Diagonal cycles and anticyclotomic twists of modular forms at inert primes

Luca Marannino

REVIEW 2 major objections 4 minor 7 references

Diagonal cycles and anticyclotomic twists of modular forms at inert primes

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that when the prime p is inert in the imaginary quadratic field, the nonvanishing of the Heegner theta element at the wild part of the anticyclotomic character forces the Bloch-Kato Selmer group to be exactly…

desk verdict The new p-inert Euler system construction is real and worth knowing, but the paper's main theorem is a conditional roadmap: the decisive Selmer-rank-one step is delegated to forthcoming Jetchev-Nekovář-Skinner theory and is not proved here. read the letter →

arxiv 2507.22755 v2 pith:Q2E6TWGP submitted 2025-07-30 math.NT

Luca Marannino This is my paper · ORCID

classification math.NT MSC 11F3311F6711R2311G40

keywords anticyclotomicEulersystemsdiagonalcyclesBloch-KatoSelmergroupsmodularformsinertprimesp-adicL-functionsexplicitreciprocitylawHeegnerthetaelements

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 establishes a p-adic criterion for the Bloch-Kato conjecture in analytic rank one, in the definite setting where a fixed prime $p$ is inert in the imaginary quadratic field $K$. The criterion is that the nonvanishing of the Heegner $\theta$ element $\Theta^{\mathrm{Heeg}}_\infty(f,\chi_t)$ at the wild part $\chi_-$ of an anticyclotomic character $\chi$ forces the Bloch-Kato Selmer group $\operatorname{Sel}^{\mathrm{BK}}(K,V_{f,\chi})$ to be exactly one-dimensional. The argument builds an anticyclotomic Euler system from diagonal cycles, following a construction that previously worked mainly when $p$ splits, and connects its bottom class to $p$-adic $L$-values through an explicit reciprocity law and a factorization of triple-product $p$-adic $L$-functions. A reader would care because a one-dimensional Selmer group is exactly the Bloch-Kato prediction in analytic rank one, and the theorem turns a computable $p$-adic $L$-value condition into that conclusion even though the usual Heegner-point and Beilinson-Flach methods are unavailable at inert primes.

What carries the argument

The mechanism is the split anticyclotomic Euler system built from improved diagonal classes. Adapting the construction of [CD23] to the case where $p$ is inert, the paper obtains classes $\kappa_{n,f,\eta_i}$ for squarefree $n$ divisible only by primes splitting in $K$, with tame norm relations at those primes; the vertical direction at $p$ is unavailable, but the bottom class $\kappa_{f,\eta_i}$ remains meaningful. The bottom class is detected by the explicit reciprocity law of [BSV20]: the Bloch-Kato logarithm of the refined diagonal class is a ratio of Petersson products, equal to a value of the improved triple-product $p$-adic $L$-function of [Mar24]. This $L$-function factorizes as $\pm A_{\mathbf f \mathbf g \mathbf h}\cdot(\alpha_-(\Theta^{\mathrm{Heeg}}_\infty(\mathbf f,\alpha_t))\hat\otimes \beta_-(\Theta^{\mathrm{Heeg}}_\infty(\mathbf f,\beta_t)))$, and the author chooses auxiliary characters so that $\alpha=\chi_t$, $\beta=\delta^2$, and $L(f/K,\delta^2,k/2)\neq 0$, which forces the nonvanishing of the bottom class exactly when $\chi_-(\Theta^{\mathrm{Heeg}}_\infty(f,\chi_t))\neq 0$.

