuspidal generalised Jacobian is concentrated in primes dividing q(q^2-1), and the full torsion is predicted by Conjecture C. The paper makes a genuine structural contribution by expressing delta on Ho's explicit basis of C(p^r) and reducing the problem to a concrete matrix computation. The reliance on independent external results by Gekeler, Ho, and Yamazaki-Yang is transparent, and no free parameters are fitted to force the conclusion. However, the proof of the decisive injectivity statement is incomplete in the present version, and the matrix argument as written does not establish the claimed theorem.","major_comments":[{"comment":"The proof of injectivity of delta|C(n) is not complete. For a homomorphism from a finite abelian group C to (Q/Z)^r, injectivity is not implied by the nonvanishing of an integer matrix determinant over Q; one needs control modulo every prime dividing the exponent of C, for example injectivity of the induced map on ell-torsion for each such ell. Corollary 5.10 only proves det(M_h^delta) = +/-1 + |p| f(|p|) is nonzero, which gives information modulo |p| but none modulo the primes dividing M(p) or N(p). These primes can occur in the exponent without dividing q(q^2-1); for instance q=2 and deg(p)=3 give |p|=8 and M(p)=21, so ell=7 divides |C(p^r)| for r>=2, while 7 does not divide q(q^2-1)=6. The unconditional statement of Theorem 1.3 therefore requires delta-injectivity on such ell-primary parts, and the congruence det = +/-1 mod |p| gives no control there. Theorem 5.3 does not follow from the determinant computation as written; a stronger statement such as det coprime to |p|^r M(p)N(p), or an equivalent Smith-normal-form computation, is needed.","section":"Section 5, Proposition 5.9"},{"comment":"The Hessenberg induction is applied to a matrix that is not Hessenberg. In the proof of Proposition 5.9, the author sets M := M_h^delta and considers the upper-left n x n blocks M(n) of M. But the matrix M_h^delta displayed in Claim 5.5 has first row (0,1,...,1), so its upper-left blocks have nonzero entries above the superdiagonal and do not satisfy the hypothesis of Theorem 5.8. The Hessenberg structure is only obtained for the (r-1)x(r-1) submatrix after deleting the last row and the first column and then moving the first row to the bottom; this is not the matrix used in the induction on M(n). Consequently the formula det(M(n)) = 1 + |p| f_n(|p|) is not justified, and Proposition 5.9 lacks a valid proof in the present version.","section":"Section 5, Proposition 5.9"},{"comment":"The exact sequence (4) and the explicit formula for delta in Lemma 3.1 are the foundation for all matrix computations in Section 5, but they are asserted rather than derived in the Drinfeld setting. Moreover, Eq. (4) writes the target as the direct sum of K(P_i)_tors tensor Q/Z, whereas Lemma 3.1 and the surrounding text use K(P_i)^x tensor Q/Z. Since Proposition 3.2 and every entry of M_delta depend on this transfer, the proof needs either a derivation of (4) and Lemma 3.1 adapted to Drinfeld modular curves or a precise reference that covers this exact situation.","section":"Section 3, Eq. (4) and Lemma 3.1"}],"minor_comments":[{"comment":"The word 'stablish' should be 'establish'.","section":"Abstract"},{"comment":"Equation (4) omits the superscript in K(P_i)^x tensor Q/Z, and Remark 3.6's statement 'ker(delta|C(n)) = ker(delta|C(n))' does not distinguish delta from the factorised map delta-bar; this makes the reduction harder to follow.","section":"Section 3, Eq. (4) and Remark 3.6"},{"comment":"Notation such as 'floor(r-3/2)' is ambiguous and should be written as floor((r-3)/2) or an equivalent explicit expression.","section":"Section 5, Step 1 and Claim 5.4"},{"comment":"In the proof of Claim 5.4 the symbol p_k appears where |p|^k is evidently intended; this should be corrected throughout the displayed computation.","section":"Section 5, Claim 5.4 proof"}],"recommendation":"major_revision","confidential_remarks":"The determinant-versus-injectivity gap is the main obstruction: the current proof only establishes nonvanishing over Q, while the finite-group statement needs arithmetic control modulo the primes in the exponent. This is potentially fixable by strengthening the determinant congruence or by computing the relevant Smith normal form. The Hessenberg induction in Proposition 5.9 also needs a genuine repair. Given the paper's clarity and the value