{"id":"f502213c-04fa-4995-be97-caeec03a2dac","arxiv_id":"2412.20174","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For elliptic curves with standard double covers and suitable reduction at a prime p, the number of common projective torsion points is at most 2p^3+8, with refinements and a conditional bad-reduction analogue.","lead":"This paper gives explicit numerical bounds, depending on a prime p, for how many torsion x-coordinates two elliptic curves can share. It is a first realistic effective answer to a question in unlikely intersections that previously had only non-effective uniform bounds.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The bad-reduction Theorem 5.9 depends on Assumption 5.2(b), an unproved finiteness condition; without it the claimed bound for multiplicative reduction is not established.","rationale":"The reader's conditional verdict is correct. The strongest proven contribution is Theorem 2.5 for good reduction; the proof there is built on Raynaud's method and, while terse, does not exhibit an obvious gap. The problematic part is the extension to bad multiplicative reduction in Section 5. Theorem 5.9 is explicitly stated under Assumption 5.2, and part (b) is a finiteness assertion that is not a standard consequence of the geometric setup in general. The authors themselves flag this in Remark 5.3, and Theorem 7.1 verifies it only in a special configuration. Since the abstract broadcasts the bad-reduction case as part of the paper's results, the unsupported assumption is the single most load-bearing point. I see no independent evidence (machine-checked proof, code, or a parameter-free derivation) that would replace the missing proof. Thus the verdict should remain CONDITIONAL: accept the good-reduction results, but require the finiteness question to be resolved before the bad-reduction claims can be deemed established. No change from the reader's verdict.","tokens_in":26988,"tokens_out":23352,"duration_ms":242550,"concrete_test":"Attempt to prove Assumption 5.2(b) in the next case not covered by Theorem 7.1: let E1 have multiplicative reduction with Tate parameter q of valuation v(q)=1 over the maximally unramified extension, and E2 good ordinary. Follow the proof of Proposition 9.2: it chooses a p-torsion point of E1(K) that moves between components of Gamma_{B,k}; such a point corresponds to a p-th root of q, which for v(q)=1 is not rational over the maximally unramified extension. Determine whether the image of Lambda cap (Gamma^o_{B,R})^{(1)}(R_1) in Gamma^o_{B,k}(k) is still Zariski dense in every component; if not, Lemma 10.2 fails and Assumption 5.2(b) is not implied by the log-geometric argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step for the bad multiplicative reduction case is Theorem 5.9, which depends on Assumption 5.2(b): the finiteness of im(pA_1^o(R_1) cap X_1^o(R_1) -> X_0^o(k)). This is not a harmless technicality; the proof of Theorem 5.9 injects the set M of common torsion pairs into exactly this image, so if the image is infinite the bound |M| <= 2p^3+2 is vacuous or false. The paper only proves Assumption 5.2(b) under the restrictive hypotheses of Theorem 7.1 (E1 has Tate parameter q = pi^p and E2 has good ordinary reduction). Remark 5.3 explicitly says the general implication from Assumption 5.2(a) to 5.2(b) is not proved. Therefore the abstract's claim of effective bounds for common projective torsion points under 'mild extra assumptions' is not supported in the multiplicative-reduction regime: Theorem 5.9 is conditional on an open finiteness conjecture.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Bogomolov-Fu-Tschinkel problem on the number of common images of torsion points of two elliptic curves under standard double covers to the projective line. It proves explicit bounds on the number of common images of torsion points of order coprime to a prime p under a good-reduction assumption (Theorem 2.5: t ≤ 2p^3+8), with refinements in the supersingular and ordinary cases (Propositions 3.2 and 3.3). It also states a bound in the case of bad multiplicative reduction (Theorem 5.9: |M| ≤ 2p^3+2) conditional on a finiteness assumption (Assumption 5.2(b)), and develops logarithmic-geometric techniques (Sections 7–10) to verify this assumption in a special case (Theorem 7.1). Section 4 attempts to combine bounds at two primes to control the full torsion