{"id":"85a93df8-0ac5-4446-8453-140ba131bbf3","arxiv_id":"2608.06286","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under GRH the average analytic rank of y²=x³-dx over odd fourth-power-free d is at most 13/6, and at most 3/2 assuming a quartic Gauss-sum conjecture.","lead":"Under the Riemann Hypothesis, this paper computes the distribution of low-lying zeros for the L-functions of the quartic twist family y²=x³-dx, for test functions with Fourier support in (-3/5,3/5). This bounds the average analytic rank by 13/6, and with an extra Gauss-sum conjecture the bound drops to 3/2, yielding positive proportions of minimal-rank curves.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The (−1,1)-support one-level density and the 3/2 average-rank bound hinge entirely on the unproved quartic Patterson conjecture (Conjecture 1.4); without it only the 3/5 support and 13/6 bound are established.","rationale":"I read the paper as a conditional one-level density computation. The GRH-only result (support <3/5) is proved by a long but structured argument: explicit formula, sieving, Poisson summation, Vaughan's identity, and a Lindelöf-on-average bound for Dirichlet series of quartic Gauss sums. I checked the main structural steps: the sieving error in Lemma 4.1 is O(D^{1+ε}/y), and with y=D^ε it contributes O(D) to the split sum, which after division by S_F(w,D) and the 1/log D in U_split is O(1/log D); this matches the claimed error. The Type I/II sum decomposition and the use of the quadratic large sieve in Z[i] are consistent with the support restriction ν<3/5. The bootstrap argument in Proposition 8.3 is delicate but internally coherent: the sup over t is finite by the convexity bound, and the choice of X absorbs the constant to close the inequality. I did not identify a concrete internal gap in the 3/5 portion. The second part, support <1 and the 3/2 bound, depends on Conjecture 1.4, which is not proved in the paper. That is precisely the reader's weakest assumption. Therefore the appropriate verdict remains CONDITIONAL: the 3/5 theorem appears sound, but the advertised improvements are conditional on a nontrivial conjecture that has not been independently verified.","tokens_in":43774,"tokens_out":43447,"duration_ms":466859,"concrete_test":"For r=1, β=1, ℓ=0, compute the Conjecture 1.4 sums over c∈Z[i], c≡1 mod 4, N(c)≤X for X=10^4,10^5,10^6, using (2.11) to evaluate g4(1,c), and fit b_{1,1}X^{3/4}. If the residual is not O(X^{1/2+ε}) with the same leading constant across ranges, Conjecture 1.4 would need revision; a clean fit would support the weaker half of the paper but would not prove the conjecture.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing point is the conditional status of Theorem 1.2's second part. The extension of the admissible support from (−3/5,3/5) to (−1,1) is obtained in Section 4 by invoking Conjecture 1.4, a quartic analogue of Patterson's conjecture imported from the authors' earlier work [11]. That conjecture supplies the cancellation in (4.11) via the asymptotic Hβ ≪ |mD/M|^{−1/2+ε}; if it fails, the support extension, the 3/2 rank upper bound, and the positive proportion of rank-0 twists (Corollary 1.3) are not supported. The GRH-only 3/5 result does not use Conjecture 1.4; it rests on GRH and on the new Lindelöf-on-average estimate, Proposition 7.6. I found no specific internal error in the proof of Proposition 7.6, but it is a long metaplectic argument (Section 8) that was not machine-checked, so a residual risk sits there as well. The conditional nature of the stronger half is exactly the reader's identified weakest assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the one-level density of low-lying zeros of the L-functions L(s,E_d) for the quartic-twist family E_d:y^2=x^3-dx with d odd and fourth-power-free. The main result, Theorem 1.2, states that under GRH for these L-functions the density equals bφ(0)+(1/2)∫bφ(u)du for test functions with Fourier support in (-3/5,3/5), and that under a quartic analogue of Patterson's conjecture (Conjecture 1.4) the support extends to (-1,1). Theorem 1.1 derives average analytic rank bounds of 13/6 and 3/2, respectively, and Corollary 1.3 gives positive proportions of rank-1 twists in F^- and, under Conjecture 1.4, rank-0 twists in F^+. The proof proceeds via the