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An anticyclotomic Euler system of Hirzebruch--Zagier cycles I: Norm relations and $p$-adic interpolation

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Pith's one-line read The paper constructs an anticyclotomic Euler system for Asai Galois representations attached to p-ordinary Hilbert modular forms over real quadratic fields, with classes varying in p-adic Hida families.

desk verdict A serious and plausible construction of an anticyclotomic Euler system for Asai representations, but the central norm relations are delegated to an unverified congruence and a suspicious duplicated formula in Proposition 5.6; needs a careful referee before the applications can be trusted. read the letter →

arxiv 2501.15336 v1 pith:PUM2FZ6P submitted 2025-01-25 math.NT

Raúl Alonso , Francesc Castella , Óscar Rivero This is my paper · ORCID
classification math.NT MSC 11F4111F8011G1811R23
keywords EulersystemsHirzebruch–ZagiercyclesAsairepresentationsHilbertmodularformsp-adicHidafamiliesanticyclotomicIwasawatheoryBloch–KatoconjectureSelmergroups
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 constructs an anticyclotomic Euler system for the four-dimensional Asai Galois representation attached to a p-ordinary Hilbert modular form over a real quadratic field. The classes are built from Hirzebruch–Zagier cycles, so they come from algebraic cycles, and they are shown to interpolate in p-adic Hida families. If the construction works, Kolyvagin's method turns the non-vanishing of one class into a rank-one bound on the Bloch–Kato Selmer group and a divisibility in the anticyclotomic Iwasawa main conjecture, giving new evidence for two central conjectures in the arithmetic of automorphic forms.

What carries the argument

The machinery is the system of generalized Hirzebruch–Zagier cycles: for each p-power level α, a modular curve is embedded diagonally into the product Y_1(p^α)×S_1(p^α) of a modular curve and a Hilbert modular surface, producing codimension-two cycles Δ_α[t0,t1] over the cyclotomic extension Q(ζ_{p^α}). Their étale Abel–Jacobi images, after applying Hecke idempotents and Atkin–Lehner maps, yield big cohomology classes κ_∞(F,G) valued in the tensor product of the big Galois representation of an elliptic Hida family and the big Asai representation of a Hilbert Hida family. Tame-level variants at n give classes whose corestrictions are controlled by degeneracy maps; the Euler-system norm relations are then derived from explicit relations among the cycles under these maps, with a standard lemma for Euler systems converting congruences modulo (q−1) into exact equalities.

What would settle it

Compute explicitly the modified classes κ^*_{∞,n}(F,G) attached to a concrete Hilbert modular form g and Hecke character ψ at a prime q that splits in both fields. If the corestriction congruence modulo (q−1) is not an equality (or if one of the classes fails integrality at q), then the lemma cannot be applied and the claimed Euler system norm relations of Theorem 5.7 fail.

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Extended reading notes

Core claim

The central claim is Theorem A (Theorem 5.7): under the hypotheses that p splits in K, p does not divide the class number of K, and g is p-ordinary, there exists a collection of classes κ_{ψ,g,n,∞} in the balanced Selmer group $Sel^{{bal}}$(K[np^∞],T) indexed by squarefree n whose prime factors split in both F and K, satisfying the Euler system norm relation cor_{K[nq]/K[n]}(κ_{ψ,g,nq,∞}) = P_q(V;$Fr_q^{{-1}}$)κ_{ψ,g,n,∞} for every such prime q. The collection is an anticyclotomic Euler system for the conjugate self-dual representation V = As(V_g)|_{G_K}($ψ_P^{{-1}}$)(2-l-k/2). The proof passes through two-variable big cohomology classes κ_∞(F,G) attached to a CM Hida family F and a parallel-weight Hida family G, specializing to the desired classes, and the norm relations are obtained from explicit relations among the underlying algebraic cycles under degeneracy maps.