set.","tokens_in":27164,"tokens_out":15939,"duration_ms":159371,"significance":"If the good-reduction bound of Theorem 2.5 is correct, it provides the first explicit, effective bound for the prime-to-p part of the common projective torsion problem, complementing the ineffective uniformity results cited in the introduction. The adaptation of Raynaud's Manin-Mumford method and the log-geometric machinery for bad reduction are original and constitute a substantive technical contribution. The main qualifications are that the bad-reduction theorem is conditional on an unproved finiteness statement and that the proposed combination step for full torsion (Section 4) is not fully justified. As a result, the paper's core unconditional contribution is the prime-to-p bound in the good-reduction case, not a complete solution of the original problem for all torsion points.","major_comments":[{"comment":"The bound |M|≤2p^3+2 in Theorem 5.9 is conditional on the finiteness of im(pA_1^∘(R_1)∩X_1^∘(R_1)→X_0^∘(k)) stated in Assumption 5.2(b). This assumption is not proved in general; as Remark 5.3 explicitly concedes, the authors only establish it under the restrictive hypotheses of Theorem 7.1 (E_1 with Tate parameter q=π^p and E_2 with good ordinary reduction). Since the proof of Theorem 5.9 injects M into exactly this image, the multiplicative-reduction case is not established unconditionally, and the abstract's phrasing \"mild extra assumptions\" overstates what is proved.","section":"Section 5, Assumption 5.2(b), Theorem 5.9"},{"comment":"The deduction of Lemma 4.4 (t_{A,X+a,p}≤8q^3) from Lemma 4.3 by \"repeating the above argument for a second prime q\" is not justified. The proof of Lemma 4.3 for a∉A(K) uses the Galois group of K, where K is the completion of the maximal unramified extension at p, and the unramifiedness of A[∞](p') at p. To obtain the analogous statement at q one would need to pass to a different base field (in general K(a)), show that the translated curve X+a has a suitable good-reduction model at q, and re-run the argument with q-primary torsion; none of these steps is addressed. Since Proposition 4.2 relies on Lemma 4.4 to combine the p'-torsion and p-primary bounds, the claimed control of the full torsion set is not proved.","section":"Section 4, Lemmas 4.3 and 4.4"},{"comment":"The estimate δ≤p^3 for the degree of the map Y'_0→X_0 is compressed into the sentence \"Since Y_0 is defined to be the reduced preimage of X_0 under this map, we get the desired bound.\" The argument should explicitly exhibit the factorization of [p]|Y_0 through the relative Frobenius and bound the degrees of the two factors, taking care that Y_0 is reduced and may have components on which the degrees differ. As written, this step is too quick for a load-bearing bound, although it is likely fixable by spelling out the standard inseparability-degree argument.","section":"Section 2, proof of Theorem 2.5"}],"minor_comments":[{"comment":"The sentence \"The curve Γ is a connected curve with N connected components each of which is a nonsingular curve of genus 3\" is contradictory in its current wording; \"connected components\" should presumably be \"irreducible components\", and the following sentence about intersections in nodes should be made consistent with that reading.","section":"Section 8, Proposition 8.1"},{"comment":"The main theorems do not state that p is odd, yet Section 8 explicitly assumes p≠2 for the arguments leading to Theorem 7.1. The paper should state clearly whether Theorems 2.5 and 5.9 are intended also for p=2, and if so, indicate what modifications are needed.","section":"Sections 2 and 5"},{"comment":"The abstract and introduction promise bounds for common projective torsion points under \"mild extra assumptions\", but the bad-reduction theorem is conditional on Assumption 5.2(b). The authors should explicitly qualify this in the abstract and introduction so that the conditional nature of the multiplicative-reduction results is clear to the reader.","section":"Introduction and abstract"},{"comment":"There is a duplicated phrase: \"we have we have only discussed\" should read \"we have only discussed\".","section":"Section 4, first paragraph"},{"comment":"The statement of Lemma 10.2 contains the typo \"Suppose that the the assumptions\", which should be corrected.","section":"Section 10, Lemma 10.2"},{"comment":"The notation \"where we denote a uniformiser of R by π\" in the log