explicit formula, a sieving step (Lemma 4.1), Poisson summation leading to quartic Gauss sums, a Vaughan-type decomposition into Type I and Type II sums, and a Lindelöf-on-average estimate (Proposition 7.6) for the Dirichlet series of quartic Gauss sums, proved in Section 8 using the quadratic large sieve over Z[i].","tokens_in":43996,"tokens_out":45257,"duration_ms":512204,"significance":"If correct, this is a substantial contribution to the study of higher-order twist families of elliptic curves: it gives the first one-level density result for quartic twists with support beyond 1/2 and, conditionally on Conjecture 1.4, reaches the optimal support (-1,1) with the expected symmetry density. The GRH-only 3/5-support result is an unconditional-in-the-conjecture statement that yields a nontrivial average-rank bound of 13/6 and a positive proportion of rank-one twists. The strategy of bounding quartic Gauss-sum sums via Vaughan's identity, the quadratic large sieve over Z[i], and a Lindelöf-on-average estimate for metaplectic L-functions is innovative and carefully structured. The paper is also transparently conditional: Conjecture 1.4 is explicitly isolated, and the main new technical ingredient, Proposition 7.6, is proved in detail. The manuscript does not contain machine-checked proofs or reproducible code, but the analytic arguments are laid out in sufficient detail for expert verification.","major_comments":[{"comment":"There is a discrepancy in the sieving step that is load-bearing for the passage from sums over F* to sums over all integers. Lemma 4.1 states a remainder of O(D^{1+ε}/y), and Remark 4.2 says to apply the lemma with y=D^ε; with that choice the stated error becomes O(D), which is not an admissible error for the target bound O(D) in Remark 3.4. The proof of Lemma 4.1 appears to yield the stronger O(D^{1+2(ν+1)ε}/y^{1+ε}) (because the sum over ℓ≥y of ℓ^{-(2+ε)} contributes y^{-(1+ε)}), and with y=D^{Cε} for a sufficiently large fixed C this would be fine. The authors should correct the stated error term in Lemma 4.1, specify the correct choice of y in Remark 4.2, and reconcile the numerology with the condition δ>30ε.","section":"§4, Lemma 4.1 and Remark 4.2"},{"comment":"The proof applies 'the quadratic large sieve [23, Theorem 1]' to a bilinear sum in which both n and v range over Gaussian integers and the character is the quadratic symbol (v/n)_2 over Z[i]. Reference [23] is a large sieve for real Dirichlet characters over the rationals; the estimate being used is precisely the Gaussian quadratic large sieve stated earlier in Section 6 as [36, Thm. 1]. Please replace the citation and state the exact theorem used, or give a direct justification from [23]. This step is central to the proof of Proposition 7.6 and hence to the 3/5-support result.","section":"§8, proof of Proposition 8.3"}],"minor_comments":[{"comment":"The sum in (5.6) runs over all integers m with |m|<16ηM D^{2ν-1}, but Hβ(X,Y,r) and R_{X,Y}(t) are defined for Y>0, while the argument mD/(16M) is negative for m<0. Please either define R_{X,Y} for all real Y or replace mD/(16M) by |m|D/(16M) and note that the m=0 term vanishes because g4(0,c)=0 for c≠1.","section":"§5, Eq. (5.6)"},{"comment":"The compatibility condition 'A≡B mod 4' in the statement of Lemma 4.1 could be made more explicit: it is the condition for the two congruences d≡A mod 16 and d≡B mod 4 to have a solution, and it is equivalent to a≡b d2 mod 4.","section":"§4, Lemma 4.1"},{"comment":"The phrase 'uniformly for all ε>0 and M<D^ε' should be read as 'for each fixed ε>0, uniformly for M<D^ε'; otherwise the existence of a single δ>0 for all ε simultaneously is not what is meant and is not what the subsequent argument proves.","section":"§4, Remark 4.6"},{"comment":"The sentence 'It remains to justify that Gβ(s,r|r,a) is c for Re(s)>1/2' appears to be missing the word 'analytic' in the source text; please correct.","section":"§7, proof of Proposition 7.4"},{"comment":"There are several typographical issues to correct, including 'V aughan' in the Section 6 heading, 'P´ olya–Vinogradov' in Remark 4.2, 'fourt -power' in Section 2.1, and 'similiar' in the proof of Corollary 1.3.