of the reduction, major revision seems appropriate rather than rejection, provided the injectivity step can be completed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my honest read of Curcó-Iranzo's paper on rational torsion of generalized Drinfeld modular Jacobians. The topic is a natural function-field analogue of Yamazaki–Yang, and the main theorem is what you'd want: for n=p^r, all ℓ-primary torsion of the cuspidally generalized Jacobian vanishes outside q(q^2−1), and under Conjecture C the full torsion is r copies of Z/(q−1). What's genuinely new is the explicit computation of the boundary map δ on Ho's basis of the rational cuspidal group C(p^r), and the conditional full torsion statement. The paper makes sensible use of external results—Ho's comparison theorem and decomposition, Gekeler's residue fields, Yamazaki–Yang's boundary formula—and does not look circular.\n\nThe problem is the injectivity argument for δ|C(n). The paper reduces injectivity to nonsingularity of an integer matrix Mδ and then proves det(Mδ)=±1+|p|f(|p|). That only makes the determinant a unit modulo |p|. To show a map from the finite group C(p^r) to (Q/Z)^r is injective, you need the determinant to be coprime to the exponent of C(p^r), meaning coprime to all primes dividing M(p) and N(p). Those primes need not divide q(q^2−1): q=2, deg p=3 gives |p|=8, M(p)=21, so ℓ=7 is exactly a prime the theorem covers and that the determinant congruence says nothing about. So Corollary 5.10 does not imply Theorem 5.3, and the main theorem is not proved as written. The Hessenberg induction in Proposition 5.9 is also mis-stated: the (r−1)×(r−1) submatrix is Hessenberg only after moving the first row, but the induction is applied to upper-left blocks of the unpermuted matrix, which are not the Hessenberg object. That needs fixing even if the determinant congruence were the right one.\n\nSmaller things: the exact sequence (4) is asserted, not derived, and has inconsistent target notation (K(P_i)_tors vs K(P_i)^×). The relation to Wei–Yamazaki's 'up to q(q^2−1)-torsion' result is never made precise; if their prior theorem already gives the ℓ-primary vanishing, the unconditional part of Theorem 1.3 is not new. The real novelty would then be the conditional structure and the explicit δ-matrix computation.\n\nBottom line: a useful and careful paper with substantial computational content, but the main proof has a load-bearing gap. I'd send it to a serious referee, because the gap may be repairable and the result is worth having. I would not cite the theorem as established in this version. Bring it to reading group if you want a concrete lesson in finite-group injectivity versus determinant nonvanishing over Q.","headline":"The intended theorem may be true, but the injectivity proof only controls determinants modulo |p|, not modulo the order of C(p^r), so the main result is not established.","tokens_in":26678,"tokens_out":8931,"would_cite":false,"duration_ms":73511,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G18","14H40","11G45","11G16","14G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For prime-power Drinfeld levels, rational torsion of the generalised Jacobian vanishes away from the primes dividing $q(q^2-1)$.","keywords":["generalised Jacobians","Drinfeld modular curves","rational points","eta-quotients","cuspidal divisor class group","rational torsion","function fields"],"falsifier":"Independently compute the determinant of the Section 5 matrix $M_\\delta$ for one fixed monic irreducible $\\mathfrak{p}$ and one $r\\ge7$ using the appendix exponents; the theorem requires $\\det(M_\\delta^h)=\\pm1+|\\mathfrak{p}|f(|\\mathfrak{p}|)$ with $|\\mathfrak{p}|\\ge3$, so a determinant divisible by $|\\mathfrak{p}|$ would disprove injectivity. At the arithmetic level, exhibiting an odd prime $\\ell\\nmid q(q^2-1)$ and a nonzero class in $J_0(\\mathfrak{p}^r)_{\\mathbf{m}}(K)[\\ell^\\infty]$ would also falsify the theorem.","tokens_in":25565,"feed_emoji":"🧮","tokens_out":11225,"duration_ms":89169,"temperature":0.7,"pith_summary":"This paper studies the rational torsion of the generalised Jacobian $J_0(\\mathfrak{p}^r)_{\\mathbf{m}}$ attached to the Drinfeld modular curve $X_0(\\mathfrak{p}^r)$, where the modulus $\\mathbf{m}$ is the sum of all cusps. The central claim is that for every odd prime $\\ell$ not dividing $q(q^2-1)$, the $\\ell$-primary rational torsion $J_0(\\mathfrak{p}^r)_{\\mathbf{m}}(K)[\\ell^\\infty]$ is trivial. The proof reduces the question to a boundary map $\\delta$ from the rational cuspidal divisor group into a sum of residue-field tori and then shows $\\delta$ is injective by an explicit matrix computation. If the paper's Conjecture C holds, the full rational torsion is exactly $r$ copies of $\\mathbb{Z}/(q-1)\\mathbb{Z}$. This gives the function-field analogue of the classical prime-power-level theorem for modular curves.","feed_headline":"Rational torsion of Drinfeld Jacobians vanishes away from q(q^2-1)","feed_subtitle":"At prime-power level, odd ℓ-torsion of the cuspidal generalized Jacobian is trivial for all ℓ outside q(q^2−1).","key_machinery":"The load-bearing object is the boundary map $\\delta$ obtained from the exact sequence $0\\to L_{\\mathbf{m}}\\to J_0(\\mathfrak{n})_{\\mathbf{m}}\\to J_0(\\mathfrak{n})\\to 0$; it sends a torsion class to $\\bigoplus_{i=0}^{r-1} K(P_i)^\\times\\otimes \\mathbb{Q}/\\mathbb{Z}$ by evaluating a function whose divisor represents the class at the cuspidal points $P_i$. After removing torsion units, $\\delta$ takes values in a lattice generated by $\\mathfrak{p}$ in each residue field, so it is represented by an $r\\times r$ integer matrix $M_\\delta$ built from exponents of the Drinfeld discriminant quotients $\\Delta_m$ at the cusps. The determinant is shown to be nonzero: row and column reductions put $M_\\delta$ into a Hessenberg form, and the Hessenberg determinant recurrence gives $\\det(M_\\delta^h)=\\pm1+|\\mathfrak{p}|f(|\\mathfrak{p}|)$ for some $f\\in\\mathbb{Z}[x]$, which cannot vanish because $|\\mathfrak{p}|\\ge q\\ge 3$.","core_discovery":"The paper establishes that for a monic irreducible $\\mathfrak{p}\\in\\mathbb{F}_q[T]$ and any positive integer $r$, every odd prime $\\ell$ with $\\ell\\nmid q(q^2-1)$ contributes no rational torsion to the generalised Jacobian of $X_0(\\mathfrak{p}^r)$ with cuspidal modulus: $J_0(\\mathfrak{p}^r)_{\\mathbf{m}}(K)[\\ell^\\infty]=0$. The mechanism is to prove that the natural boundary map $\\delta$ from torsion to $\\bigoplus_{i=0}^{r-1} K(P_i)^\\times\\otimes \\mathbb{Q}/\\mathbb{Z}$ is injective on the rational cuspidal subgroup $C(\\mathfrak{p}^r)$, after quotienting out roots of unity. The proof uses the basis of $C(\\mathfrak{p}^r)$ supplied by [10] and represents $\\delta$ by an explicit $r\\times r$ matrix $M_\\delta$ whose entries are exponents of $\\mathfrak{p}$ appearing in $\\Delta$-quotients at the cusps. Since a known result [11] identifies $C(\\mathfrak{p}^r)[\\ell^\\infty]$ with $J_0(\\mathfrak{p}^r)(K)_{\\mathrm{tors}}[\\ell^\\infty]$ for primes $\\ell\\nmid q(q-1)$, this injectivity kills the $\\ell$-primary part for the allowed $\\ell$. Conditional on Conjecture C, the residue-field terms each contribute $\\mathbb{Z}/(q-1)\\mathbb{Z}$, yielding $\\prod_{i=0}^{r-1}\\mathbb{Z}/(q-1)\\mathbb{Z}$ as the full torsion group.","pith_inferences":["Not in the paper: once Conjecture C is settled for primes dividing $q(q^2-1)$, the same degeneration sequence would pin down the full torsion group, so any counterexample to the conjecture would have to live in those exceptional primes.","Not in the paper: the block structure of the exponent table suggests the method could extend from prime-power level to composite levels if an explicit basis of the rational cuspidal divisor group analogous to the one used here is constructed.","Not in the paper: because the determinant has the form $\\pm1+|\\mathfrak{p}|f(|\\mathfrak{p}|)$, the injectivity claim is a statement about the integers $q$ and $|\\mathfrak{p}|$; it could be tested computationally for many small primes without computing the Jacobian itself."],"forward_implications":["For level $\\mathfrak{p}^r$, rational $\\ell$-torsion of the cuspidal generalised Jacobian is confined to primes dividing $q(q^2-1)$; in particular, the semi-abelian part contributes nothing for the allowed primes.","If Conjecture C holds, the whole rational torsion group is finite of order $(q-1)^r$ and is isomorphic to $\\prod_{i=0}^{r-1}\\mathbb{Z}/(q-1)\\mathbb{Z}$.","The determinant criterion gives an effective check: injectivity of $\\delta$ on $C(\\mathfrak{p}^r)$ is equivalent to $\\det M_\\delta\\neq 0$, so the same matrix computation can be reused for other moduli or other base fields.","Together with the equality stated in [11], the theorem completes the function-field analogue of