A Note on the Growth of Sha in Dihedral Extensions

Jamie Bell

REVIEW 1 major objections 3 minor 10 references

A Note on the Growth of Sha in Dihedral Extensions

T0 review · 1 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For elliptic curves over dihedral extensions, the size of the Tate–Shafarevich group is fixed up to fourth powers by its sizes over quadratic subfields.

desk verdict A clean, largely correct short note that packages known Brauer-relation isogenies and Cassels squareness into a modulo-fourth-powers formula for Sha in dihedral extensions, with a couple of fixable rigour gaps. read the letter →

arxiv 2411.15663 v1 pith:V6VKBHXF submitted 2024-11-23 math.NT

Jamie Bell This is my paper · ORCID

classification math.NT MSC 11G0511G40

keywords Tate–ShafarevichgroupellipticcurvesdihedralextensionsBrauerrelationsWeilrestrictionp-primaryShaGaloismodulestructuresquareorder

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 proves that for an elliptic curve over a number field, the size of the p-primary Tate–Shafarevich group over a dihedral extension $F/k$ of degree $2n$ is controlled up to fourth powers by the size over the quadratic subextensions, for any prime $p$ not dividing $n$. When $n$ is odd, the order over $F$ differs from the order over the unique quadratic subfield by a fourth power; when $n$ is even, it differs from the product of the orders over the three quadratic subfields by a fourth power. The proof uses Brauer relations in the dihedral group to build isogenies between products of Weil restrictions, transferring the classical square-order theorem for finite Tate–Shafarevich groups into a fourth-power comparison. A second theorem describes the Galois module structure of the p-primary Sha, showing it is a direct sum of two isomorphic submodules. This gives a new structural constraint on how Sha can grow in dihedral towers and a practical tool for computing it up to fourth powers.

What carries the argument

The central object is a Brauer relation in the dihedral group $D_{2n}$: a formal sum of subgroups $\sum H_i - \sum H'_j$ whose permutation modules are isomorphic over $\mathbb{Q}$. The paper uses two such relations, one for odd $n$ and one for even $n$, which become $\mathbb{Z}_{(p)}$-relations for $p \nmid n$. These induce isogenies between products of Weil restrictions of $E$ to the fixed fields, of degree coprime to $p$, so the p-primary Tate–Shafarevich orders of the two products are equal. Comparing the factors and applying the square-order theorem for finite Sha gives the fourth-power formula. For the module structure, the key is that $\mathbb{Z}_p[C_n]$ splits as a product of local rings with principal maximal ideals, which forces the p-primary Sha over $F$ to be a direct sum of two isomorphic modules.

What would settle it

A direct descent computation of $\mathrm{Sha}(E/F)[5^\infty]$ for a curve over a dihedral extension of degree 6 (so $n=3$, $p=5 \nmid n$) would settle Theorem 1: the ratio to the quadratic subfield's Sha order must be exactly a fourth power of 5, and any other pure power would disprove it.

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Extended reading notes

Core claim

Let $E$ be an elliptic curve over a number field $k$ and let $F/k$ be a dihedral extension of degree $2n$. Fix a prime $p$ not dividing $n$ and assume the p-primary Tate–Shafarevich group $\mathrm{Sha}(E/F)[p^\infty]$ is finite. The paper establishes that for odd $n$, $|\mathrm{Sha}(E/F)[p^\infty]| / |\mathrm{Sha}(E/K)[p^\infty]| = p^{4t}$ for some integer $t$, where $K$ is the quadratic subextension; for even $n$, $|\mathrm{Sha}(E/F)[p^\infty]|$ equals the product $|\mathrm{Sha}(E/K_1)[p^\infty]| |\mathrm{Sha}(E/K_2)[p^\infty]| |\mathrm{Sha}(E/K_3)[p^\infty]|$ times $p^{4t}$, where $K_1, K_2, K_3$ are the three quadratic subextensions. The argument works by applying dihedral Brauer relations to obtain isogenies of Weil restrictions with degree coprime to $p$, so the two sides have identical p-primary Sha orders, and then using the fact that a finite Sha has square order to compare the extra factors. The same relations yield a Galois module statement: over the cyclic subgroup $H$ of order $n$, $\mathrm{Sha}(E/F)[p^\infty] \cong X \oplus X$ and $\mathrm{Sha}(E/K)[p^\infty] \cong X^H \oplus X^H$, and in the odd case with $p^a \equiv -1 \bmod n$ solvable, $|X|/|X^H|$ is a square.