What would settle it

Take a concrete weight-2 newform $f$, an imaginary quadratic field $K$, and a prime $p$ inert in $K$ satisfying Assumption 1.1, compute $\chi_-(\Theta^{\mathrm{Heeg}}_\infty(f,\chi_t))$ by a modular-symbol or overconvergent computation, and independently compute $\dim_E\operatorname{Sel}^{\mathrm{BK}}(K,V_{f,\chi})$ by descent or an existing algorithm; a single instance with a nonzero $\theta$ value and Selmer dimension different from 1 would refute Theorem 1.2.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.2: under Assumption 1.1 (which includes $p$ inert in $K$, $p$ large, $f$ ordinary with big image and non-CM, $p \nmid h_K$, $N_f$ squarefree and definite, and a conductor condition on $\chi$), if $\chi_-(\Theta^{\mathrm{Heeg}}_\infty(f,\chi_t)) \neq 0$, then $\dim_E \operatorname{Sel}^{\mathrm{BK}}(K,V_{f,\chi}) = 1$. The proof chains three ingredients: the construction of a split anticyclotomic Euler system $\{\kappa_{n,f,\eta_i}\}$ from improved diagonal classes, with tame norm relations for primes splitting in $K$; the assertion (Corollary 3.29, cited to forthcoming work on split anticyclotomic Euler systems) that nonvanishing of the bottom class $\kappa_{f,\eta_i}$ forces the corresponding Bloch-Kato Selmer group to be one-dimensional; and an explicit reciprocity law showing that $\kappa_{f,\eta_1}\neq 0$ follows from the nonvanishing of a specialized triple-product $p$-adic $L$-function, which in turn factorizes as an essentially nonzero factor times $\alpha_-(\Theta^{\mathrm{Heeg}}_\infty(f,\alpha_t)) \hat\otimes \beta_-(\Theta^{\mathrm{Heeg}}_\infty(f,\beta_t))$. Choosing auxiliary characters so that $\alpha=\chi_t$ and $\beta=\delta^2$ with $L(f/K,\delta^2,k/2)\neq 0$ isolates the Heegner $\theta$ element in the statement.

Load-bearing premise

The load-bearing premise is the unproved consequence of a forthcoming general theory of split anticyclotomic Euler systems that a nonzero bottom class forces the corresponding Bloch-Kato Selmer group to be one-dimensional; if that theory does not appear as assumed, the paper's main theorem does not follow from the arguments given.

Editorial extensions

If this is right

If the theorem is correct, then for every triple $(f,\chi,p)$ satisfying Assumption 1.1, the nonvanishing of $\chi_-(\Theta^{\mathrm{Heeg}}_\infty(f,\chi_t))$ gives a proof that the Bloch-Kato Selmer group in analytic rank one has dimension exactly one.
The explicit reciprocity chain gives a practical way to verify rank-one Bloch-Kato: compute a $p$-adic Heegner theta value rather than compute a Selmer group directly.
The construction provides anticyclotomic Euler system classes at inert primes, at least in the split direction, and the author notes that the squarefree and conductor assumptions can be relaxed, so the method should extend to broader classes of levels and characters.
The appearance of the Heegner theta element at the wild part as the sole arithmetic input suggests that the Iwasawa main conjecture for anticyclotomic twists in the $p$-inert case can be approached through this Euler system, a direction the paper explicitly hopes to pursue.

Reading between the lines

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

A direct extension would be to run the same nonvanishing criterion through the $\eta_2$ component: since Corollary 3.29 applies to both $\kappa_{f,\eta_1}$ and $\kappa_{f,\eta_2}$, a choice of auxiliary characters with $\xi_1\xi_2$ equal to a different tame ring class character would yield a one-dimensional Selmer statement for the conjugate twist, not isolated in the main theorem.
Remark 3.21 computes the would-be norm relation at inert primes and finds a congruence modulo $q^2-1$; if this congruence can be promoted to an actual norm relation, inert primes could be added to the Euler system, turning the split system into a full anticyclotomic Euler system and possibly giving control of the vertical direction at $p$ as well.
The author states that Assumptions 4.3(ii)-(iii) can be relaxed; a testable extension would be to redo the argument with $N_f^+$ not squarefree or with inert primes allowed in the conductor of $\xi_i$, which would widen the theorem's range.
Because the criterion is a concrete $p$-adic $L$-value condition, a computational implementation (for instance on elliptic curves) could provide numerical evidence or counterexamples for the Bloch-Kato conjecture in this setting before the full split anticyclotomic Euler system theory is written down.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a solid research announcement with a load-bearing missing proof. The genuinely new content is the direct construction of the anticyclotomic Euler system for general balanced triples without p-adic interpolation, explicitly to handle the p-inert case, plus the nonvanishing-to-Selmer-rank-one criterion that comes out of it. Theorem 3.20's tame norm relations at split primes are actually proved, with denominators and test vectors handled carefully. The explicit reciprocity law section is a clean adaptation of BSV20, and the paper is honest about what it imports and where the gaps are. Credit where due: the construction is coherent, the write-up is clear, and the author does not hide the dependencies.