the classical prime-power-level torsion theorem, with $q(q^2-1)$ playing the role of the exceptional prime set in the classical case."],"supporting_citations":[{"why":"Supplies Lemma 3.1, the explicit boundary-map formula for $\\delta$ that the whole matrix computation starts from.","marker":"[25]"},{"why":"Provides the basis of $C(\\mathfrak{p}^r)$ and the $\\Delta$-quotients whose images under $\\delta$ become the rows of the matrix.","marker":"[10]"},{"why":"Identifies $C(\\mathfrak{p}^r)[\\ell^\\infty]$ with $J_0(\\mathfrak{p}^r)(K)_{\\mathrm{tors}}[\\ell^\\infty]$ for primes not dividing $q(q-1)$, reducing the target of $\\delta$ to cuspidal divisors.","marker":"[11]"},{"why":"Gives the earlier computation of $J_0(\\mathfrak{n})_{\\mathbf{m}}(K)_{\\mathrm{tors}}$ up to $q(q^2-1)$-torsion that the present theorem sharpens.","marker":"[24]"},{"why":"Supplies the Drinfeld discriminant function, its product expansion, and the cusp data used to compute the exponents in Proposition 3.2.","marker":"[5]"},{"why":"Provides the Hessenberg determinant recurrence used to prove $\\det(M_\\delta^h)=\\pm1+|\\mathfrak{p}|f(|\\mathfrak{p}|)$.","marker":"[17]"}],"fun_headline_variants":["Drinfeld Jacobians: all odd torsion dies outside q(q^2-1)","Prime-power level: ℓ-torsion trivial for ℓ∤q(q^2-1)","Cuspidal Drinfeld Jacobians: ℓ-torsion zero for ℓ∤q(q^2-1)","Function field analogue: odd torsion vanishes for ℓ∤q(q^2-1)","Rational torsion trivial for Drinfeld Jacobians outside q(q^2-1)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof inherits, without deriving it in the Drinfeld setting, the explicit formula for the boundary map $\\delta$ (Lemma 3.1, transferred from [25]) and the exact sequence it comes from; if that formula or the exact sequence is wrong, every matrix entry in Section 5 changes and the injectivity conclusion no longer follows.","fun_headline_variants_meta":{"raw":{"variants":["Drinfeld Jacobians: all odd torsion dies outside q(q^2-1)","Prime-power level: ℓ-torsion trivial for ℓ∤q(q^2-1)","Cuspidal Drinfeld Jacobians: ℓ-torsion zero for ℓ∤q(q^2-1)","Function field analogue: odd torsion vanishes for ℓ∤q(q^2-1)","Rational torsion trivial for Drinfeld Jacobians outside q(q^2-1)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002136,"raw_usage":{"total_tokens":8343,"prompt_tokens":1050,"completion_tokens":7293,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":666,"completion_tokens_details":{"reasoning_tokens":7168}},"tokens_in":666,"tokens_out":7293,"duration_ms":45825,"temperature":1.0,"reasoning_tokens":7168,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:21:57.593704+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Independently compute the determinant of the Section 5 matrix $M_\\delta$ for one fixed monic irreducible $\\mathfrak{p}$ and one $r\\ge7$ using the appendix exponents; the theorem requires $\\det(M_\\delta^h)=\\pm1+|\\mathfrak{p}|f(|\\mathfrak{p}|)$ with $|\\mathfrak{p}|\\ge3$, so a determinant divisible by $|\\mathfrak{p}|$ would disprove injectivity. At the arithmetic level, exhibiting an odd prime $\\ell\\nmid q(q^2-1)$ and a nonzero class in $J_0(\\mathfrak{p}^r)_{\\mathbf{m}}(K)[\\ell^\\infty]$ would also falsify the theorem.","supporting_citations":[{"cited_title":"Yamazaki and Y","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 3.1, the explicit boundary-map formula for $\\delta$ that the whole matrix computation starts from."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the basis of $C(\\mathfrak{p}^r)$ and the $\\Delta$-quotients whose images under $\\delta$ become the rows of the matrix."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies $C(\\mathfrak{p}^r)[\\ell^\\infty]$ with $J_0(\\mathfrak{p}^r)(K)_{\\mathrm{tors}}[\\ell^\\infty]$ for primes not dividing $q(q-1)$, reducing the target of $\\delta$ to cuspidal divisors."},{"cited_title":"Wei and T","cited_arxiv_id":null,"evidence_quote":"Gives the earlier computation of $J_0(\\mathfrak{n})_{\\mathbf{m}}(K)_{\\mathrm{tors}}$ up to $q(q^2-1)$-torsion that the present theorem sharpens."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Drinfeld discriminant function, its product expansion, and the cusp data used to compute the exponents in Proposition 3.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Hessenberg determinant recurrence used to prove $\\det(M_\\delta^h)=\\pm1+|\\mathfrak{p}|f(|\\mathfrak{p}|)$."}],"review_version":1}