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"This is a serious and substantial paper. The central theorems appear defensible, and the main issue I found is a fixable but load-bearing error-term mismatch in Section 4; the proof of Lemma 4.1 already contains the margin needed for the repair. The Section 8 citation issue with the quadratic large sieve is also straightforward to correct. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a real result, not a repackaging. Under GRH, they compute the one-level density for the quartic twist family y^2=x^3-dx over odd fourth-power-free d, for test functions with Fourier support in (-3/5,3/5), and derive an average analytic rank upper bound of 13/6 plus positive proportions of rank 1. That extends what was known for cubic and quartic Dirichlet families and for the authors' earlier CM-character framework. The technical core is a long, careful treatment of quartic Gauss sums over Z[i]: Vaughan's identity splits the sums into Type I and Type II, Type II is handled by the quadratic large sieve, Type I by a Lindelof-on-average bound on metaplectic L-functions. I did not find internal errors; the structure is coherent and the reductions are explicit.\n\nThe paper is honest about its conditionality. The stronger half—support (-1,1), average rank <= 3/2, positive proportion of rank 0—uses Conjecture 1.4, a quartic analogue of Patterson's conjecture that is imported from their own earlier paper and is not proved here. The stress-test note is right: without that conjecture, those results do not stand; the 3/5 result does not use it. That is a soft spot only in the sense that the headline 'support 1' is conditional on a conjecture. They do not hide it, and the abstract says so.\n\nThe main residual risk sits in Section 8, the proof of the Lindelof-on-average proposition. It is a long metaplectic argument, and I did not machine-check it. Nothing looks wrong, but it is the kind of place where a missing factor or a bad contour shift could lurk. A referee should go through that section carefully.\n\nMinor notes: the conductor approximation by D^2 is justified by Lemma 3.3; the root number calculation in Section 9 checks out on the cases I tried. The citation pattern is fine; the self-citation to [11] is appropriate since that is the source of the conjecture.\n\nWho is this for: people working on low-lying zeros, CM elliptic curves, and Gauss sums. It deserves a serious referee. My recommendation: send it to review, with the understanding that it will likely be accepted conditionally, and the authors should keep the distinction between GRH-only and GRH+Conjecture 1.4 results prominent.","headline":"A serious conditional theorem paper: under GRH it gets one-level density at support 3/5 for quartic twists and rank bound 13/6; the stronger support-1 and 3/2 results rest on an unproved conjecture from the authors' own earlier work.","tokens_in":44546,"tokens_out":2310,"would_cite":true,"duration_ms":27024,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","11M50","11L05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under the Generalized Riemann Hypothesis, the average analytic rank of the quartic twists $y^2=x^3-dx$ is at most $13/6$, and with a quartic Patterson conjecture it improves to $3/2$.","keywords":["average analytic rank","quartic twists","elliptic curves","one-level density","low-lying zeros","quartic Gauss sums","Patterson conjecture","generalized Riemann hypothesis"],"falsifier":"For a fixed residue class $\\beta\\in\\{1,1+\\lambda^3\\}$, compute the sums in Conjecture 1.4 for Gaussian integers $c\\equiv\\beta\\bmod 4$ up to large $X$; if the $\\ell=0$ sum fails to grow like a constant times $X^{3/4}$, the $3/2$ average rank bound collapses. Separately, evaluate the one-level density with a test function whose Fourier support crosses $\\nu=0.6$: under the paper's claims it must match the Katz–Sarnak density up to $O(1/\\log D)$ as $D\\to\\infty$.","tokens_in":43560,"feed_emoji":"🔢","tokens_out":5774,"duration_ms":50888,"temperature":0.7,"pith_summary":"This paper studies the family of elliptic curves $E_d: y^2=x^3-dx$ as $d$ ranges over odd fourth-power-free integers, and asks how large the average analytic rank can be. Under the Generalized Riemann Hypothesis, the authors compute the one-level density of low-lying zeros for the associated $L$-functions and deduce that the average analytic rank is at most $13/6$. Assuming, in