{"id":"03c05a70-1326-407a-b733-15d2750f9b62","arxiv_id":"1908.04684","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Ordinary curves of even genus over odd characteristic have automorphism groups of order less than 821.37 g^{7/4}, with Hurwitz's bound restored for Alt7 and M11 except for one exceptional M11 case.","lead":"This paper proves a new upper bound on the size of automorphism groups of ordinary algebraic curves of even genus over fields of odd characteristic. It shows that such groups have order below 821.37 g^{7/4}, and that the classical Hurwitz bound holds for the exceptional groups Alt7 and M11, except for one specific case realized by the modular curve X(11) in characteristic 3.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main bound is only as sound as the quoted classification Lemma 3.4 from [4]; an incomplete or misstated lemma would leave Theorem 3.10 without a case that violates the bound, and the present paper supplies no independent verification.","rationale":"The reader's conditional verdict focuses on exactly this point, and I agree. The paper's internal arithmetic is mostly consistent, and the case-by-case bounds once Lemma 3.4 is granted appear sufficient; the small local gaps, such as the unstated q ≥ 53 justification in Proposition 3.11, are patchable and do not threaten the numerical bound. The genuinely load-bearing step is the transfer from the external classification [4] to the case split in Theorem 3.10. Because the authors do not reproduce or prove Lemma 3.4 and [4] is an unpublished preprint, the central claim is conditional. This does not move the verdict: it remains CONDITIONAL as the reader said. If the classification check succeeds, the same evidence supports ACCEPT; if it fails and an ordinary even-genus curve with an unlisted automorphism group exists, the theorem would need revision. The proposed test targets exactly that completeness.","tokens_in":18149,"tokens_out":35827,"duration_ms":376433,"concrete_test":"Independently derive Lemma 3.4 from [4] and from the classification of finite groups with Sylow 2-subgroups of 2-rank at most 2 (Gorenstein-Walter for dihedral and Alperin-Brauer-Gorenstein for semidihedral). Verify explicitly that an odd-core-free automorphism group of a curve of even genus in odd characteristic has Sylow 2-rank at most 2, and that no other simple group (e.g. PSL(2,2^f), PSL(3,2^f), PSU(3,2^f), or ^2G2(3^{2m+1})) survives the even-genus ordinary-curve hypotheses. If all such groups are exactly the five listed, recompute Theorem 3.10; if any unlisted group remains, test whether it can act on an ordinary even-genus curve and whether its order exceeds 821.37 g^{7/4}.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.10 is proved by removing the solvable/elementary-abelian cases with Theorem 3.1 and then splitting all remaining odd core-free non-solvable automorphism groups according to Lemma 3.4. The lemma is quoted, not proved: it asserts that the only possible non-abelian minimal normal subgroups are PSL(2,q) (q odd), PSL(3,q) (q ≡ 3 mod 4), PSU(3,q) (q ≡ 1 mod 4), Alt7, or M11, with G inside the corresponding PGammaL-group. Every one of Propositions 3.11, 3.12, 3.14, and 3.18 begins with 'according to Remark 3.5', so any additional simple group whose Sylow 2-subgroup has 2-rank 2 (or any failure of the step from even genus to the Sylow 2-rank condition) would escape the case analysis and could in principle violate the claimed exponent 7/4. Since [4] is itself an arXiv preprint, the completeness of Lemma 3.4 is an external, unresolved premise. The paper's Remark 3.6 explicitly labels the list 'complete', but that completeness is exactly what needs proof. This is the most load-bearing assumption; if it is wrong the main theorem has no support.