{"id":"628c6dc9-a7ba-4732-8687-fe643e902841","arxiv_id":"2502.08760","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An explicit family of weight 2 CM Hecke eigenforms is built from hypergeometric data, with exact L-values and Jacobi sum formulas for the Fourier coefficients.","lead":"This paper constructs an explicit family of weight 2 modular forms with complex multiplication from 2F1(1) hypergeometric series, and computes their exact central L-values as algebraic multiples of beta values and Chowla-Selberg periods. The values connect to the Birch and Swinnerton-Dyer conjecture and to a finite-field model of Fourier coefficients via Jacobi sums.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's eigenform construction is the load-bearing hinge, and its proof is deferred to an omitted Lemma 3.5; the Appendix's explicit candidates make the gap finitely checkable.","rationale":"The paper makes explicit, numerically plausible claims, and the Appendix gives enough data for a decisive check; no internal inconsistency is apparent. However, Theorem 3.1 is the single hinge of the paper: every exact L-value in Section 5.3 is computed by integrating the identity f = Σ β_i K1, and the proof of that identity is not present here, being deferred to an omitted lemma and to an unreviewed preprint. The reader's weakest_assumption already identifies this as the primary fragile premise; I agree. The other gap, Theorem 4.2's relation (4.4) for Jacobi sums, matters for the coefficient formula and the Galois-representation claim, but the L-value theorem can survive even if the Jacobi-sum equality is weakened, whereas it cannot survive a failure of Theorem 3.1. The good news is that the gap is eminently checkable: comparing q-expansions against LMFDB up to the Sturm bound would verify the central construction for all listed cases, turning the conditional acceptance into a solid result for the finite set S1. I therefore see no reason to change the conditionality: the verdict stays as the reader gave it, pending that verification.","tokens_in":15208,"tokens_out":4554,"duration_ms":46457,"concrete_test":"For each row of Table 1, expand the eta-product combination on the left using Proposition 2.1 and compare its q-expansion with the LMFDB q-expansion of the listed newform. The two forms are equal if the first M coefficients agree, with M the Sturm bound for the relevant weight-2 level (the largest level is 2304, so M is finite and computationally modest). Because the LMFDB form is a newform, coefficient agreement proves the combination is a Hecke eigenform, closing the omitted part of Theorem 3.1 for every case used in the L-value tables. The same check should be run for the irrational-coefficient rows (3.4, 3.5, 8.x, 24.x) using the stated algebraic β_i. A mismatch in any row would disprove the corresponding entry in Theorem 1.1; a full match would make the central L-value claims rigorous for the finite set S1 without relying on [25].","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1's existence and L-value formula both rest on Theorem 3.1: the Hecke eigenform f_{r,s} is defined as a finite linear combination of K1(ri,si), and the exact value L(f,1) = Σ β_i L(K1(ri,si),1) is obtained in Corollary 5.1 by integrating that identity term by term. But the proof of Theorem 3.1 is not in this manuscript. Section 3.2 gives only a geometric sketch (modular-curve realization of the underlying Jacobian and CM) and then stops at Lemma 3.5, whose proof is 'virtually identical' to Lemma 3.2 of [25] and is omitted; the subsequent diagonalization of the Hecke matrix is likewise referred to [25], an unreviewed preprint. Consequently, the identification of the Appendix combinations with the listed LMFDB newforms is not derived from a written Hecke calculation. If Lemma 3.5 or the diagonalization fails for any of the listed families, the corresponding L(f,1) would not be the stated multiple of a beta value. Theorem 4.2's unproved Galois-conjugacy relation (4.4) is also a gap in the Jacobi-sum formula