addition, a quartic analogue of Patterson's conjecture on the distribution of quartic Gauss sums at prime elements, the support of the density computation widens to $(-1,1)$ and the average rank bound improves to $3/2$. Both results imply that a positive proportion of the twists have analytic rank $1$, and the conjectural half also yields a positive proportion of twists with analytic rank $0$. If the paper is right, the low-lying zeros of this CM family match the expected Katz–Sarnak symmetry in the admissible support ranges, and the average rank is significantly smaller than the trivial bound.","feed_headline":"Quartic twist ranks: 13/6 under GRH, 3/2 with a conjecture","feed_subtitle":"One-level density of low-lying zeros pins down the average analytic rank of y^2=x^3-dx twists.","key_machinery":"The machinery is the explicit formula for the Hecke-character $L$-functions $L(s,\\xi_d)$, which splits the one-level density into an archimedean term, an inert-primes term, and a split-primes term; the split term is the hard part. The split term is reduced, through Poisson summation and a sieving step, to bounding sums $H_\\beta(X,Y,r)$ of quartic Gauss sums $g_4(r,c)$ weighted by von Mangoldt's function over $\\mathbb Z[i]$. Vaughan's identity decomposes these sums into Type I and Type II pieces; Type I pieces are handled by a Lindelöf-on-average bound across metaplectic $L$-functions (obtained via the quadratic large sieve), and Type II pieces by factoring the quartic Gauss sums and exploiting the oscillation of quadratic characters through the same sieve. The support constraint $\\nu<3/5$ arises solely from the Type II sums.","core_discovery":"The central discovery is that the one-level density of the family $\\{L(s,E_d)\\}$ for $d$ odd fourth-power-free obeys, on average, the formula $D_{\\mathcal F^*}(\\phi,w,D)=\\hat\\phi(0)+\\frac12\\int_{\\mathbb R}\\hat\\phi(u)\\,du+O(1/\\log D)$, where the Fourier support of $\\phi$ is contained in $(-3/5,3/5)$ under GRH and in $(-1,1)$ under a quartic Patterson conjecture. This matches the expected Katz–Sarnak symmetry for these supports. From this, the authors derive the average analytic rank bounds of $13/6$ (under GRH) and $3/2$ (under the conjecture), and consequently positive proportions of twists with minimal analytic rank consistent with parity.","pith_inferences":["The quartic Patterson conjecture could be tested numerically for Gaussian primes of moderate norm; a failure of the predicted $X^{3/4}$ growth in Conjecture 1.4 would directly invalidate the $3/2$ rank bound.","The Type II limitation suggests that a different factorization or sieve for quartic Gauss sums could push the admissible support further, which would also benefit non-vanishing results for quartic Hecke characters.","The bounds for quartic twists parallel Heath-Brown's results for quadratic twists, strengthening the heuristic expectation that Goldfeld's minimalist conjecture holds across all twist families, not only the quadratic one.","A natural extension is to apply the same Vaughan-identity and large-sieve decomposition to cubic twists over $\\mathbb Q(\\sqrt{-3})$, where the analogue of Conjecture 1.4 is Patterson's original cubic conjecture and a similar rank bound could be pursued."],"forward_implications":["Under GRH alone, at least $5/12$ of the negative-root-number twists have analytic rank $1$.","If Conjecture 1.4 holds, the proportions improve: at least $3/4$ of the negative-root-number twists have rank $1$, and at least $1/4$ of the positive-root-number twists have rank $0$.","The low-lying zeros of the family follow the expected Katz–Sarnak symmetry for test functions whose Fourier transform is supported in $(-3/5,3/5)$ under GRH, and in $(-1,1)$ under the conjecture.","As the paper notes, if the density formula held for every even Schwartz test function, the average analytic rank would be $1/2$, the minimalist value predicted by Goldfeld's conjecture.","The average rank bounds of $13/6$ and $3/2$ are direct consequences of Theorem 1.2, obtained by substituting admissible test functions into the density formula."],"supporting_citations":[{"why":"Supplies Conjecture 1.4 and the analytic properties of quartic Gauss sums and metaplectic theta functions used throughout.","marker":"[11]"},{"why":"Provides the quadratic large sieve