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A $p$-Converse theorem for Real Quadratic Fields

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Pith's one-line read The paper proves that for an elliptic curve over a real quadratic field with split multiplicative reduction at an inert prime p>5, rank-one Mordell–Weil group plus finite p-primary Tate–Shafarevich group forces the Hasse–Weil L-function…

desk verdict First p-converse over a real quadratic field in the multiplicative case, built on a coherent chain of deep published results, but the genuinely new non-base-change case rests on an unproved minimal modular lifting hypothesis. read the letter →

arxiv 2504.21799 v3 pith:R7DHG3DT submitted 2025-04-30 math.NT

Muskan Bansal , Somnath Jha , Aprameyo Pal , Guhan Venkat This is my paper · ORCID
classification math.NT MSC 11G4011G0511R23
keywords ellipticcurvesrealquadraticfieldsp-conversetheoremp-adicL-functionsIwasawamainconjectureSelmercomplexesTate-ShafarevichgroupHilbertmodularforms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to prove a p-converse theorem for elliptic curves over real quadratic fields: if the Mordell–Weil group has rank one and the p-primary Tate–Shafarevich group is finite, then the complex L-function vanishes to order one at s=1. The proof works when p>5 is inert in the field, the curve has split multiplicative reduction above p, and two representation-theoretic hypotheses hold. Because the reverse implication is already known for totally real fields, the result upgrades rank-one-plus-finite-Sha to an actual equivalence with analytic rank one. This is the first such converse established for a number field beyond the rationals.

What carries the argument

The carrying object is the Hida family, a p-adic analytic family of Hilbert modular forms that specializes to the eigenform attached to E. Associated to it are a four-variable p-adic L-function and a Selmer group whose dual has a characteristic ideal. The argument is a four-inequality chain: the Iwasawa main conjecture divisibility bounds the order of the p-adic L-function above by a Selmer length; a second-derivative formula bounds that order below by four; Selmer complexes with a p-adic weight pairing, plus Kummer theory, pin the relevant Selmer length to two; and a comparison lemma shows the imprimitive Selmer length is at most the strict length plus two. These force the p-adic order to be exactly four, which is equivalent to order-two vanishing of the complex L-function over a chosen quadratic extension, and the desired order-one vanishing over F follows from a suitable choice of that extension.

What would settle it

Take the paper's explicit example, the base change of the elliptic curve X0(11) to Q(\sqrt{2}) with p=11, and compute the Hasse-Weil L-function near s=1; the theorem predicts L(E/F,1)=0 with a nonzero first derivative, so a computed order of vanishing different from 1 would refute the theorem. Alternatively, compute the order of vanishing at k=2 of the central critical p-adic L-function L_cc_p(f/K,k) for the auxiliary K, which the proof forces to be exactly 4.

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Extended reading notes

Core claim

The central discovery is a conditional proof that, under the hypotheses (irred) and (MML), the conditions rank_Z E(F)=1 and #III(E/F)_{p^\infty}<\infty imply ord_{s=1}L(E/F,s)=1. The proof moves to a carefully chosen imaginary quadratic extension K/F, proves that the central critical p-adic L-function of the associated Hida family vanishes to order exactly four at weight 2, translates this into order-two vanishing of the complex L-function over K, and then uses the factorization L(E/K,s)=L(E/F,s)L(E^K/F,s) together with a nonvanishing twist to conclude order-one vanishing over F. The paper also applies this to elliptic curves over Q, removing a technical hypothesis from an earlier p-converse theorem.

Load-bearing premise

Everything hinges on a technical existence hypothesis called (MML): a certain p-adic modular form must lift the residual Galois representation in a minimal way, and without it the Iwasawa-main-conjecture input cannot be invoked.