for a_p, but the L-value table depends on Theorem 3.1, making that the critical missing step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines weight-2 modular functions K1(r,s) attached to 2F1(1) hypergeometric data and, for a finite set S1, claims that certain finite linear combinations of Galois-conjugate K1's form explicit Hecke eigenforms with complex multiplication. It then derives exact formulas for the central L-values of these forms as algebraic multiples of beta values (equivalently, of Chowla-Selberg periods), gives a Jacobi-sum expression for the Fourier coefficients, and tabulates the resulting L-values together with LMFDB labels and inner-twist relations. The main theorem, Theorem 1.1, packages these assertions as the existence of an explicit CM newform f_{r,s} with a prescribed Deligne representation and L(f_{r,s},1) = α_{r,s} B(r, s-r-1/2).","tokens_in":15520,"tokens_out":9467,"duration_ms":92453,"significance":"If the construction is correct, the paper provides a genuinely explicit family of weight-2 CM Hecke eigenforms realized as linear combinations of K1-functions, with exact central L-values that are numerically verified against LMFDB and of direct interest for BSD-type computations and for the hypergeometric-modularity program of [1], [2], [25]. The explicit table of eigenforms and their L-values is a useful computational contribution. However, the central existence theorem is not proved in the manuscript; the proof is deferred to an unreviewed preprint and to an omitted case-by-case check. The significance is therefore conditional on filling those gaps.","major_comments":[{"comment":"Theorem 3.1 is the load-bearing assertion that a finite linear combination of the K1(ri,si) is a Hecke eigenform. The proof is not given: Lemma 3.5 is stated with the statement that its proof is 'virtually identical' to Lemma 3.2 of [25] and is omitted, and the subsequent diagonalization of the Hecke matrix is referred to [25] 'almost word for word'. This matters because Corollary 5.1 obtains L(f,1) by integrating the identity in Theorem 3.1 term by term, and Table 1 identifies the resulting f with specific LMFDB newforms. As the manuscript stands, the existence and L-value formula in Theorem 1.1 are unproven, depending on an unreviewed preprint. Since the set S1 is finite, the missing Hecke-operator calculation is checkable and should be supplied in the revision.","section":"§3.2, Theorem 3.1 and Lemma 3.5"},{"comment":"The proof of the exact Jacobi-sum formula for a_p(f_{r,s}) relies on the Galois-conjugacy relation (4.4), which is asserted but not proved. The text explicitly says that because there is no explicit formula for I_{r,s}, 'there is not an easy way to show that the automorphisms σ_j switch as in equation (4.4) explicitly as in [2], but we can check this case by case as well.' No such checks are presented. Without (4.4), the argument that v_{r,s}=0 from p | b_i and the bound |v_{r,s}| < 4√p is incomplete, because the congruence v_{r,s}≡0 mod p is needed for all Galois conjugates, which is exactly what (4.4) supplies. This gap should be closed for the finite list of pairs in S1.","section":"§4, Theorem 4.2 and Eq. (4.4)"},{"comment":"Lemma 5.2 states that Jnew(R,S) is isogenous over Q to E^{φ(M)} for an elliptic curve E with CM by Q(√-d), citing Shimura's Theorem 1.6 [26]. This is a strong claim: a Q-simple CM abelian variety of dimension greater than 1 is not generally isogenous to a power of an elliptic curve over Q; its endomorphism algebra is typically a CM field of degree 2·dim. For M=24, where φ(M)/2=4, the surrounding discussion (e.g., the Hecke field Q(ζ12) for the form 576.2.d.c) suggests that the relevant abelian variety has quartic endomorphisms, not a matrix algebra over an imaginary quadratic field. The cited theorem does not obviously yield the claimed isogeny in all cases needed, and the period conclusion B(r,s-r-1/2) ∈ Ω_{-D}·Qbar depends on this step. The proof of Lemma 5.2 needs to be substantiated or restricted to the cases where the