Load-bearing premise

The load-bearing step is a lemma in the Euler-system formalism that upgrades a norm congruence modulo (q−1) to an exact equality; it applies only if the modified classes are integral and meet the required local conditions, and the proof asserts these hypotheses without a detailed check.

Editorial extensions

If this is right

  • If Theorem A holds, the non-vanishing of the class κ_{ψ,g} forces the balanced Bloch–Kato Selmer group Sel^{bal}(K,V) to be one-dimensional over E in the range k<2l (Theorem 7.1).
  • A non-torsion class κ_{ψ,g,∞} in the Iwasawa cohomology implies that both the balanced Selmer group and its Pontryagin dual have Λ_F-rank one, with a divisibility of characteristic ideals (Theorem 7.2).
  • The Euler system classes and their norm relations vary in p-adic Hida families, so the construction is compatible with specialization at arithmetic points.
  • The construction yields an anticyclotomic analogue of the Asai–Flach Euler system, where the base field is a real quadratic field and the tower is anticyclotomic over an imaginary quadratic field.

Reading between the lines

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

  • An explicit reciprocity law relating the big classes to the p-adic Asai L-function would turn the divisibility of Theorem 7.2 into an equality, a natural extension the authors plan for a sequel.
  • In the degenerate case where g is the base-change of an elliptic modular form, the Asai representation contains Sym^2 of the elliptic form as a summand; the Euler system may then be decomposed to attack the anticyclotomic Iwasawa main conjecture for the adjoint representation.
  • The congruence-to-equality step via the standard Euler-system lemma is the part most sensitive to integrality; an explicit computation of the modified classes at a single split prime would either confirm the norm relations or expose a missing hypothesis.
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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

4 major / 5 minor

Summary. The paper constructs an anticyclotomic Euler system for the Asai Galois representation attached to a p-ordinary Hilbert modular form g over a real quadratic field F, twisted by a CM Hecke character ψ. The construction uses generalized Hirzebruch–Zagier cycles obtained from modular curves of varying level diagonally embedded into products with Hilbert modular surfaces, following the approach of Darmon–Rotger. The main theorems are: (A) the existence of an anticyclotomic Euler system {κ_{ψ,g,n,∞}} with the expected norm relations, (B) applications to the Bloch–Kato conjecture in rank one, and (C) applications to the Iwasawa main conjecture, conditional on big-image hypotheses and on the framework of the unpublished Jetchev–Nekovář–Skinner manuscript. The paper also proves p-adic interpolation of the classes along Hida families.

Significance. If the main construction is correct, this is an important new Euler system: it is the first anticyclotomic Euler system for Asai representations over real quadratic fields, and the p-adic interpolation result is a substantial new ingredient. The geometric lemmas in Sections 4 and 5.2 (Lemmas 4.1–4.4, 5.3, 5.4) are proven in detail, and the construction of the big cohomology classes is clearly presented. The applications, while conditional on the unpublished work [JNS24], are natural and would be significant. The paper is well written and carefully signposted. However, the central norm-relation step in Theorem 5.7 contains a deferred 'one can check' congruence whose hypotheses and proof are not supplied, and Proposition 5.6 shows evidence of a transcription error. These issues are load-bearing for the main theorem.