Editorial extensions

If this is right

  • If the theorem is correct, then for every curve satisfying the hypotheses, rank-one plus finite p-primary Sha is equivalent to ord_{s=1}L(E/F,s)=1, not merely implied by it.
  • The proof yields a precise computation: the central critical p-adic L-function has order exactly 4 at weight 2 for the auxiliary quadratic extension K.
  • Over Q, the result removes the auxiliary-prime hypothesis from earlier p-converse theorems for split multiplicative primes p>5: rank-one plus finite Sha suffices to conclude analytic rank one.
  • The strategy provides a template for extending p-converse theorems to totally real fields when the corresponding modularity and Leopoldt-type hypotheses hold.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: The same proof scheme should adapt to any totally real field satisfying Leopoldt's conjecture and modularity, so the real-quadratic restriction is probably not essential.
  • Inference: The p-adic weight pairing constructed here might give a route to rank-one p-converse statements for curves with additive or good reduction at p, once the local pairings are replaced appropriately.
  • Inference: The condition (MML) is the real bottleneck; if minimal modular liftings are proven to exist in greater generality, the theorem immediately covers all such curves without further change.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proves a rank-one p-converse for elliptic curves over real quadratic fields, under two global hypotheses called (irred) and (MML). Assuming E/F has split multiplicative reduction at an inert prime p>5, rank one, and finite Tate-Shafarevich group, the authors show ord_{s=1} L(E/F,s)=1. The proof combines Wan's Iwasawa main conjecture for Hilbert modular forms, Mok's p-adic L-function and Heegner point results, and Nekovár's Selmer complexes. The authors also deduce a p-converse over Q and give a worked example with F=Q(√2), p=11, E=X0(11).

Significance. If the hypotheses are met, the result is a genuine extension of the Venerucci/Skinner-Zhang circle of ideas from Q to a real quadratic base field, and the chain of inequalities in §§4–9 is coherent and uses deep published results rather than fitted parameters. The paper's main contribution is the exact computation of a Selmer-complex length (Theorem 7.1), which combined with Wan's divisibility forces the order of vanishing to be exactly 4. The significance is tempered by two limitations: the theorem is explicitly conditional on the unproved minimal modular lifting hypothesis (MML), and the only numerical example verifying (MML) is a base change from Q, so the genuinely new real-quadratic case is not yet demonstrated. I find no internal contradiction in the main chain, but one load-bearing step in the application over Q needs additional justification.

major comments (3)
  1. [Section 10, proof of Theorem 10.1] The deduction that the hypotheses of Theorem 1.3 hold for the base-changed curve E/F is incomplete. Theorem 1.3 requires (irred) for E/F, but the proof only assumes (irred) for E/Q. Absolute irreducibility of a mod p representation does not automatically survive restriction to a quadratic field: a dihedral representation induced from a quadratic field becomes reducible over that field. The proof should either prove that ρ̄_{E,F} is absolutely irreducible for the particular F constructed, or explicitly choose F so as to avoid the finitely many quadratic fields over which the residual representation becomes reducible. This is load-bearing because Proposition 4.1 uses (irred) to obtain the vanishing of an H^0 that is essential for the control theorem.
  2. [Section 3.3, Theorem 3.1(3), and Section 11] The main theorem is conditional on (MML), the existence of a minimal modular lifting of ρ̄_f, and this hypothesis is not proved in any genuinely new setting. The example in Section 11 is the base change of X0(11) from Q, so the required minimal lifting is inherited from Q. Since (MML) is exactly the input that supplies Wan's divisibility char ⊂ (L) and hence the inequality in Corollary 4.2, the advertised 'beyond Q' theorem remains conditional on an unverified modular lifting input for non-base-change real quadratic curves. This is not an internal contradiction, but the limitation should be stated prominently in the introduction and abstract, and the paper would be substantially strengthened by an unconditional instance of (MML) for a curve that does not arise as a base change.
  3. [Section 8, Remark 8.2(c), and Section 9] The proof of Theorem 9.2 uses the assertion that p splits in K, which is equivalent to Ǩ(p)=1. This does follow from the conditions in Lemma 8.1 because nE=np, so the set of primes l dividing nED_F includes p and the condition ξ'(l)=1 for all such l forces Ǩ(p)=1. The manuscript should say this explicitly; as written, the reader is left to infer that the notation nE includes the inert prime p, and a misreading here would invalidate both Corollary 5.2 and the hypotheses of Theorem 7.1.
minor comments (4)
  1. [Abstract and Section 1] The Tate-Shafarevich group is typeset as 'ΠΠ' throughout the abstract and introduction; this should be corrected to a standard symbol such as III.
  2. [Corollary 4.2] The step invoking [SU14, Corollary 3.8] to pass characteristic ideals modulo the ideal pcc is very terse; a short explanation of why the reduction is compatible for these Selmer groups would improve readability.
  3. [Section 11] The example relies on LMFDB data for the rank and the finiteness of the Tate-Shafarevich group and for the 'maximal image' assertion. Please indicate whether these are rigorous proven computations or numerical/conjectural data, since the example is meant to certify all hypotheses.
  4. [Section 10] In the proof of Theorem 10.1, the notation NE=Np makes the condition ξ(q)=1 for q|NE include q=p; given that the later argument requires ξ(p)=1 and ξ'(p)=-1, a parenthetical clarification would prevent confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the p-converse is a conditional derivation from Wan's external Iwasawa main conjecture and an independent Selmer-length computation; the unproved (MML) hypothesis is a declared scope limitation, not a circular step.

