REVIEW
$p$-parity conjecture for Panchishkin Galois representations
Number Theory · Submitted 5 Nov 2026 · 74 references
Pith's verdict: The main relative p-parity theorem is substantial and probably sound, but the flagship numerical example does not verify one of the hypotheses it needs.
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Referee report
The paper
The paper claims
For residually isomorphic, Panchishkin Galois representations, the ratio of global root numbers equals the sign of the difference of their Bloch-Kato Selmer ranks, with no restriction on dimension.
The reading
What carries the argument
The carrying objects are local invariants over the cyclotomic $\mathbb{Z}_p$-extension $\mathbb{Q}_\infty$. For a finite prime $\ell \neq p$, $\delta(A_m,\ell)$ is the $\mathcal{O}$-rank of the Pontryagin dual of $H^1(\mathbb{Q}_{\infty,\eta}, A_m)$ at a prime $\eta \mid \ell$; its parity, read off a Frobenius characteristic polynomial by a result of [GV], matches the local root-number ratio at $\ell$. At $p$, the new invariant $\delta(A_m,p)$ measures in the residue field how much the strict condition $H^{str}(\mathbb{Q}_{\infty,p}, A_m[\pi])$ exceeds the intersection of the two strict conditions for the congruent pair, and its mod-2 difference equals the difference of $\dim_F H^0(\mathbb{Q}_{\infty,p}, V_m^-)$, which Corollary 3.3 identifies modulo 2 with the corresponding dimension over $\mathbb{Q}_p$. The bridge to Selmer theory is Theorem 3.7: over the deeply ramified cyclotomic extension the strict local condition has uniformly bounded Cartesian defect (a boundedness variant of the Cartesian condition of [MR2]) and the Bloch-Kato and strict local conditions coincide, so Proposition 5.4 expresses the Selmer-corank difference over $\mathbb{Q}_\infty$ through the same $\delta$'s. The Panchishkin condition — the representation has a $G_{\mathbb{Q}_p}$-stable subrepresentation $V^+$ whose Hodge-Tate weights are all positive, with all non-positive weights living in the quotient $V^-$ — is what makes the 'minus' part $V^-$ control both the local conditions and the parity computations.
Extended reading notes
Core claim
The central claim, Theorem 1.4, is a conditional proof of the relative $p$-parity conjecture: let $V_1$ and $V_2$ be symplectic self-dual de Rham representations of $G_{\mathbb{Q}}$ that are pure of weight $-1$, satisfy the Panchishkin condition at $p$, and satisfy $A_1[\pi] \cong A_2[\pi]$ as $G_{\mathbb{Q}}$-modules. If the dual Bloch-Kato Selmer group $S_{BK}(A_1/\mathbb{Q}_\infty)^\vee$ is torsion over the Iwasawa algebra with $\mu = 0$, and the residual isomorphism restricts to an isomorphism $A_1^-[\pi] \cong A_2^-[\pi]$ of the 'minus' Panchishkin parts at $p$, then $$W(V_1)/W(V_2) = (-1)^{\mathrm{rk}_{\mathcal{O}}\, S_{BK}(A_1/\mathbb{Q})^\vee - \mathrm{rk}_{\mathcal{O}}\, S_{BK}(A_2/\mathbb{Q})^\vee}.$$ Unconditionally (Theorem 1.3), the root-number ratio is $(-1)^{\sum_\ell (\delta(A_1,\ell) - \delta(A_2,\ell))}$, where $\delta(-,p)$ is a new local invariant built from strict local conditions over the cyclotomic $\mathbb{Z}_p$-extension and $\delta(-,\ell)$ is a local Iwasawa $\lambda$-invariant for $\ell \neq p$; both are compatible with the arithmetic local constants of [MR1] and [Ne1]. The key input, Theorem 1.5, states that the Bloch-Kato and strict local conditions at $p$ coincide over $\mathbb{Q}_{\infty,p}$ for every de Rham, Panchishkin, pure-weight $-1$ representation; its proof uses p-adic Hodge theory to control the Galois action on $H^0(\mathbb{Q}_{\infty,p}, V^-)$ and to compare its dimension with the same space over $\mathbb{Q}_p$ modulo 2.