The soft spot is exactly where the stress test lands. Corollary 3.29 is the step that turns a nonzero bottom class into one-dimensionality of the Bloch–Kato Selmer group, and it is not proved. It is introduced with “Forthcoming work of Jetchev–Nekovář–Skinner … should lead to the following result,” citing an overview rather than a theorem. That is not a proof. Without it, Theorem 1.2 does not follow from the arguments in this paper. The paper itself notes in Section 1 that because p is inert, it cannot construct the vertical direction, so it is genuinely unclear whether the announced JNS theory will cover this setting. This is a serious condition, not a cosmetic one. The reliance on the unpublished [Mar24, Thm 5.25] for the factorization in Theorem 4.5 is a second dependency, though that paper is on arXiv and the argument there is at least concrete enough to check.

If I were refereeing this, I would not desk-reject. The Euler system construction itself is worth having, and the p-inert case is exactly where the subject needs new tools. But I would send it back with a clear instruction: either prove Corollary 3.29, or reframe the paper as a conditional result with that implication explicitly labelled as an assumption. As written, the main theorem is a roadmap, not a proof. The paper is for specialists in anticyclotomic Iwasawa theory and diagonal cycles; a careful reader will get value from the construction and the explicit norm relations. I would not cite Theorem 1.2 as established until the missing bridge appears. Yes to peer review, with conditions.

Referee Report

2 major / 4 minor

Summary. The paper constructs an anticyclotomic Euler system over an imaginary quadratic field K at a prime p inert in K, starting from diagonal classes in the style of Castella–Do, and uses it to prove Theorem 1.2: under Assumption 1.1, the nonvanishing of the evaluation of the Chida–Hsieh Heegner theta element at the wild part χ_− of an anticyclotomic character χ implies dim_E Sel^BK(K, V_{f,χ}) = 1. The proof chain is: build classes with tame norm relations (Theorem 3.20, Proposition 3.19), show they lie in Bloch–Kato Selmer groups (Proposition 3.24), detect nonvanishing of the bottom class through the explicit reciprocity law for diagonal classes (Corollary 5.8) and the factorization of triple product p-adic L-functions (Theorem 4.5, Lemma 6.3), and finally convert nonvanishing of the bottom class into one-dimensionality of the Selmer group via Corollary 3.29, which is stated as a consequence of forthcoming work of Jetchev–Nekovář–Skinner.

Significance. If the deferred inputs are supplied, the paper would give a genuinely new criterion in the p-inert anticyclotomic setting, a case where full Euler systems with a vertical direction are not available. The paper's concrete construction of the split anticyclotomic collection in general balanced weights, including the modification of diagonal classes to remove the ϕ(n) factor (Remark 3.15), and its explicit norm relations (Theorem 3.20) are useful contributions. The final criterion is falsifiable and parameter-free, and the paper is honest about many of its technical hypotheses. However, the advertised main theorem is not self-contained: the decisive Selmer-rank-one step and the factorization of the triple product p-adic L-function are both external inputs, one explicitly forthcoming and one from an unpublished preprint. The result is therefore best regarded as conditional until those inputs are available or proved.

major comments (2)