full rationale

The paper's main implication, rank_E(E(F))=1 plus finite III(E/F)_p^∞ implies ord_{s=1} L(E/F,s)=1, is obtained by combining external results with an explicit computation, not by fitting a parameter to the target. Wan's Iwasawa main conjecture (Theorem 3.1) supplies the one-sided divisibility char ⊂ (L_S_K), Corollary 4.2 converts this into the inequality ord_{k=2} L^cc_p ≤ len_P(X^{S,cc}_{F∞}), and Mok's theorem gives the order-of-vanishing comparison with complex L-functions. The Selmer length len_P(X^cc_Gr)=2 is computed independently via Nekovar's p-adic weight pairing and Kummer theory, using the rank-one and finite-III hypotheses; no step defines the analytic objects in terms of the algebraic conclusion. The main theorem is explicitly conditional on (MML), the existence of a minimal modular lifting, which is stated as an assumption in Theorem 3.1(3) and is not proved for a general real quadratic field; Section 11 verifies it only in a base-change example. This is a genuine scope limitation and a correctness risk for the advertised unconditional setting, but it is not circularity because the paper never claims to prove (MML) and does not sneak it in as a prediction. All load-bearing cited theorems are by other authors, so there is no self-citation chain forcing the result. Accordingly, no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim is conditional on (irred) and (MML) and on several published theorems. No numerical free parameters are fitted to the target result, and no new entities are postulated. The main external inputs are Wan's Iwasawa main conjecture, Mok's p-adic L-function results, Nekovar's Selmer complex theory, Friedberg-Hoffstein nonvanishing, and Zhang's Gross-Zagier theorem.

assumptions (6)
  • standard math Elliptic curves over real quadratic fields are modular (FLHS15).
    Used to associate a Hilbert modular form f to E/F in Section 1.2.
  • domain assumption Wan's Iwasawa main conjecture for Hilbert modular forms, in the form of Theorem 3.1.
    Provides the divisibility char^S_{K-infinity}(f) subset (L^S_K(f)) used in Corollary 4.2.
  • standard math Mok's results on the two-variable p-adic L-function and its central derivative (Theorem 5.1).
    Gives the lower bound ord>=4 and the equivalence ord=4 iff complex order 2 in Corollary 5.2.
  • standard math Nekovar's Selmer complex formalism, including Cassels-Tate pairings (Nek06).
    Underpins the p-adic weight pairing and Theorem 7.1.
  • ad hoc to paper The residual representation rho_f is absolutely irreducible.
    Hypothesis (irred), used in Proposition 4.1 and elsewhere to ensure vanishing of H^0.
  • ad hoc to paper There exists a minimal modular lifting of rho_f.
    Hypothesis (MML), one of the conditions of Theorem 3.1; not proven in general.

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Pith. "Pith review of A $p$-Converse theorem for Real Quadratic Fields." pith.science (2026). https://pith.science/paper/R7DHG3DT

@misc{pith2026250421799,
  author       = {Pith},
  title        = {Pith review of: A $p$-Converse theorem for Real Quadratic Fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R7DHG3DT}},
  note         = {Machine review of arXiv:2504.21799}
}
abstract

Let $E$ be an elliptic curve defined over a real quadratic field $F$. Let $p > 5$ be a rational prime that is inert in $F$ and assume that $E$ has split multiplicative reduction at the prime $\mathfrak{p}$ of $F$ dividing $p$. Let $\underline{III}(E/F)$ denote the Tate-Shafarevich group of $E$ over $F$ and $ L(E/F,s) $ be the Hasse-Weil complex $L$-function of $E$ over $F$. Under some technical assumptions, we show that when $rank_{\mathbb{Z}} \hspace{0.01mm} \hspace{1mm} E(F) = 1$ and $\#\Big(\underline{III}(E/F)_ {p^\infty}\Big) < \infty$, then $ord_{s=1} \ L(E/F,s) = 1$. Further, we give an application to a $p$-converse theorem over $\mathbb{Q}$.

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