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies automorphism groups G of ordinary projective curves X of even genus g ≥ 2 over an algebraically closed field K of odd characteristic p. Its main result, Theorem 3.10, asserts that every such G satisfies |G| < 821.37·g^{7/4}, and that for the two sporadic possibilities G ≅ Alt_7 and G ≅ M_11 the classical Hurwitz bound |G| < 84(g−1) holds unless p = 3, g = 26, and G ≅ M_11. The proof splits the problem into solvable and non-solvable cases: solvable groups are handled by earlier work of Korchmáros–Montanucci (Theorem 3.1), while non-solvable odd core-free groups are classified by Lemma 3.4, quoted from the preprint [4], into cases with minimal normal subgroup PSL(2,q), PSL(3,q), PSU(3,q), Alt_7, or M_11. The PSL(2,q), PSL(3,q), and PSU(3,q) cases are treated in Propositions 3.11 and 3.12 using subgroup bounds and Lemma 3.3. The Alt_7 and M_11 cases are treated by a ramification analysis in Propositions 3.14 and 3.18, with a non-existence proof for the possible genus-10 Alt_7 curve (Propositions 3.15 and 3.16). Section 4 gives the modular curve X(11) in characteristic 3 as an ordinary genus-26 curve with M_11 automorphism group.","tokens_in":18458,"tokens_out":8654,"duration_ms":88932,"significance":"If the quoted classification is accepted, this is a strong and quantitatively explicit result: it improves Nakajima's general bound from O(g^2) to O(g^{7/4}) for ordinary curves of even genus in odd characteristic, and it gives a complete exceptional analysis for the two sporadic groups, including a concrete ordinary curve attaining the exceptional case. The case analysis in Propositions 3.9, 3.11, 3.12, 3.14, and 3.18 is detailed and largely self-contained beyond the external classification; the use of Deuring–Shafarevich and Hurwitz formulas is careful and appropriate. The paper also makes good use of structural restrictions imposed by ordinary even genus, such as Theorem 2.2, to narrow the possible ramification filtrations. The main weaknesses are the unproved completeness of Lemma 3.4, which is load-bearing, and several essential computer checks that are neither documented by code nor replaced by human-checkable proofs.","major_comments":[{"comment":"The proof of Theorem 3.10 is logically dependent on the completeness of the classification stated in Lemma 3.4 and summarized in Remark 3.6, which is quoted from the preprint [4]. This lemma asserts that every odd core-free non-solvable automorphism group of an even-genus curve in odd characteristic has a simple minimal normal subgroup among PSL(2,q), PSL(3,q), PSU(3,q), Alt_7, and M_11, and that G is contained in the corresponding PΓL-group. Every subsequent proposition begins with an 'according to Remark 3.5' assumption, so if the classification is incomplete or misstated, Theorem 3.10 has no support. Since [4] is an arXiv preprint, the completeness part is not verified by a peer-reviewed source. Please either cite a published version of [4], or include a proof of Lemma 3.4, or make explicit in Theorem 3.10 that the result is conditional on this classification; the current presentation gives no independent verification of the most load-bearing input.","section":"§3, Lemma 3.4 and Remark 3.6"},{"comment":"Several essential steps rely on MAGMA computations for which no code, certificates, or reproducible scripts are provided. Specifically: Proposition 3.14 uses 'it can be verified by MAGMA that the largest solvable subgroup of Alt_7 whose Sylow p-subgroup (p ∈ {3,5,7}) has a cyclic complement has size 36'; Proposition 3.15 uses the claim that for a ≠ −1 the curve Z_a is isomorphic to y^2 = a_5 x^5 + a_3 x^3 + a_0 with 5-rank 2; and Section 4 uses 'by direct MAGMA computation' the equality γ(X(11)) = g(X(11)) = 26. These facts are load-bearing: the first excludes the Alt_7 Hurwitz-bound violations, the second is needed to rule out the genus-10 Alt_7 case, and the third establishes that the constructed exceptional example is ordinary. The authors should supply the MAGMA scripts and outputs (or a precise algorithmic description sufficient for independent verification), or replace these computations by mathematical proofs.","section":"§3, Propositions 3.14 and 3.15; §4"},{"comment":"In the PSL(3,q) case, the argument excludes the possibility |H| ≥ 30(g−1) by using Lemma 3.3 to conclude d = p and then Theorem 3.2 to obtain that the Sylow p-subgroup of H is elementary abelian, contradicting the structure of a Sylow p-subgroup of PSL(3,q) for the relevant q. This step is sound, but the presentation compresses the reason why the normalizer subgroup H is a solvable subgroup of the automorphism group of an ordinary curve to which Theorem 3.2 applies. Since