isogeny actually holds.","section":"§5, Lemma 5.2"}],"minor_comments":[{"comment":"The proof of Proposition 2.1, which establishes the eta-product expression, holomorphy, and level of K1(r,s), is omitted with the comment that it is 'very similar' to the K2(r,s) case in [1]. Because this proposition underlies the modularity and the L-value computation, a proof or a precise reference to a published proof should be included.","section":"§2, Proposition 2.1"},{"comment":"There are several notation inconsistencies in the statement of Theorem 3.1 and Corollary 5.1: the final term is written as 'β_{n-1}K2(r_n, q_n)' although all functions are K1, and the notation 'q' appears in place of 's' in the L-values (e.g., 'L(r1,q1)' is listed twice). The sums should use a consistent indexing over the conjugates (ri,si).","section":"Theorem 3.1 and Corollary 5.1"},{"comment":"The entry for 3.2 is written as 'K2(1/6, 5/6)', but only K1 functions are defined in the paper; the intended expression is presumably K1(1/6, 5/6). This needs to be corrected to avoid ambiguity.","section":"Table 1, row 3.2"},{"comment":"In the proof of Lemma 4.1, the statement 'π_p^{p-1} = -p by definition' is imprecise; this identity is a standard consequence of the Gross-Koblitz formula, and should be stated as such.","section":"§4, proof of Lemma 4.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is effectively an extended announcement: the key existence theorem (Theorem 3.1) and the Galois-conjugacy relation (4.4) are deferred to an unreviewed preprint [25] and to an unperformed case check. I recommend requiring the author to include a complete proof of Theorem 3.1 for the finite list in S1 and to display the case-by-case verification of (4.4), or otherwise to restrict the claims to the cases that are fully proved. The LMFDB numerical agreement is a useful sanity check but does not replace the missing arguments. Also, the isogeny claim in Lemma 5.2 needs careful verification; the current citation to Shimura's theorem appears insufficient for the generality in which it is stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper does something genuinely useful. It gives explicit weight-2 CM newforms as combinations of K1 hypergeometric functions, a Jacobi-sum formula for their Fourier coefficients, and exact central L-values expressed as algebraic multiples of beta values and Chowla-Selberg periods. The modularity itself is already known from Darmon and Ribet, and the author says so plainly. The new content is the explicit construction, the Jacobi-sum description, and the L-value table, which was checked numerically against LMFDB. That verification is real evidence and makes the claims concrete and finitely checkable.\n\nThe main structural issue is that Theorem 3.1, the load-bearing result that the K1 combinations are Hecke eigenforms, is not proved in this manuscript. Proposition 2.1 and Lemma 3.5 are stated without proof; the diagonalization step is referred to the author's earlier preprint [25], which is not peer-reviewed. Since Corollary 5.1 integrates Theorem 3.1 term by term to get the L-values, the entire L-value table rests on that missing calculation. This is not a circularity—the L-values are derived from explicit beta integrals and known period relations—but it is an omitted proof of a central claim. The appendix's explicit candidates and LMFDB labels make the gap checkable, and the paper says the L-values were verified numerically, which is genuine supporting evidence. Still, the written proof is not there.\n\nA smaller but real gap is Theorem 4.2: the Galois-conjugacy relation (4.4) for Jacobi sums is asserted, and the paper says it can be checked case by case, but no checks are shown. That affects the coefficient formula, not the L-values directly, so it is less critical.