[Section 3.4, Corollary 3.29] Corollary 3.29 is the only step that passes from nonvanishing of the bottom class to one-dimensionality of the Bloch–Kato Selmer group, yet it is not proved in this paper. It is introduced by the sentence "Forthcoming work of Jetchev–Nekovář–Skinner ... should lead to the following result" and the supporting citation is [ACR23, Section 8.1], an overview rather than a theorem. What is proved directly is weaker: Theorem 3.20 gives tame norm relations for split primes only, and Proposition 3.24 puts the classes in the relevant Selmer groups. Neither statement implies dim_E Sel^BK(K, V_{f,η_i}) = 1. Moreover, because p is inert, the paper itself notes in Section 1 that it cannot construct a full Euler system with vertical direction; if the JNS theory requires vertical classes, the hypothesis of Corollary 3.29 is not met. Please either prove Corollary 3.29, or state Theorem 1.2 explicitly as conditional on this implication, or wait for and cite the JNS theory.
[Section 4.3, Theorem 4.5] The factorization identity L^f_{p,ac}(f,g,h) = ±A_{fgh}(α_-(Θ^Heeg_∞(f,α_t)) ⊗ β_-(Θ^Heeg_∞(f,β_t))) is imported as "cf. Theorem 5.25 of [Mar24]", where [Mar24] is an unpublished preprint by the author. This identity is load-bearing: Lemma 6.3 uses it to convert nonvanishing of the specialized triple product p-adic L-function into nonvanishing of the Chida–Hsieh theta element, and Section 6 then combines Lemma 6.3, Corollary 5.8, and Corollary 3.29 to prove Theorem 1.2. Since [Mar24] is not yet published and its proof is not reproduced here, the main theorem is conditional on an external, non-refereed input. Please state clearly the status of [Mar24], or include a proof of the factorization under the exact hypotheses used in this paper.

minor comments (4)

[Theorem 3.20, eq. (3.9)] In the norm relation (3.9), the class on the right-hand side should be κ_{n,f,η_i}, not κ_{nq,f,η_i}; as printed the relation would be trivial and inconsistent with the proof that follows.
[Section 6, first paragraph] The phrase "the p-adic avatar of j" should read "the p-adic avatar of χ", since j is the infinity-type exponent and not a character.
[Notation 3.9] The symbol m is used both for the weight m in Section 3.3 and for the ideal m = cn_+(n_-) in Notation 3.9; this clash is confusing and the ideal should be renamed.
[Page 1 header] The running header on the first page reads "A T INER T PRIMES"; it should be "AT INERT PRIMES".

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.2 is derived from independent ingredients; the deferred JNS implication in Corollary 3.29 is a missing proof, not a circular reduction.

full rationale

The claimed derivation chain is: nonvanishing of the Heegner theta element at chi_- (Lemma 6.3) is equivalent, via the factorization Theorem 4.5, to nonvanishing of the specialized triple-product p-adic L-function; Corollary 5.8 (from the explicit reciprocity law of [BSV20]) turns that nonvanishing into nonvanishing of the bottom class kappa_{f,eta1}; Corollary 3.29 would then convert kappa_{f,eta1} != 0 into dim_E Sel^BK(K, V_{f,eta1}) = 1. Inspecting the equations, no step is circular: the theta element and the Selmer group are not defined in terms of each other, no parameter is fitted to the predicted Selmer dimension, and the Euler-system classes of Theorem 3.20 are constructed directly rather than chosen to force the conclusion. The two dependencies that deserve flagging are: (1) Corollary 3.29 is explicitly not proved here - the paper says 'Forthcoming work of Jetchev-Nekovar-Skinner ... should lead to the following result' - so the Selmer-rank-one implication rests on an unpublished theory; this is an omitted proof/gap in the derivation, but it is not circularity. (2) The factorization Theorem 4.5 is imported from the author's own [Mar24, Thm 5.25] and is load-bearing for Lemma 6.3; however, it is a parameter-free theorem with stated hypotheses that do not include Theorem 1.2, and the current paper uses it as a black box. Under the stated rules, such independent prior-work citations do not raise the circularity score. No step reduces to its own input by construction, so the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central theorem rests on a stack of imported results: higher-weight Ihara/Wiles, [Mar24]'s factorization, [CH18]'s nonvanishing of an auxiliary L-value, and the forthcoming JNS Euler-system theory. The most fragile input is the JNS theory, because it is not available in any citable form.