Load-bearing premise

The argument depends on the p-primary square-order theorem for finite Tate–Shafarevich groups, but the paper assumes finiteness only for $\mathrm{Sha}(E/F)[p^\infty]$ and does not state or cite the p-primary version it applies.

Editorial extensions

If this is right

For odd dihedral extensions, the p-primary part of Sha over the full field is, up to a fourth power, the same as over the quadratic subfield; so any new Sha in the extension is constrained to a very specific size.
For even dihedral extensions, the same holds with the three quadratic subfields playing the role of the base; the product of their Sha orders determines the big Sha up to fourth powers.
The Galois module splitting $\mathrm{Sha}(E/F)[p^\infty] \cong X \oplus X$ means the group is even as a $\mathbb{Z}_p[H]$-module, and the fixed part is the same module twice, giving constraints on possible module structures beyond mere sizes.
When $n$ is odd and $p^a \equiv -1 \pmod{n}$ has a solution, the sharper ratio $|X|/|X^H|$ is a square, so the fourth-power formula can be upgraded to a square condition.
The result is sharp: the paper gives explicit examples showing the formula fails for general cyclic extensions of odd degree and when $p$ divides $n$.

Reading between the lines

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

A likely next step is to supply the missing p-primary Cassels squareness theorem explicitly; once stated, the proof of Theorem 1 becomes self-contained and could be adapted to other p-primary pairings.
The same Brauer-relation technique might apply to other non-dihedral groups whose rational permutation modules coincide, producing analogous fourth-power formulas for other solvable extensions.
One testable prediction: for a fixed curve and a dihedral tower with $p$ not dividing any of the degrees, the $p$-adic valuation of $|\mathrm{Sha}|$ modulo 4 should follow a deterministic pattern determined by the quadratic subfields; a computational search over quadratic twists could confirm or refute this across many examples.
If the analytic order of Sha is used via the Birch–Swinnerton-Dyer conjecture, the fourth-power formula translates into a constraint on the special values of $L$-functions of elliptic curves over dihedral extensions.

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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

Worth a look if you care about Sha growth in dihedral extensions. The note proves a modulo-fourth-power formula for |Sha(E/F)[p∞]| in terms of smaller fields when F/k is dihedral of degree 2n and p∤n: for odd n it matches the quadratic subfield, and for even n it matches the product of the three quadratic subfields. That is a genuinely new packaged statement, though every ingredient is known: Dokchitser–Dokchitser isogenies from Brauer relations, Bartel's Z_(p)-relation result, and Cassels squareness. The two numerical examples in Remarks 2 and 3 are a nice touch and show the hypotheses are not decorative.

What the paper does well: Lemma 6 writes down the explicit dihedral Brauer relations, Lemmas 7 and 8 convert them into p-primary Sha order equalities via isogenies, and the proof of Theorem 1 is structurally sound. Theorem 4 is a reasonable bonus: under the same finiteness, it gives a Z_p[H]-module decomposition X(E/F)[p∞] ≅ X ⊕ X and identifies the K-invariants, with a square-ratio statement when p^a ≡ −1 mod n. The appeal to Chetty's lemma is terse but plausible.

The soft spots are real but minor. The proof of Theorem 1 says 'when X is finite it has square order' and applies that to the p-primary groups of the subfields k, F^{C2}, and (in the even case) the quadratic subfields. The theorem only assumes Sha(E/F)[p∞] finite; finiteness over the subfields is footnoted to Dokchitser–Dokchitser, but the square-order conclusion for the individual p-primary groups needs the p-primary Cassels–Tate pairing, which is not stated or cited. This is an easy fix—cite the p-primary version or note it explicitly—but as written it is an unstated assumption. Also, the displayed congruence in the even-n case has a typo: it writes F^{D_{2n}} and F^{D_{2n}'} where the intended fields are the fixed fields of the two order-n subgroups from Lemma 6, i.e. two of the three quadratic subfields. With the corrected subgroups the final formula does follow.