\n\nOn balance, the work is honest and mostly clearly written. It credits prior results properly, and the method is a legitimate extension of the hypergeometric modularity framework to the weight-2 case. The soft spots are omissions, not obviously wrong mathematics. For a reader interested in hypergeometric L-values, BSD computations, or CM forms, this is worth engaging with. It deserves peer review, but the referee should demand that the Hecke calculation be written out or supplied as a verified computation, and that the case-by-case checks for (4.4) be documented.","headline":"Explicit weight-2 CM eigenforms and exact L-values, but the central Hecke-eigenform proof is deferred to an unreviewed preprint, making the paper a strong checkable claim awaiting completion.","tokens_in":16017,"tokens_out":2321,"would_cite":true,"duration_ms":22366,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F11","11F30","11F67","11G15","33C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs an explicit family of weight-2 Hecke eigenforms with complex multiplication whose central L-values are algebraic multiples of beta values.","keywords":["hypergeometric 2F1 series","modular forms","complex multiplication","Hecke eigenforms","central L-values","Jacobi sums","Chowla-Selberg periods","Fermat curves"],"falsifier":"Take the listed pair $(r,s)=(1/24,23/24)$, so $M=24$, and a prime $p\\equiv1\\pmod{24}$ such as $p=73$. Compute both sides of the claimed equality $a_p(f_{r,s})=-\\omega_p^{-4(p-1)r}(2)J_p(r,s-r-1/2)-\\omega_p^{-4(p-1)(1-r)}(2)J_p(1-r,1/2-s+r)$ exactly in $\\mathbb{Q}(\\zeta_{24})$, and separately compute $a_p$ from the explicit linear combination in the appendix after checking that it is an eigenform; any mismatch falsifies the central claim. Alternatively, verify the omitted Hecke step directly by applying $T_p$ to the four $K_1$ functions in that family for several small $p$ and confirming the stated eigenvector coefficients.","tokens_in":14977,"feed_emoji":"🔢","tokens_out":12562,"duration_ms":108236,"temperature":0.7,"pith_summary":"This paper establishes a finite dictionary between certain ${}_2F_1(1)$ hypergeometric series and weight-2 modular forms with complex multiplication. For each pair $(r,s)$ in the finite set $S_1$, it constructs an explicit Hecke eigenform $f_{r,s}$ whose Fourier coefficients are expressed through Jacobi sums and whose attached Galois representation is a piece of an induced Grössencharakter. The exact central value formula is $L(f_{r,s},1)=\\alpha_{r,s}B(r,s-r-1/2)$, an algebraic multiple of a $\\beta$ value and hence of a Chowla-Selberg period $\\Omega_{-D}$. If the construction is right, these are closed-form values rather than numerical approximations, and the tables in the paper record them for 32 twists of CM newforms.","feed_headline":"Exact L-values for 32 CM modular forms from hypergeometric series","feed_subtitle":"A finite family of hypergeometric periods yields explicit eigenforms whose central L-values are algebraic beta multiples.","key_machinery":"The central object is the family $K_1(r,s)(\\tau)$, a weight-2 modular differential built from the hypergeometric integrand $\\lambda^r(1-\\lambda)^{s-r-3/2}\\,d\\lambda/\\lambda$ by substituting the modular $\\lambda$ function $\\lambda(\\tau)$; it has the eta-product form $\\eta(\\tau/2)^{16s-8r-16}\\eta(2\\tau)^{8r+8s-12}/\\eta(\\tau)^{24s-32}$. The argument is carried by the Hecke-orbit machinery: conjugate families of these functions are preserved by the operators $T_p$ for primes coprime to 6, and diagonalizing the finite matrix of their action produces the eigenform $f_{r,s}$. The underlying geometric identification is that the associated new Jacobian $J_{\\mathrm{new}}(R,S)$ is a $\\mathbb{Q}$-simple factor of a Fermat-curve Jacobian and hence of a modular-curve Jacobian, which gives the CM structure and the Galois representation. On the finite-field side, the Jacobi sum $J_p(r,s)=\\sum_{k=1}^{p-1}\\iota(r)(k)\\iota(s)(1-k)$ is the étale realization of the same motive, and a p-adic gamma identity converts its values into the Fourier coefficients and the $\\beta$ periods.","core_discovery":"On the paper's own terms, the discovery is a complete explicit dictionary for a family of weight-2 CM newforms built from