smoothness discussion conflicts with the use of π for the projections π_i, π_{i,R}; using a different symbol such as ϖ for the uniformiser would avoid confusion.","section":"Section 10, notation"}],"recommendation":"major_revision","confidential_remarks":"The good-reduction result (Theorem 2.5) is the paper's main unconditional contribution and appears plausible, though the degree estimate δ≤p^3 would benefit from a more detailed proof. The bad-reduction theorem is conditional on an unproved finiteness statement and should be presented as such; the Section 4 combination argument is currently a gap rather than a minor omission. I recommend major revision: the authors should either prove or clearly isolate the finiteness assumption, complete or curtail the Section 4 claims, and adjust the abstract to match what is actually established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, the good-reduction part is a genuine step forward: Theorem 2.5 gives t <= 2p^3+8 for common images of torsion coprime to p, and the refinements for supersingular and ordinary cases (2p^2+8 and 2p+8) are natural and plausible. Second, the bad-reduction Theorem 5.9 is conditional on Assumption 5.2(b), a finiteness statement that is close to the target conclusion; the authors prove it only under special hypotheses (Theorem 7.1) and admit in Remark 5.3 that the general implication is open. The abstract's phrase \"mild extra assumptions\" does not convey that the multiplicative-reduction case rests on an unproved hypothesis.\n\nWhat is actually new: the extension of Raynaud's method to bad multiplicative reduction using log algebraic geometry, the admissible factorization of multiplication-by-p in Proposition 7.4, and the bound in Proposition 6.3 on sections of rational maps with zero differential. The paper builds on Raynaud's classical intersection-counting framework rather than the ineffective uniformity results, so it is a substantial technical extension rather than a new paradigm. The degree estimate for the reduced preimage under multiplication by p (the p^3 bound) is a bit quick but plausible; the overall proof of Theorem 2.5 is coherent.\n\nThe soft spots, in proportion: (1) Assumption 5.2(b) is the load-bearing issue in the bad reduction case. If it fails, the bound |M| <= 2p^3+2 collapses. The paper is honest about the gap, but the packaging is not. (2) The full torsion bound in Section 4 involves a constant c = 8p^{4r+3} where r depends on the Galois action, so it is explicit in principle but not \"realistic\" in the sense advertised. (3) Lemma 4.3 has a terse step for a not rational; the Galois-conjugate argument is standard but the intersection bound could use a line of detail. These are minor relative to Assumption 5.2(b).\n\nThe citation pattern is fine; there is no dependence on the authors' own prior results, and no data fitting or fabrication concerns. The paper deserves a serious referee. I would send it to a good journal, but the referee should push for either a proof of Assumption 5.2(b) in greater generality or a clear separation of unconditional and conditional results, and the abstract should be rewritten to state the finiteness hypothesis explicitly. This is not a desk reject.","headline":"The good-reduction bound is a real effective result, but the bad-reduction theorem is conditional on an unproved finiteness assumption that the abstract understates.","tokens_in":27717,"tokens_out":2054,"would_cite":true,"duration_ms":21391,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","14H52","14K12","14G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"When two elliptic curves with standard double covers have good reduction at a prime $p$, the paper bounds their common torsion images by $2p^3+8$, with refinements down to $2p+8$; in bad multiplicative reduction it gives the conditional…","keywords":["common projective torsion points","elliptic curves","double covers","unlikely intersections","effective bounds","logarithmic algebraic geometry","Witt vectors","torsion points on curves"],"falsifier":"A direct calculation for an explicit pair satisfying the good-reduction assumptions at a small prime $p$ would settle the main bound: the theorem predicts at most $2p^3+8$ common projective torsion points of order coprime to $p$, so a pair with more would refute it. For the bad-reduction theorem, a concrete bad-multiplicative-reduction model with a $p$-th power