assumptions (7)

standard math Higher-weight Ihara's lemma (Diamond, [Dia91, Lemma 3.2])
Used in the proof of Theorem 3.4 to extend the patching isomorphisms of [LLZ15] from weight 2 to higher weights.
standard math Higher-weight Wiles theorem on Galois representations (Faltings-Jordan, [FJ95, Theorem 2.1])
Invoked in Theorem 3.4 under the assumption p > nu to generalize [LLZ15]'s key theorem to higher weight modular forms.
standard math Rubin's Euler system norm-relation lemmas ([Rub00, Lemmas 9.6.1 and 9.6.3])
Used in Theorem 3.20 to repair the norm relations and to lift the classes kappa_{n,f,eta_i} to integral coefficients.
domain assumption Existence of an auxiliary split ring class character delta with L(f/K, delta^2, k/2) != 0 ([CH18, Theorem D])
Used in Section 6 to ensure the second factor in the factorization theorem does not vanish, which is needed to link the triple product L-value to Theta^Heeg.
domain assumption Factorization theorem for generalized triple product p-adic L-functions ([Mar24, Theorem 5.25])
Theorem 4.5 is quoted verbatim from the author's previous work [Mar24]; it is a nontrivial unpublished input on which Lemma 6.3 and Theorem 1.2 rest.
ad hoc to paper Split anticyclotomic (JNS) Euler system theory implying dim Sel = 1 for nonvanishing bottom class
Corollary 3.29 is explicitly stated as a consequence of the forthcoming work of Jetchev-Nekovar-Skinner; no proof or available reference is given, making this the central unproved input of the paper.
domain assumption Standing technical hypotheses (not CM, big image, p-distinguished, p not dividing h_K, p > max{k-2,j+1}, N_f squarefree, conductor conditions)
Assumption 1.1 lists the constraints under which Theorem 1.2 is proven; they are imposed rather than derived.

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Pith. "Pith review of Diagonal cycles and anticyclotomic twists of modular forms at inert primes." pith.science (2026). https://pith.science/paper/Q2E6TWGP

@misc{pith2026250722755,
  author       = {Pith},
  title        = {Pith review of: Diagonal cycles and anticyclotomic twists of modular forms at inert primes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q2E6TWGP}},
  note         = {Machine review of arXiv:2507.22755}
}

abstract

We revisit the construction of Castella and Do of an anticyclotomic Euler system for the $p$-adic Galois representation of a modular form, using diagonal classes. Combining this construction and some previous results of ours, we obtain new results towards the Bloch--Kato conjecture in analytic rank one, assuming that the fixed prime $p$ is inert in the relevant imaginary quadratic field.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

7 extracted references · 4 canonical work pages

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Iwasawa theory of twists of elliptic modular forms over imaginary quadratic fields at inert primes

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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Iwasawa theory of twists of elliptic modular forms over imaginary quadratic fields at inert primes

arXiv: 2008.08411 [math.NT]. url: https://arxiv.org/abs/2008.08411 (cit. on p. 2). [BSV20] Massimo Bertolini, Marco Adamo Seveso, and Rodolfo Venerucci. “Diagonal classes and the Bloch-Kato conjecture”. In: M¨ unster J. Math.13.2 (2020), pp. 317–352. issn: 1867-5778 (cit. on pp. 2, 22, 23). [BSV22] Massimo Bertolini, Marco Adamo Seveso, and Rodolfo Veneru...

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