used for Type II sums and for the Lindelöf-on-average bound.","marker":"[23]"},{"why":"Supplies Vaughan's identity in the form used to decompose the Gauss-sum sums into Type I and Type II pieces.","marker":"[24]"},{"why":"Gives the meromorphic continuation and functional equation of the Dirichlet series of quartic Gauss sums.","marker":"[15]"},{"why":"Establishes the Hecke-character description of $L(s,E_d)$ and the explicit formula, including the conductor computation.","marker":"[10]"},{"why":"Formulates the original Patterson conjecture for cubic Gauss sums, the model for Conjecture 1.4.","marker":"[37]"},{"why":"States the Katz–Sarnak symmetry predictions for low-lying zeros that the paper's density formula matches.","marker":"[31]"},{"why":"Provides the quadratic large sieve over $\\mathbb Z[i]$ used explicitly in bounding the Type II sums.","marker":"[36]"}],"fun_headline_variants":["Average rank of quartic twists capped at 13/6 under GRH, 3/2 under conjecture","Quartic twist ranks: density yields 13/6 bound, 3/2 with conjecture","Positive proportions of quartic twists with rank 1, and rank 0 under extra conjecture","From 13/6 to 3/2: refining average rank bounds for quartic twists"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For the stronger rank bound, the load-bearing premise is Conjecture 1.4, which asserts that quartic Gauss sums at prime elements in $\\mathbb Z[i]$ are equidistributed with main term of size $X^{3/4}$; if that distribution is wrong, only the $13/6$ bound under GRH remains.","fun_headline_variants_meta":{"raw":{"variants":["Average rank of quartic twists capped at 13/6 under GRH, 3/2 under conjecture","Quartic twist ranks: density yields 13/6 bound, 3/2 with conjecture","Positive proportions of quartic twists with rank 1, and rank 0 under extra conjecture","From 13/6 to 3/2: refining average rank bounds for quartic twists"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00107,"raw_usage":{"total_tokens":4494,"prompt_tokens":968,"completion_tokens":3526,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":584,"completion_tokens_details":{"reasoning_tokens":3425}},"tokens_in":584,"tokens_out":3526,"duration_ms":25089,"temperature":1.0,"reasoning_tokens":3425,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:24:24.688459+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed residue class $\\beta\\in\\{1,1+\\lambda^3\\}$, compute the sums in Conjecture 1.4 for Gaussian integers $c\\equiv\\beta\\bmod 4$ up to large $X$; if the $\\ell=0$ sum fails to grow like a constant times $X^{3/4}$, the $3/2$ average rank bound collapses. Separately, evaluate the one-level density with a test function whose Fourier support crosses $\\nu=0.6$: under the paper's claims it must match the Katz–Sarnak density up to $O(1/\\log D)$ as $D\\to\\infty$.","supporting_citations":[{"cited_title":"David, A","cited_arxiv_id":null,"evidence_quote":"Supplies Conjecture 1.4 and the analytic properties of quartic Gauss sums and metaplectic theta functions used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the quadratic large sieve used for Type II sums and for the Lindelöf-on-average bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Vaughan's identity in the form used to decompose the Gauss-sum sums into Type I and Type II pieces."},{"cited_title":"Diaconu,Mean square values of HeckeL–series formed withr–th order characters, Invent","cited_arxiv_id":null,"evidence_quote":"Gives the meromorphic continuation and functional equation of the Dirichlet series of quartic Gauss sums."},{"cited_title":"David, L","cited_arxiv_id":null,"evidence_quote":"Establishes the Hecke-character description of $L(s,E_d)$ and the explicit formula, including the conductor computation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Formulates the original Patterson conjecture for cubic Gauss sums, the model for Conjecture 1.4."},{"cited_title":"Katz and P","cited_arxiv_id":null,"evidence_quote":"States the Katz–Sarnak symmetry predictions for low-lying zeros that the paper's density formula matches."},{"cited_title":"Onodera,Bound for the sum involving the Jacobi symbol inZ[i], Funct","cited_arxiv_id":null,"evidence_quote":"Provides the quadratic large sieve over $\\mathbb Z[i]$ used explicitly in bounding the Type II sums."}],"review_version":1}