Load-bearing premise
The load-bearing premise is the pair of hypotheses in Theorem 1.4: the dual Bloch-Kato Selmer group of $A_1$ over the cyclotomic $\mathbb{Z}_p$-extension must be torsion over the Iwasawa algebra with vanishing $\mu$-invariant, and the residual isomorphism $A_1[\pi] \cong A_2[\pi]$ must restrict to an isomorphism $A_1^-[\pi] \cong A_2^-[\pi]$ of the minus Panchishkin parts at $p$; the $\mu = 0$ condition is conjectural in most applications, and the authors state that even after extending scalars to make the residual representations symplectically equivalent at $p$, it is not clear the isomorphism carries $A_1^-[\pi]$ to $A_2^-[\pi]$.
Editorial extensions
If this is right
- Parity transfers along congruences: if the $p$-parity conjecture holds for one member of a residually isomorphic Panchishkin family satisfying the hypotheses of Theorem 1.4, it holds for every other member, so each known case seeds an infinite family of new ones.
- Rank greater than two becomes accessible: the applications cover Artin twists of CM Hida families, orthogonal Artin twists of ordinary modular forms, symmetric cubes along Hida families, tensor products of symmetric powers, and families of abelian surfaces, with numerical examples where all hypotheses are verified unconditionally.
- The local comparison at $p$ no longer needs potentially Barsotti-Tate representations or vanishing monodromy: arbitrary Hodge-Tate weights, nontrivial monodromy operators, and exceptional-zero phenomena are allowed.
- The residual isomorphism is not required to be symplectic, so the theorems cover pairs for which no symplectic similitude between residual representations exists at all, such as the elliptic-curve construction of Example 4.9.
- The invariant $\delta(-,p)$ is compatible with the arithmetic local constants at $p$ (Proposition 6.3), and the results apply to families over reducible or higher-dimensional bases, including different branches of a Hida family.
Reading between the lines
- Because $\delta(-,p)$ is defined purely from strict conditions over the cyclotomic tower, it should be computable directly from the filtered $(\varphi,N)$-module of a congruent pair; a closed local formula of that kind would make the root-number comparison at $p$ algorithmic and numerically checkable beyond the paper's examples.
- Remark 4.12 notes that after extending scalars the residual representations can be made symplectically equivalent at $p$ without it being clear that the isomorphism preserves the minus parts; locating a concrete pair where $A_1^-[\pi] \not\cong A_2^-[\pi]$ despite residual equivalence would map the exact boundary of the method.
- If the conjectural $\mu = 0$ statements quoted in Section 6.3 become theorems for irreducible residual representations, the relative theorem upgrades to an unconditional parity statement on entire Hida families, effectively reducing the $p$-parity conjecture to finding one parity-known member per congruence class.