Theorem 3.2 is stated for X ordinary and H solvable, and H is indeed a subgroup of Aut(X), the step is valid; however, the reader is left to fill in the verification that the normalizer N_{PSL(3,q)}(Q) described by [11, Theorem 2.4] is the subgroup H used in Lemma 3.3. The authors should spell out this identification, preferably with a sentence or a short lemma, so that the application of Lemma 3.3 is fully transparent.","section":"§3, Proposition 3.12, PSL(3,q) case"}],"minor_comments":[{"comment":"In Case 2 of Proposition 3.11, the text refers to 'Remark 3.9'; the intended reference is Proposition 3.9, where the bound |U| < sqrt(4g+4) is proved.","section":"§3, Proposition 3.11, Case 2"},{"comment":"The cross-references 'Equation (3.14)' and 'Equation (3.18)' should be to the displayed Hurwitz formulas numbered (20) and (23), respectively; the current numbering is confusing.","section":"§3, Propositions 3.14 and 3.18"},{"comment":"In Case 4 of Proposition 3.11, the expression 'PGL(2, 53) = 5859000' appears to be a typesetting or OCR error for 'PGL(2, 5^3)' (i.e., q = 125), since |PGL(2,125)| = 1,953,000 and three times that is 5,859,000; the printed 'q ≥ 53' likewise should read 'q ≥ 5^3'. Please correct this.","section":"§3, Proposition 3.11, Case 4"},{"comment":"There are several minor typos and stylistic inconsistencies that should be fixed in a revision: 'wich' for 'which' in Propositions 3.11 and 3.12; 'pagg 600-601' in the reference to [19]; and the notation 'G(2)_P' in the proof of Proposition 3.9 is used without definition (it presumably denotes the second ramification group of the stabilizer G_P).","section":"Throughout"},{"comment":"Reference [4] is cited only as an arXiv preprint. If a peer-reviewed or final version of this classification paper has appeared, the authors should update the citation; if not, this strengthens the need for the major comment above.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's central claim is conditional on the classification in [4], which the authors quote but do not prove. I would recommend that the editors ask the authors to either verify that [4] has appeared in a refereed venue, or include the needed part of the classification in this paper. The MAGMA dependence is also a serious reproducibility issue for a journal that values machine-checked computation; the authors should be asked to provide code. The underlying mathematics, conditional on those inputs, appears coherent and the constants are plausible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper delivers a real result: a quantitative bound |G| < 821.37 g^{7/4} for automorphism groups of ordinary curves of even genus in odd characteristic, improving Nakajima's quadratic bound, plus a clean treatment of the exceptional Alt7 and M11 cases. The Hurwitz bound holds for both except for p=3, g=26, M11, with X(11) as an example. That is a solid, non-routine contribution.\n\nThe proof is a careful case analysis, and Propositions 3.11, 3.12, 3.14, and 3.18 are worked out in real detail. The authors also rule out a potential genus-10 Alt7 example (Theorem 3.16) and verify ordinariness of X(11) by direct computation. This is honest, competent work.\n\nThe soft spot is exactly where the stress-test note points: the main theorem leans on Lemma 3.4, quoted from the preprint [4], which classifies odd core-free non-solvable automorphism groups of curves of even genus in odd characteristic. The paper calls that list 'complete' in Remark 3.6 but gives no independent verification. If [4] is incomplete or misstated, the case analysis could miss a group that violates the claimed exponent. That is a genuine load-bearing assumption, and the paper cannot fix it by itself. I also note that several essential finite checks are asserted as MAGMA computations without code or certificates (e.g., the largest solvable subgroup in Alt7, the 5-rank of some genus-2 curves), which makes full reproducibility difficult. Minor typos ('wich', a stray 'a a non-trivial subgroup') do not affect the mathematics.