uniformiser and a good ordinary partner for which the first-order lift set reduces to infinitely many points on the special fibre would disprove the finiteness claim.","tokens_in":26720,"feed_emoji":"🔢","tokens_out":15984,"duration_ms":146900,"temperature":0.7,"pith_summary":"This paper asks a quantitative version of a uniformity question in unlikely intersections: for two elliptic curves with the standard double cover that identifies a point with its inverse, how many torsion points of one curve can have the same image in the projective line as torsion points of the other? Earlier work proved that the number is bounded independently of the curves, but did not provide realistic constants. The main new result is an explicit bound: if both curves have good reduction at a prime $p$ and the branch loci of the two covers are disjoint on the special fibre, then the number of common images of torsion points of order coprime to $p$ is at most $2p^3+8$. Refinements improve this to $2p^2+8$ or $2p+8$ depending on whether the reductions are supersingular or ordinary and whether certain $p$-torsion group schemes split. For bad multiplicative reduction the paper gives the analogous bound $2p^3+2$, under a finiteness assumption that it verifies only in special cases; these are the first effective bounds in reach of numerical testing.","feed_headline":"Two elliptic curves share at most 2p^3+8 projective torsion points","feed_subtitle":"First realistic effective bound for common torsion images when both curves reduce well at a chosen prime p.","key_machinery":"The load-bearing construction is the first-order infinitesimal deformation bundle. One works with the abelian scheme $A=E_1\\times E_2$ over the Witt vectors $R=W(\\mathbb{F}_p)$ and with $X=(\\pi_1\\times\\pi_2)^{-1}(\\Delta)$, the preimage of the diagonal in $\\mathbb{P}^1\\times\\mathbb{P}^1$. Torsion of order coprime to $p$ specialises injectively into the central fibre, and the paper bounds its image by $|\\mathrm{im}(pA_1(R_1)\\cap X_1(R_1)\\to X_0(k))|$, where $R_1=R/p^2$. Points of $X_1(R_1)$ lifting a point of $X_0$ determine normal directions, encoded in the affine bundle $V_0=\\mathbb{P}(N_{X_0/A})\\setminus\\mathbb{P}(N_{X_0/A_0})$; the image of $X_1(R_1)$ is a curve $X'_0$ over $X_0$, and multiplication by $p$ on $A_0$, which factors through the relative Frobenius morphism, produces a curve $Y'_0$ that is numerically $\\delta X'_0$. Intersection theory gives $X'_0\\cdot Y'_0=8\\delta$, and the Frobenius factorisation gives $\\delta\\leq p^3$. In bad multiplicative reduction the same counting problem is moved to $\\mathbb{P}^1\\times\\mathbb{P}^1$ through the rational multiplication-by-$p$ maps, and a bidegree computation with lifted polynomials (Proposition 6.3) supplies the $p^3$ bound.","core_discovery":"On the paper's own terms, the central discovery is Theorem 2.5: under Assumptions 1.2 and 2.4, $t(E_1,\\pi_1,E_2,\\pi_2,p') \\leq 2p^3+8$, where $t$ counts points of the projective line that are images, under both standard double covers, of torsion points of order not divisible by $p$. The proof routes the count through torsion points of the abelian surface $A=E_1\\times E_2$ lying on the curve $X=(\\pi_1\\times\\pi_2)^{-1}(\\Delta)$, the preimage of the diagonal, and then through first-order deformations over the Witt vectors; the geometry of an affine bundle over the special fibre reduces the problem to bounding an intersection number by $8\\delta$ with $\\delta\\leq p^3$. Refinements in Section 3 sharpen $\\delta$ to $p^2$ or $p$ in the supersingular and split-ordinary cases. In bad multiplicative reduction, Theorem 5.9 transfers the argument to a product of projective lines and obtains $|M|\\leq 2p^3+2$ conditional on Assumption 5.2(b), and Theorem 7.1 proves that finiteness when the first curve's bad-reduction model has a $p$-th power uniformiser and the second has good ordinary reduction.","pith_inferences":["A natural testable extension, not carried out in the paper, is to run the algorithmic canonical-lift criterion of Section 3 on explicit curves for small primes and compare the refined bounds $2p^2+8$ and $2p+8$ with brute-force torsion computations.","If the finiteness in Assumption 5.2(b) is proved in general, the bad-multiplicative-reduction theorem becomes unconditional, and the restriction to a $p$-th-power uniformiser is likely removable.","The affine-bundle intersection technique