Bottom line: the central claim is correct in spirit and the gaps are fixable. This is a solid short note, not a landmark. I'd send it to a competent referee rather than desk-reject; the referee should ask for the p-primary Cassels statement and a corrected even-n congruence. I'd probably cite it if I were working on parity or arithmetic statistics in dihedral towers, and I might bring it to a reading group as a clean example of Brauer-relation isogenies in action.

Referee Report

1 major / 3 minor

Summary. The paper proves a congruence modulo fourth powers for the size of the p-primary Tate-Shafarevich group of an elliptic curve in a dihedral extension F/k of degree 2n, for primes p not dividing n. Theorem 1 asserts that, assuming X(E/F)[p^\infty] is finite, |X(E/F)[p^\infty]|/|X(E/K)[p^\infty]| is a fourth power when n is odd, and for even n the analogous ratio with the product of the three quadratic subextensions is a fourth power. The proof uses Brauer relations in the dihedral group to construct isogenies between products of Weil restrictions whose degrees are coprime to p, then applies a lemma equating p-primary Tate-Shafarevich orders and discards square factors using Cassels' squareness theorem. Theorem 4 gives a Z_p[H]-module decomposition X(E/F)[p^\infty] \cong X\oplus X and X(E/K)[p^\infty] \cong X^H\oplus X^H, with a squareness criterion for |X|/|X^H| under a congruence condition modulo n. The paper also gives Magma examples showing that the hypotheses on dihedrality and on p not dividing n are necessary.

Significance. If Theorem 1 is established, it is a clean and useful structural result: it severely constrains the growth of the p-primary Tate-Shafarevich group in dihedral extensions for primes not dividing n, expressing it up to fourth powers in terms of the quadratic subextensions. The proof is mostly a transparent application of existing machinery (Bartel, Dokchitser-Dokchitser, Chetty, Cassels) rather than an introduction of fundamentally new ideas, and the explicit counterexamples for cyclic non-dihedral extensions and for p dividing n are valuable because they show the statement is sharp. The paper is honest about its limitations and cites its dependencies clearly. The main gap, the need for the p-primary Cassels-Tate squareness statement, is local and fixable; it does not affect the conceptual contribution.

major comments (1)

[Proof of Theorem 1, Section 2] The squareness step in the proof of Theorem 1 invokes 'When X of an elliptic curve is finite, it has square order' to discard factors such as |X(E/k)[p^\infty]|^2 and |X(E/F^{C_2})[p^\infty]|^2 in the odd case, and the analogous factors in the even case. But Theorem 1 assumes only that X(E/F)[p^\infty] is finite; the footnote citing [5, Remark 2.10] gives finiteness of the p-primary groups over subfields, not squareness of their orders. The proof therefore needs the p-primary version of Cassels' theorem: if X(E/L)[p^\infty] is finite, then the Cassels-Tate pairing restricts to a nondegenerate alternating pairing on it, so its order is a square. This version is not stated or cited in the paper. Without it, the step 'so |X(E/F)[p^\infty]| \equiv |X(E/K)[p^\infty]| (mod Q^{*4})' is unjustified as written. The fix is local: add the p-primary statement as a lemma or a citation and apply it to each subfield.

minor comments (3)