\n\nNone of this undermines the internal logic: the argument is coherent, and the reliance on [4] is explicit. I would send this to a serious referee, with the request that the authors either prove or precisely cite the classification lemma, and provide the MAGMA code or certificates. Specialists in algebraic curves and finite group actions will get genuine value from it.","headline":"New bound with a heavy but explicit caveat: Theorem 3.10 lives or dies with the quoted classification from [4].","tokens_in":18950,"tokens_out":2468,"would_cite":true,"duration_ms":25710,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14H37","14H05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every even-genus ordinary curve has fewer than $821.37g^{7/4}$ automorphisms","keywords":["ordinary curves","p-rank","automorphism groups","even genus","Hurwitz bound","positive characteristic","Mathieu group M11","alternating group Alt7"],"falsifier":"Run the ramification-data check of Theorem 3.13 for every group in Lemma 3.4, or equivalently search computationally for an ordinary curve of even genus over an algebraically closed field of odd characteristic whose automorphism group has order at least $821.37g^{7/4}$. A concrete target is the plane model $y^{10}(y+1)^9 = x^{22} - y(y+1)^4x^{11}(y^3+2y+1)$ in characteristic $3$: if it fails to be ordinary of genus $26$ with automorphism group $M_{11}$, the exceptional case would be empty; if a curve with $|G|\\ge821.37g^{7/4}$ exists, the main theorem would be false.","tokens_in":17896,"feed_emoji":"📐","tokens_out":12931,"duration_ms":118278,"temperature":0.7,"pith_summary":"The paper proves a uniform bound on the size of automorphism groups of ordinary curves of even genus in odd characteristic. If $\\mathcal{X}$ is such a curve of genus $g \\ge 2$ over an algebraically closed field of odd characteristic, every automorphism group $G$ satisfies $|G| < 821.37\\,g^{7/4}$. Here ordinary means that the $p$-rank $\\gamma(\\mathcal{X})$ equals the genus $g$, a condition under which the classical Hurwitz bound $|G|\\le 84(g-1)$ can fail in positive characteristic. The result replaces Nakajima's general bound $|G|\\le 84g(g-1)$ with a polynomial bound for this family and sharpens it in the two sporadic non-solvable cases: if $G\\cong \\operatorname{Alt}_7$ or $G\\cong M_{11}$, the Hurwitz bound holds unless $p=3$, $g=26$, and $G\\cong M_{11}$, an exception realized by the modular curve $X(11)$ in characteristic $3$.","feed_headline":"Every even-genus ordinary curve has < 821.37g^{7/4} automorphisms","feed_subtitle":"Hurwitz's 84(g-1) bound returns for Alt7 and M11, except one characteristic-3 case built from X(11).","key_machinery":"The load-bearing object is the classification quoted in Lemma 3.4: an odd core-free non-solvable automorphism group of a curve of even genus in odd characteristic has a simple minimal normal subgroup $N$ of one of five types, $\\operatorname{PSL}(2,q)$, $\\operatorname{PSL}(3,q)$, $\\operatorname{PSU}(3,q)$, $\\operatorname{Alt}_7$, or $M_{11}$, and $G$ is contained in the corresponding projective semilinear automorphism group. Around this classification the proof uses two quantitative mechanisms. First, ordinary curves inherit Nakajima's bound $|\\operatorname{Aut}(\\mathcal{X})|\\le 84g(g-1)$ and the vanishing of second ramification groups $G_P^{(2)}=1$ at every point $P$. Second, a structural lemma for large solvable groups, Lemma 3.8 and Proposition 3.9, shows that such a group is a semidirect product $Q\\rtimes U$ with rational quotient curve, exactly two non-tame short orbits, and a cyclic complement of order $|U|<\\sqrt{4g+4}$. The Hurwitz genus formula and the Deuring--Shafarevich formula convert these structural facts into the explicit inequalities that bound $q$, and hence $|G|$, in terms of $g$.","core_discovery":"The central claim, Theorem 3.10, is that for an ordinary curve $\\mathcal{X}$ of even genus $g\\ge 2$ over an algebraically closed field of odd characteristic, every automorphism group $G$ has order below $821.37g^{7/4}$. The authors split the problem by group structure. A solvable group, or one with an elementary abelian minimal normal subgroup, already satisfies the stronger bound $|G|\\le 34(g+1)^{3/2}$ by Theorem 3.1. The remaining odd core-free non-solvable groups are forced by the quoted classification to contain a simple minimal normal subgroup $N$ isomorphic to $\\operatorname{PSL}(2,q)$, $\\operatorname{PSL}(3,q)$, $\\operatorname{PSU}(3,q)$, $\\operatorname{Alt}_7$, or $M_{11}$, with $G$ lying inside the corresponding automorphism group. For the projective families the proof bounds $q$ in terms of $g$ using the Hurwitz and Deuring--Shafarevich formulas together with a structural lemma for large solvable subgroups, and obtains $|G|<821.37g^{7/4}$. The sporadic cases are eliminated by enumerating all ramification data that could violate $|G|<84(g-1)$; the only surviving data are $p=3$, $g=26$, $G\\cong M_{11}$, and the paper exhibits $X(11)$ in characteristic $3$ as an ordinary curve of genus $26$ with this automorphism group.","pith_inferences":["Beyond the paper, the same two-mechanism proof could be run with sharper constants: the $821.37$ arises from worst-case estimates, and an exhaustive search over the allowed ramification data for small $g$ would show how far the true maximum lies below the bound.","Beyond the paper, the dichotomy between solvable and non-solvable cases is tied to Sylow $2$-subgroups of even-genus curves; for odd genus the same classification input would not be available, so a similar bound would need a new structural statement before the conjectured $O(g^{8/5})$ growth could be tested.","Beyond the paper, the authors leave open whether $X(11)$ is the only ordinary genus-$26$ curve in characteristic $3$ with automorphism group $M_{11}$; a computation over plane models of that genus and characteristic could settle the uniqueness question."],"forward_implications":["For every ordinary curve of even genus $g\\ge2$ over an algebraically closed field of odd characteristic, every automorphism group has order below $821.37\\,g^{7/4}$.","The Hurwitz bound $|G|<84(g-1)$ is restored for $G\\cong \\operatorname{Alt}_7$ and $G\\cong M_{11}$, with exactly one exception: $p=3$, $g=26$, $G\\cong M_{11}$.","There is no ordinary curve of genus $10$ in characteristic $5$ with automorphism group $\\operatorname{Alt}_7$, eliminating the only candidate that the ramification data allow.","In the exceptional $M_{11}$ case, the quotient of the curve by its Sylow $3$-subgroup is the ordinary genus-$2$ hyperelliptic curve $y^2=x^5-x$.","The exceptional case is nonempty: the modular curve $X(11)$ in characteristic $3$ is ordinary of genus $26$ and has automorphism group $M_{11}$."],"supporting_citations":[{"why":"Supplies Lemma 3.4, the classification of odd core-free non-solvable automorphism groups of even-genus curves in odd characteristic; every non-solvable case in Theorem 3.10 is taken from this list.","marker":"[4]"},{"why":"Gives the general bound $|\\operatorname{Aut}(\\mathcal{X})|\\le84g(g-1)$, the vanishing of second ramification groups for ordinary curves, and the ramification-counting lemmas used for the sporadic groups.","marker":"[19]"},{"why":"Provides the solvable-group bound $|G|\\le34(g+1)^{3/2}$, the elementary-abelian Sylow $p$-subgroup theorem, and the improved structural lemma for large solvable groups of ordinary curves.","marker":"[12]"},{"why":"Standard source for the Hurwitz genus formula, the Deuring--Shafarevich formula, ramification groups, and the bound $|Q|\\le4g+4$ used in Proposition 3.9.","marker":"[8]"},{"why":"Supplies the large solvable subgroup $S_q\\rtimes C_{(q-1)/2}$ inside $\\operatorname{PSL}(2,q)$ used to apply Lemma 3.3 in the $\\operatorname{PSL}(2,q)$ case.","marker":"[11]"},{"why":"Supplies the corresponding solvable normalizer structure inside $\\operatorname{PSU}(3,q)$ used in Proposition 3.12.","marker":"[18]"},{"why":"Shows that the modular curve $X(11)$ in characteristic $3$ has automorphism group $M_{11}$, making the exceptional case concrete.","marker":"[1]"},{"why":"Confirms the automorphism group of $X(11)$ over characteristic $3$ is $M_{11}$; cited with [1] to certify the example.","marker":"[20]"},{"why":"Classifies genus-$2$ curves with the automorphism group needed to identify the quotient $X/E$ in Proposition 4.1 as $y^2=x^5-x$.","marker":"[22]"}],"fun_headline_variants":["Odd-char ordinary curves: automorphism bound 821.37g^(7/4)","Hurwitz 84(g-1) holds except M11 in char 3 with g=26","Ordinary even genus: |Aut| < 821.37g^(7/4) over odd char","Only M11 on X(11) escapes Hurwitz bound for even genus","Even-genus ordinary