suggests a route to higher-dimensional abelian varieties or higher-genus curves, with the degree of multiplication-by-$p$ on the relevant subvariety replacing the role of $p^3$.","The explicit constant opens up a computational search for the true optimal value: enumerating torsion images for many pairs at a fixed small $p$ could show whether $2p^3+8$ is close to the maximum."],"forward_implications":["For any pair of elliptic curves with standard double covers satisfying the good-reduction assumptions at a prime $p$, the number of common projective torsion images of order coprime to $p$ is at most $2p^3+8$, a constant small enough to be compared with computational torsion data.","When both special fibres are supersingular, the bound improves to $2p^2+8$; when both are ordinary and the relevant connected-étale sequences split, it improves to $2p+8$.","Combining the $p$-prime bound with a second prime $q$ and a large-Galois-orbit condition yields a finite total bound $c=8p^{4r+3}$ on all torsion points of $A=E_1\\times E_2$ lying on $X$.","In the bad multiplicative reduction case, under Assumption 5.2 the number of coprime-to-$p$ torsion pairs with a common projection is at most $2p^3+2$.","The finiteness needed for the bad-reduction bound is proved unconditionally when the first curve's model has a $p$-th power uniformiser and the second curve has good ordinary reduction."],"supporting_citations":[{"why":"Its first-order deformation and intersection method is the backbone of Theorem 2.5: it converts a torsion count into an intersection number in an affine bundle.","marker":"[Ray83-1]"},{"why":"Provides the Frobenius non-liftability argument that Theorem 7.1 generalises to log smooth curves, making the bad-reduction finiteness theorem work in special cases.","marker":"[Ray83-2]"},{"why":"Raises the uniform boundedness question for common projective torsion points, which this paper turns into an effective bound.","marker":"[BFT18]"},{"why":"Supplies the known lower-bound examples (up to 34 common torsion points) that show the constants are in the right range.","marker":"[FS19]"},{"why":"Its appendix gives the canonical-lift criterion used in Lemma 3.4 to decide when the ordinary-reduction bound improves.","marker":"[MS87]"},{"why":"Starts the log smooth deformation theory used to prove the finiteness assertion in the bad multiplicative reduction case.","marker":"[Kato96]"},{"why":"Gives the Galois-orbit growth for p-torsion of supersingular reduction used to show the Galois action is large when combining primes.","marker":"[Se72]"}],"fun_headline_variants":["Explicit bound: two elliptic curves share at most 2p^3+8 projective torsion points","At most 2p^3+8 common projective torsion points on elliptic curves","First effective bound for common projective torsion on elliptic curves"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Assumption 5.2(b), which says that the set of first-order lift points lying on the common-torsion curve after multiplication by $p$ is finite; the paper proves this only under extra hypotheses and leaves the general bad-reduction case open.","fun_headline_variants_meta":{"raw":{"variants":["Explicit bound: two elliptic curves share at most 2p^3+8 projective torsion points","At most 2p^3+8 common projective torsion points on elliptic curves","First effective bound for common projective torsion on elliptic curves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000891,"raw_usage":{"total_tokens":3861,"prompt_tokens":982,"completion_tokens":2879,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":598,"completion_tokens_details":{"reasoning_tokens":2813}},"tokens_in":598,"tokens_out":2879,"duration_ms":23784,"temperature":1.0,"reasoning_tokens":2813,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:29:52.035015+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct calculation for an explicit pair satisfying the good-reduction assumptions at a small prime $p$ would settle the main bound: the theorem predicts at most $2p^3+8$ common projective torsion points of order coprime to $p$, so a pair with more would refute it. For the bad-reduction theorem, a concrete bad-multiplicative-reduction model with a $p$-th power uniformiser and a good ordinary partner for which the first-order lift set reduces to infinitely many points on the special fibre would disprove the finiteness claim.","supporting_citations":[],"review_version":1}