[Proof of Theorem 1, even-n case] The displayed congruence in the even-n case uses F^{D_{2n}} and F^{D_{2n}'}; since D_{2n} is the whole group, F^{D_{2n}} = k, and D_{2n}' is undefined. The intended subgroups are D_n and D_n' from Lemma 6, so the congruence should read |X(E/F)[p^\infty]| |X(E/F^{D_n})[p^\infty]| \equiv |X(E/F^{D_n'})[p^\infty]| |X(E/F^{C_n})[p^\infty]| (mod Q^{*4}). This typo should be corrected, as the current display does not follow from Lemma 8.
[Proof of Theorem 4, Section 3] In the sentence 'This condition implies ord_p(d) is even for all d|n', the notation should be ord_d(p), the multiplicative order of p modulo d, rather than ord_p(d).
[Remark 2 and Remark 3] The two Magma examples are helpful, but the text should state explicitly that the computed quantities are analytic orders of Sha, so that the conclusion 'changes by a non-fourth-power' is understood to be conditional on the relevant cases of the Birch-Swinnerton-Dyer conjecture.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1 is derived from external Brauer-relation, isogeny, and Cassels theorems, not from its conclusion.

full rationale

The paper's central result, Theorem 1, is a congruence modulo fourth powers relating Tate-Shafarevich p-primary orders. Its proof uses Lemma 8, which follows from Brauer relations and isogenies of degree coprime to p, and then cancels square-order factors using Cassels' squareness theorem. These are external results (Bartel, Dokchitser-Dokchitser, Cassels, Milne), and the paper supplies its own proof of the Brauer relation in Lemma 6. No parameter is fitted to data, no subset of the result is assumed, and no uniqueness or ansatz is imported from prior work of the same authors. The only personal link is that the supervisor, V. Dokchitser, is an author of one cited paper [5], but that paper is published, independent mathematics, and the derivation does not reduce to it by construction. The possible need for the p-primary Cassels-Tate pairing to justify squareness of the auxiliary p-primary Sha orders is an unstated assumption or correctness gap, not circularity, since it adds external input rather than presupposing Theorem 1. The apparent F^{D_{2n}}/F^{D_n} typo in the even-n display is likewise a typographical issue, not a circular step. I therefore find no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper contributes no free parameters or invented entities; it derives the result from standard Brauer relation machinery and cited theorems. The main external inputs are Cassels squareness, the Dokchitser-Dokchitser isogeny theorem, Bartel's Z_(p)-relation result, and Chetty's lemma.

assumptions (5)

domain assumption Cassels' squareness theorem: a finite Tate-Shafarevich group has square order (p-primary analogue invoked implicitly).
Used in the proof of Theorem 1 to treat |Sha(E/k)[p∞]|^2 and |Sha(E/F^{C2})[p∞]|^2 as fourth powers modulo Q*4.
domain assumption Brauer relations induce isogenies between products of Weil restrictions (Dokchitser-Dokchitser [5] Theorem 2.3).
Basis of Lemma 8 and hence of the equality of p-primary Sha orders.
domain assumption The dihedral Brauer relations of Lemma 6 are Z_(p)-relations for p not dividing n (Bartel [1] Lemma 3.9).
Ensures the isogeny degree is coprime to p, so the p-primary Sha orders are equal.
domain assumption Finiteness of Sha(E/L)[p∞] over subfields L of F follows from finiteness over F (Dokchitser-Dokchitser [5] Remark 2.10).
Needed so Cassels-type squareness can be applied to the subfield Sha groups.
domain assumption Chetty's Lemma 2.8 applies to the Cassels-Tate pairing twisted by the involution s.
Used to prove the Galois-module decomposition X(E/F)[p∞] isomorphic to X ⊕ X in Theorem 4.

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Cite this review

Pith. "Pith review of A Note on the Growth of Sha in Dihedral Extensions." pith.science (2026). https://pith.science/paper/V6VKBHXF

@misc{pith2026241115663,
  author       = {Pith},
  title        = {Pith review of: A Note on the Growth of Sha in Dihedral Extensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V6VKBHXF}},
  note         = {Machine review of arXiv:2411.15663}
}

abstract

We provide a formula for the order of the Tate--Shafarevich group of elliptic curves over dihedral extensions of number fields of order $2n$, up to $4^{th}$ powers and primes dividing $n$. Specifically, for odd $n$ it is equal to the order of the Tate--Shafarevich group over the quadratic subextension. A similar formula holds for even $n$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

A. Bartel. On Brauer–Kuroda type relations of S-class numbers in dihed ral extensions. Journal für die reine und angewandte Mathematik, vol. 2012, no. 668, pp.211–244 ( 2012)

work page 2012
[2]