curves: automorphism groups bounded"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that the classification quoted as Lemma 3.4 is complete: every non-solvable automorphism group of a curve of even genus in odd characteristic, once reduced by its largest normal subgroup of odd order, has a simple minimal normal subgroup of one of the five listed types. If a group type is missing from that list, the case analysis could miss a curve that violates the bound.","fun_headline_variants_meta":{"raw":{"variants":["Odd-char ordinary curves: automorphism bound 821.37g^(7/4)","Hurwitz 84(g-1) holds except M11 in char 3 with g=26","Ordinary even genus: |Aut| < 821.37g^(7/4) over odd char","Only M11 on X(11) escapes Hurwitz bound for even genus","Even-genus ordinary curves: automorphism groups bounded"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000549,"raw_usage":{"total_tokens":2724,"prompt_tokens":1147,"completion_tokens":1577,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":763,"completion_tokens_details":{"reasoning_tokens":1466}},"tokens_in":763,"tokens_out":1577,"duration_ms":13495,"temperature":1.0,"reasoning_tokens":1466,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:57:48.262903+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the ramification-data check of Theorem 3.13 for every group in Lemma 3.4, or equivalently search computationally for an ordinary curve of even genus over an algebraically closed field of odd characteristic whose automorphism group has order at least $821.37g^{7/4}$. A concrete target is the plane model $y^{10}(y+1)^9 = x^{22} - y(y+1)^4x^{11}(y^3+2y+1)$ in characteristic $3$: if it fails to be ordinary of genus $26$ with automorphism group $M_{11}$, the exceptional case would be empty; if a curve with $|G|\\ge821.37g^{7/4}$ exists, the main theorem would be false.","supporting_citations":[{"cited_title":"Algebraic curves with many automorphisms","cited_arxiv_id":"1702.08812","evidence_quote":"Supplies Lemma 3.4, the classification of odd core-free non-solvable automorphism groups of even-genus curves in odd characteristic; every non-solvable case in Theorem 3.10 is taken from this list."},{"cited_title":"Nakajima, p-ranks and automorphism groups of algebraic curves, Trans","cited_arxiv_id":null,"evidence_quote":"Gives the general bound $|\\operatorname{Aut}(\\mathcal{X})|\\le84g(g-1)$, the vanishing of second ramification groups for ordinary curves, and the ramification-counting lemmas used for the sporadic groups."},{"cited_title":"Ordinary algebraic curves with many automorphisms in positive characteristic","cited_arxiv_id":"1610.05252","evidence_quote":"Provides the solvable-group bound $|G|\\le34(g+1)^{3/2}$, the elementary-abelian Sylow $p$-subgroup theorem, and the improved structural lemma for large solvable groups of ordinary curves."},{"cited_title":"Hirschfeld, G","cited_arxiv_id":null,"evidence_quote":"Standard source for the Hurwitz genus formula, the Deuring--Shafarevich formula, ramification groups, and the bound $|Q|\\le4g+4$ used in Proposition 3.9."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the large solvable subgroup $S_q\\rtimes C_{(q-1)/2}$ inside $\\operatorname{PSL}(2,q)$ used to apply Lemma 3.3 in the $\\operatorname{PSL}(2,q)$ case."},{"cited_title":"Mitchell: Determination of the ordinary and modular ternary linear groups, Trans","cited_arxiv_id":null,"evidence_quote":"Supplies the corresponding solvable normalizer structure inside $\\operatorname{PSU}(3,q)$ used in Proposition 3.12."},{"cited_title":"Adler, The Mathieu Group M11 and the modular curve X(11), Proc","cited_arxiv_id":null,"evidence_quote":"Shows that the modular curve $X(11)$ in characteristic $3$ has automorphism group $M_{11}$, making the exceptional case concrete."},{"cited_title":"Rajan, Automorphisms of X(11) over characteristic 3, and the Mathieu Group M11, J","cited_arxiv_id":null,"evidence_quote":"Confirms the automorphism group of $X(11)$ over characteristic $3$ is $M_{11}$; cited with [1] to certify the example."},{"cited_title":"Shaska, H","cited_arxiv_id":null,"evidence_quote":"Classifies genus-$2$ curves with the automorphism group needed to identify the quotient $X/E$ in Proposition 4.1 as $y^2=x^5-x$."}],"review_version":1}