major comments (4)
  1. [§5.2, Proposition 5.6] The displayed formula for cor_{K[nq]/K[n]}κ_{∞,n}(F,G) contains a likely duplication: after the term '-q^{-1}(q^2+1)' the expression repeats the first two summands exactly (the terms with -η1(q)κ_1^{-1/2}(q)a_{q1}(G)a_{q2}(G)(...) and +qω1(q)χ_G(̟_{q1})χ_G(̟_{q2})(...)^2). This suggests a copying error in the computation, which is delegated to 'somewhat tedious computation'. Since this formula is the basis for the congruence used in Theorem 5.7, the formula must be corrected and the computation either supplied in detail or independently verified.
  2. [§5.2, proof of Theorem 5.7] The proof asserts 'One can check' the congruence cor_{K[nq]/K[n]}κ*_{∞,nq}(F,G) ≡ P_q(VG,ψ(κ^{k-2}_{ac}); Fr_q^{-1})κ*_{∞,n}(F,G) mod (q-1), and then invokes [Rub00, Lem. 9.6.1] to replace this congruence by an equality. However, the hypotheses of Rubin's lemma are not verified: the classes κ*_{∞,n}(F,G) are not shown to lie in the integral Iwasawa cohomology H^1_Iw(K[np^∞], T) for a fixed O-lattice T, and the required local conditions are not checked. Without these checks, the exact norm relations of Theorem 5.7 do not follow.
  3. [§5.2, proof of Theorem 5.7] The application of [Rub00, Thm. 6.3.5] to remove the twist by κ^{k-2}_{ac} is not justified. The character κ^{k-2}_{ac} is an infinite-order anticyclotomic character with values in Λ^×, and the hypotheses of the twisting theorem (which typically requires finite-order twists or specific integrality conditions) are not discussed in the proof. This is a load-bearing step in obtaining the untwisted norm relations with P_q(VG,ψ; Fr_q^{-1}).
  4. [§5.2, Proposition 5.5 to Theorem 5.7] The transition from Proposition 5.5 to Theorem 5.7 is not fully demonstrated. In particular, the definition of the modified classes κ*_{∞,n}(F,G) involves multiplication by a product over q|n of terms depending on ψ_P(Fr_q) and Fr_q, and it is not shown that these operations preserve the integral classes required for Rubin's lemma. The congruence modulo (q-1) is stated without a derivation from the (currently defective) Proposition 5.6, so the proof of Theorem 5.7 is incomplete at this point.
minor comments (5)
  1. [Abstract] The abstract contains the typo 'emdedded' for 'embedded'.
  2. [Introduction, Theorem A] In the statement of Theorem A, the collection is written as {κψ,g,n,∞ : m∈S}; the dummy variable should be n, not m.
  3. [§5.2, equation (3) and following] In the definition of κ_{∞,n}(F,G), the notation [ξ_{̟_{n1}}^{-1}] is used without prior definition; the notation for the diamond operator on the Hilbert modular form side should be clarified.
  4. [§5.2, Lemma 5.4] In the third displayed relation of Lemma 5.4, the term '(q+1)(⟨1,⟨̟_{q1}⟩)' should presumably be '(q+1)(1,⟨̟_{q1}⟩)'; the ⟨1 is likely a typo.
  5. [References] The reference [NN16] is corrupted: 'Wies/suppress lawa Nizio/suppress l' should read 'Wiesława Nizioł'. The reference [ACR23b] is listed as 'to appear' but is used for several technical results; if it is not yet published, the dependence should be clearly flagged in the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the Euler system classes are constructed geometrically and the norm relations are derived from cycle-theoretic computations, with self-citations used only as ancillary references.

full rationale

The central derivation is self-contained: the classes κ∞,n(F,G) are produced in Sections 4–5 as p-adic étale Abel–Jacobi images of Hirzebruch–Zagier cycles, and the tame norm relations in Theorem 5.7 are not assumed, fitted, or renamed from an input. Lemma 5.3 computes degeneracy pushforwards of cycles; Lemma 5.4 and Proposition 5.5 convert these into cohomology operator relations; and Proposition 5.6, though proved tersely 'after a somewhat tedious computation', compares the corestriction to the Euler factor P_q(VG,ψ; Fr_q^{-1}). The modified classes κ* are defined precisely to absorb the extra local factors, so the congruence in Theorem 5.7 follows from Proposition 5.5 by construction in the legitimate sense of a construction, not as a disguised fit. The genuinely load-bearing unverified point is the sentence 'One can check' in the proof of Theorem 5.7, together with the unchecked applicability of [Rub00, Lem. 9.6.1] and the integrality/Selmer conditions; this is an omitted-verification or correctness risk, not circularity, because it does not identify the predicted norm relation with an input. Similarly, the apparent duplication of terms in Proposition 5.6's displayed formula is a possible algebra or copying error and should be checked, but it is a correctness issue, not a circularity. Self-citations to [ACR23b] occur in Propositions 5.1 and 5.2, Theorem 5.7, and Theorems 7.1–7.2, but in each case the primary support is external ([LLZ15, Prop. 3.2.1 and Cor. 5.2.6], [NN16, Thm. 5.9], [JNS24]); [ACR23b] is used only as 'as in' or 'same argument as', so the load-bearing content does not reduce to an unverified self-citation. No self-definitional identification, no uniqueness imported from the authors' own prior work, and no renaming of a known result as organization was found.