A. Bartel. Large Selmer groups over number ﬁelds . Mathematical Proceedings of the Cambridge Philosoph- ical Society, Vol. 148 (No.1). pp. 73-86. (2010)

work page 2010
[3]

S. Chetty. Arithmetic local constants for abelian varieties with extr a endomorphisms . Funct. Approx. Com- ment. Math. 55(1). pp. 59–81 (2016)

work page 2016
[4]

Wieb Bosma, John Cannon, and Catherine Playoust The Magma algebra system. I. The user language J. Symbolic Comput., 24, pp. 235–265 (1997)

work page 1997
[5]

Dokchitser, V.Dokchitser On the Birch–Swinnerton-Dyer quotients modulo squares

T. Dokchitser, V.Dokchitser On the Birch–Swinnerton-Dyer quotients modulo squares . Annals of Math. 172 no. 1 (2010), 567-596

work page 2010
[6]

Cassels Arithmetic on Curves of Genus 1

J.W.S. Cassels Arithmetic on Curves of Genus 1. IV. Proof of the Hauptvermut ung. Journal für die reine und angewandte Mathematik, vol. 1962, no. 211, pp. 95-112 (1 962)

work page 1962
[7]

Milne On the Arithmetic of Abelian Varieties

J.S. Milne On the Arithmetic of Abelian Varieties . Inventiones math. 17, pp. 177-190 (1972)

work page 1972
[8]

Serre Linear Representations of Finite Groups

J.-P. Serre Linear Representations of Finite Groups . GTM 42, Springer Verlag 1977

work page 1977

Show all 10 references

[9]

Serre Local Fields

J.-P. Serre Local Fields. GTM 67, Springer Verlag 1979

1979
[10]

Park Relations among Shafarevich–Tate Groups S¯ urikaisekikenky¯ usho K¯ oky¯ uroku 998, pp

H. Park Relations among Shafarevich–Tate Groups S¯ urikaisekikenky¯ usho K¯ oky¯ uroku 998, pp. 117–125 (1997) University College London, Gower Street, London, WC1E 6BT, UK Email address : james.bell.20@ucl.ac.uk

1997

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

A. Bartel. On Brauer–Kuroda type relations of S-class numbers in dihed ral extensions. Journal für die reine und angewandte Mathematik, vol. 2012, no. 668, pp.211–244 ( 2012)

work page 2012

[2] [2]

A. Bartel. Large Selmer groups over number ﬁelds . Mathematical Proceedings of the Cambridge Philosoph- ical Society, Vol. 148 (No.1). pp. 73-86. (2010)

work page 2010

[3] [3]

S. Chetty. Arithmetic local constants for abelian varieties with extr a endomorphisms . Funct. Approx. Com- ment. Math. 55(1). pp. 59–81 (2016)

work page 2016

[4] [4]

Wieb Bosma, John Cannon, and Catherine Playoust The Magma algebra system. I. The user language J. Symbolic Comput., 24, pp. 235–265 (1997)

work page 1997

[5] [5]

Dokchitser, V.Dokchitser On the Birch–Swinnerton-Dyer quotients modulo squares

T. Dokchitser, V.Dokchitser On the Birch–Swinnerton-Dyer quotients modulo squares . Annals of Math. 172 no. 1 (2010), 567-596

work page 2010

[6] [6]

Cassels Arithmetic on Curves of Genus 1

J.W.S. Cassels Arithmetic on Curves of Genus 1. IV. Proof of the Hauptvermut ung. Journal für die reine und angewandte Mathematik, vol. 1962, no. 211, pp. 95-112 (1 962)

work page 1962

[7] [7]

Milne On the Arithmetic of Abelian Varieties

J.S. Milne On the Arithmetic of Abelian Varieties . Inventiones math. 17, pp. 177-190 (1972)

work page 1972

[8] [8]

Serre Linear Representations of Finite Groups

J.-P. Serre Linear Representations of Finite Groups . GTM 42, Springer Verlag 1977

work page 1977