the functions $K_1(r,s)(\\tau)=2^{1-4r}\\lambda(\\tau)^r(1-\\lambda(\\tau))^{s-r-1/2}\\theta_3^4(\\tau)$, where $\\lambda$ is the modular $\\lambda$ function. A finite Galois-conjugate family of these functions is asserted to lie in one Hecke orbit, and diagonalizing the Hecke action yields the eigenform $f_{r,s}$. The Deligne representation attached to $f_{r,s}$ is a subrepresentation of $\\operatorname{Ind}_{G_M}^{G_{\\mathbb{Q}}}\\chi_{r,s}$, and for primes $p\\equiv 1\\pmod M$ the $p$-th Fourier coefficient is identified with a sum of two Jacobi sums, $a_p(f_{r,s})=-\\omega_p^{-4(p-1)r}(2)J_p(r,s-r-1/2)-\\omega_p^{-4(p-1)(1-r)}(2)J_p(1-r,1/2-s+r)$. Integrating the Hecke-orbit identity term by term gives $L(f_{r,s},1)=\\alpha_{r,s}B(r,s-r-1/2)$ with $\\alpha_{r,s}$ explicit, and Lemma 5.2 places this value in $\\Omega_{-D}\\overline{\\mathbb{Q}}$ by identifying the period with a period of a CM abelian variety.","pith_inferences":["If the omitted Hecke diagonalization is completed, the same method should produce eigenforms for every conjugate family in $S_1$; the table already suggests the pattern is systematic rather than a list of coincidences.","The Jacobi-sum formula gives a direct finite-field computation of $a_p$ that does not require constructing the eigenform, so the equality can be tested for primes far beyond the paper's examples; this would check the motivic dictionary independently of the unproved Hecke calculation.","The eta-product expression for $K_1(r,s)$ is valid for a wider range of rational exponents, so the construction plausibly extends outside $S_1$ whenever the form stays holomorphic; the obstruction is the level and cusp condition, not the algebraic mechanism.","In the rank-zero cases, each closed-form central value can be combined with known formulas for elliptic curve invariants to pin down the full Birch and Swinnerton-Dyer prediction, turning the table into a wholesale verification of the conjecture for the attached abelian varieties."],"forward_implications":["For every listed pair, $L(f_{r,s},1)$ is an algebraic multiple of a single Chowla-Selberg period, so the central value is a period of the attached CM abelian variety and can be written as an explicit product of gamma values.","The Fourier-coefficient formula gives a finite-field description of $a_p$ for primes $p\\equiv1\\pmod M$: the coefficient is a rational combination of Jacobi sums rather than a quantity found only by computing the form.","The method recovers four of the five weight-2 CM eta-product Hecke eigenforms and extends them to linear combinations of eta products that are themselves eigenforms, so the table supplies newforms not reached by a single eta product.","The exact L-values feed into the Birch and Swinnerton-Dyer framework: for the rank-zero curve attached to example 4.1, the value $\\Omega_{-4}$ with the general formula predicts the order of the Tate-Shafarevich group is 1.","Twist relations show that many of the listed forms differ by quadratic or character twists, so the 32 table entries organize into a smaller number of twist classes."],"supporting_citations":[{"why":"Supplies the hypergeometric modularity method and the definition of $K_1(r,s)$ and its conjugates.","marker":"[1]"},{"why":"Provides the weight-3 analogue and the special $K_1$ modularity result that this paper generalizes.","marker":"[2]"},{"why":"Contains the Hecke-operator proof to which Section 3 refers for the omitted diagonalization argument.","marker":"[25]"},{"why":"Lists the twelve weight-2 eta-product Hecke eigenforms, five with CM, extended here to linear combinations.","marker":"[20]"},{"why":"Realizes Fermat curves as modular curves, the step that identifies $J_{\\mathrm{new}}$ with a modular Jacobian factor.","marker":"[24]"},{"why":"Provides the hypergeometric abelian varieties $J_{\\mathrm{new}}$ and the basis of differentials used to relate periods to L-values.","marker":"[5]"},{"why":"Gives the period theorem that identifies the beta periods with algebraic multiples of CM elliptic-curve