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

All inputs are standard external results or explicitly stated hypotheses; no free parameters are fitted to data. The main dependency is the unpublished Jetchev-Nekovar-Skinner preprint, whose results are used as a black box for the applications.

assumptions (7)
  • standard math Existence and local properties of p-adic Galois representations and big Galois representations attached to Hilbert modular forms and Hida families.
    Invoked in Sections 2.4-2.6 to define V and V_{FG}; proof relies on work of Eichler-Shimura, Deligne, Carayol, Wiles, Taylor, and Hida.
  • standard math Three-step filtration on the Asai representation with graded pieces of dimensions 1, 2, and 1, as in LLZ18 Cor 9.2.2.
    Used to define the balanced Selmer conditions and the local filtration in Sections 2.4.1 and 2.6.1.
  • standard math Rubin's Euler system formalism, specifically Lemma 9.6.1 and Theorem 6.3.5 in [Rub00].
    Used in the proof of Theorem 5.7 to convert congruence norm relations into exact Euler system norm relations and to remove the twist by κ_{ac}.
  • domain assumption The split anticyclotomic Euler system results of Jetchev-Nekovar-Skinner [JNS24], as recalled in ACR23b Thms 8.3 and 8.5.
    Underpins Theorems B and C; [JNS24] is cited as a preprint and not yet published.
  • standard math Loeffler's adelic image theorem [Loe17, Thm 3.2.2], extended to Hilbert modular forms, is used in Proposition 6.3.
    Provides the big-image hypothesis (HS) needed for the applications.
  • standard math Nekovar-Niziol [NN16, Thm 5.9] is used to show the classes lie in the balanced Selmer group.
    Cited at the end of the proof of Theorem 5.7 to identify the image of the classes under the Selmer condition.
  • standard math The correspondence ℓ+1-(T_ℓ,id) annihilates H^4_et(Z_α,Q,Z_p(2)), as in [FJ24, Prop. 5.10].
    Used in Section 4.2 to make the Hirzebruch-Zagier cycles null-homologous.

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Pith. "Pith review of An anticyclotomic Euler system of Hirzebruch--Zagier cycles I: Norm relations and $p$-adic interpolation." pith.science (2026). https://pith.science/paper/PUM2FZ6P

@misc{pith2026250115336,
  author       = {Pith},
  title        = {Pith review of: An anticyclotomic Euler system of Hirzebruch--Zagier cycles I: Norm relations and $p$-adic interpolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PUM2FZ6P}},
  note         = {Machine review of arXiv:2501.15336}
}
abstract

We construct an anticyclotomic Euler system for the Asai Galois representation associated to $p$-ordinary Hilbert modular forms over real quadratic fields. We also show that our Euler system classes vary in $p$-adic Hida families. The construction is based on the study of certain Hirzebruch--Zagier cycles obtained from modular curves of varying level diagonally emdedded into the product with a Hilbert modular surface. By Kolyvagin's methods, in the form developed by Jetchev--Nekov\'{a}\v{r}--Skinner in the anticyclotomic setting, the construction yields new applications to the Bloch--Kato conjecture and the Iwasawa Main Conjecture.

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