Original abstract
Let $p$ be an odd prime, and let $V_1$ and $V_2$ be symplectic self-dual $p$-adic de Rham representations of $G_{\mathbb{Q}}$ that are pure (of weight $-1$) and satisfy the Panchishkin condition at $p$. Suppose that $V_1$ and $V_2$ are residually isomorphic, compatibly with their Panchishkin filtrations. We prove the relative $p$-parity conjecture relating the ratio of the global root numbers of $V_1$ and $V_2$ to the difference of their Bloch-Kato Selmer dimensions. We have no restriction on the dimension of the Galois representation, the Hodge-Tate weights or the monodromy operator at $p$, and the residual isomorphism need not be symplectic. The proof expresses ratios of local root numbers in terms of local invariants over the cyclotomic $\mathbb{Z}_p$-extension: a new invariant introduced at $p$, and local Iwasawa $\lambda$-invariants at $\ell\neq p$. Both are compatible with the arithmetic local constants of Mazur-Rubin and Nekov\'a\v{r}. A key input, proved using Sen theory, is that the Bloch-Kato and strict local conditions at $p$ coincide over the cyclotomic $\mathbb{Z}_p$-extension, generalising results of Coates-Greenberg and Nekov\'a\v{r}. As applications, we obtain new cases of the $p$-parity conjecture, including cases of rank greater than two: Artin twists of CM Hida families, orthogonal Artin twists of ordinary modular forms, symmetric cubes along Hida families, and families of abelian surfaces, together with unconditional numerical examples.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Let $p$ be an odd prime, and let $V_1$ and $V_2$ be symplectic self-dual $p$-adic de Rham representations of $G_{\mathbb{Q}}$ that are pure (of weight $-1$) and satisfy the Panchishkin condition at $p$. Suppose that $V_1$ and $V_2$ are residually isomorphic, compatibly with their Panchishkin filtrations. We prove the relative $p$-parity conjecture relating the ratio of the global root numbers of $V_1$ and $V_2$ to the difference of their Bloch-Kato Selmer dimensions. We have no restriction on the dimension of the Galois representation, the Hodge-Tate weights or the monodromy operator at $p$, and the residual isomorphism need not be symplectic. The proof expresses ratios of local root numbers in terms of local invariants over the cyclotomic $\mathbb{Z}_p$-extension: a new invariant introduced at $p$, and local Iwasawa $\lambda$-invariants at $\ell\neq p$. Both are compatible with the arithmetic local constants of Mazur-Rubin and Nekov\'a\v{r}. A key input, proved using Sen theory, is that the Bloch-Kato and strict local conditions at $p$ coincide over the cyclotomic $\mathbb{Z}_p$-extension, generalising results of Coates-Greenberg and Nekov\'a\v{r}. As applications, we obtain new cases of the $p$-parity conjecture, including cases of rank greater than two: Artin twists of CM Hida families, orthogonal Artin twists of ordinary modular forms, symmetric cubes along Hida families, and families of abelian surfaces, together with unconditional numerical examples.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $p$ be an odd prime, and let $V_1$ and $V_2$ be symplectic self-dual $p$-adic de Rham representations of $G_{\mathbb{Q}}$ that are pure (of weight $-1$) and satisfy the Panchishkin condition at $p$. | background 0.99 | 0.09 | 0.02 | asserted 0.81 |
| s1 | Suppose that $V_1$ and $V_2$ are residually isomorphic, compatibly with their Panchishkin filtrations. | background 0.96 | 0.12 | 0.02 | asserted 0.71 |
| s2 | We prove the relative $p$-parity conjecture relating the ratio of the global root numbers of $V_1$ and $V_2$ to the difference of their Bloch-Kato Selmer dimensions. | result 0.72 | 0.96 | 0.95 | asserted 1.00 |
| s3 | We have no restriction on the dimension of the Galois representation, the Hodge-Tate weights or the monodromy operator at $p$, and the residual isomorphism need not be symplectic. | contribution 0.66 | 0.59 | 0.04 | asserted 1.00 |
| s4 | The proof expresses ratios of local root numbers in terms of local invariants over the cyclotomic $\mathbb{Z}_p$-extension: a new invariant introduced at $p$, and local Iwasawa $\lambda$-invariants at $\ell\neq p$. | method 0.96 | 0.92 | 0.05 | asserted 1.00 |
| s5 | Both are compatible with the arithmetic local constants of Mazur-Rubin and Nekov\'a\v{r}. | interpretation 0.40 | 0.52 | 0.02 | asserted 0.99 |
| s6 | A key input, proved using Sen theory, is that the Bloch-Kato and strict local conditions at $p$ coincide over the cyclotomic $\mathbb{Z}_p$-extension, generalising results of Coates-Greenberg and Nekov\'a\v{r}. | method 0.66 | 0.87 | 0.03 | asserted 1.00 |
| s7 | As applications, we obtain new cases of the $p$-parity conjecture, including cases of rank greater than two: Artin twists of CM Hida families, orthogonal Artin twists of ordinary modular forms, symmetric cubes along Hida families, and families of abelian surfaces, together with unconditional numerical examples. | result 0.66 | 0.95 | 0.12 | asserted 1.00 |
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Discussion
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