periods.","marker":"[26]"},{"why":"Shows that Jacobi sums are Grössencharaktere, the Galois-side input for the representation-theoretic statement.","marker":"[30]"},{"why":"Provides the p-adic gamma identity that converts Jacobi sums into Fourier-coefficient congruences.","marker":"[14]"}],"fun_headline_variants":["Exact L-values from hypergeometric series for CM forms","Linking hypergeometric series to modular forms: exact L-values","Jacobi sums and beta values give exact central L-values for CM forms","Weight-2 CM eigenforms: explicit L-values via hypergeometric series","Hypergeometric series yield exact L-values for CM modular forms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on the unproved Hecke-operator calculation that each finite conjugate family of hypergeometric functions diagonalizes to one eigenform, and on a Galois-conjugacy relation for Jacobi sums that the paper checks only case by case; if either fails for any listed pair, the L-value and Fourier-coefficient formulas are unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Exact L-values from hypergeometric series for CM forms","Linking hypergeometric series to modular forms: exact L-values","Jacobi sums and beta values give exact central L-values for CM forms","Weight-2 CM eigenforms: explicit L-values via hypergeometric series","Hypergeometric series yield exact L-values for CM modular forms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000589,"raw_usage":{"total_tokens":2764,"prompt_tokens":947,"completion_tokens":1817,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":563,"completion_tokens_details":{"reasoning_tokens":1725}},"tokens_in":563,"tokens_out":1817,"duration_ms":12091,"temperature":1.0,"reasoning_tokens":1725,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T23:45:47.304925+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the listed pair $(r,s)=(1/24,23/24)$, so $M=24$, and a prime $p\\equiv1\\pmod{24}$ such as $p=73$. Compute both sides of the claimed equality $a_p(f_{r,s})=-\\omega_p^{-4(p-1)r}(2)J_p(r,s-r-1/2)-\\omega_p^{-4(p-1)(1-r)}(2)J_p(1-r,1/2-s+r)$ exactly in $\\mathbb{Q}(\\zeta_{24})$, and separately compute $a_p$ from the explicit linear combination in the appendix after checking that it is an eigenform; any mismatch falsifies the central claim. Alternatively, verify the omitted Hecke step directly by applying $T_p$ to the four $K_1$ functions in that family for several small $p$ and confirming the stated eigenvector coefficients.","supporting_citations":[{"cited_title":"The Explicit Hypergeometric-Modularity Method I","cited_arxiv_id":"2404.00711","evidence_quote":"Supplies the hypergeometric modularity method and the definition of $K_1(r,s)$ and its conjugates."},{"cited_title":"Eta-quotients and elliptic cur ves","cited_arxiv_id":null,"evidence_quote":"Lists the twelve weight-2 eta-product Hecke eigenforms, five with CM, extended here to linear combinations."},{"cited_title":"Modular Functions and the Fermat Curves","cited_arxiv_id":null,"evidence_quote":"Realizes Fermat curves as modular curves, the step that identifies $J_{\\mathrm{new}}$ with a modular Jacobian factor."},{"cited_title":"Hypergeometric abelian varieties","cited_arxiv_id":null,"evidence_quote":"Provides the hypergeometric abelian varieties $J_{\\mathrm{new}}$ and the basis of differentials used to relate periods to L-values."},{"cited_title":"On elliptic curves with complex multipli cation as factors of the Jacobians of modular function fields","cited_arxiv_id":null,"evidence_quote":"Gives the period theorem that identifies the beta periods with algebraic multiples of CM elliptic-curve periods."},{"cited_title":"Gr¨ ossencharaktere","cited_arxiv_id":null,"evidence_quote":"Shows that Jacobi sums are Grössencharaktere, the Galois-side input for the representation-theoretic statement."},{"cited_title":"Gross and Neal Koblitz","cited_arxiv_id":null,"evidence_quote":"Provides the p-adic gamma identity that converts Jacobi sums into Fourier-coefficient congruences."}],"review_version":1}