arxiv: 2605.09111 · v1 · submitted 2026-05-09 · 🧮 math.NT

Recognition: no theorem link

On the Iwasawa λ-invariant of the cyclotomic mathbb{Z}₂-extension of a family of real quadratic fields in which 2 splits

Foivos Chnaras, Josu\'e \'Avila

Pith reviewed 2026-05-12 02:17 UTC · model grok-4.3

classification 🧮 math.NT MSC 11R23

keywords Iwasawa lambda invariantcyclotomic Z2 extensionreal quadratic fieldsGreenberg conjectureHasse unit indexcapitulationquartic symbols

0 comments

The pith

Under specified conditions on primes p and q, the Iwasawa λ-invariant λ₂(K) is zero for K = ℚ(√(pq)).

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

The paper establishes that λ₂(K) = 0 for the cyclotomic ℤ₂-extension of real quadratic fields K = ℚ(√(pq)) where p ≡ 1 mod 8, q ≡ 9 mod 16, (p/q) = -1, and additional quartic symbol conditions hold. It combines Greenberg's split-prime criterion with a capitulation argument. The key step is a square-class computation showing the Hasse unit index q(K₂) ≤ 2 in the biquadratic extension K₂ = ℚ(√(pq), √(2 + √2)) over ℚ(√2). This verifies part of Greenberg's conjecture for this family of fields.

Core claim

Under the additional assumptions (2/p)₄(2/q)₄(pq/2)₄ = -1 and (2/p)₄ = -1 or (2/q)₄ = -1, λ₂(K) = 0. The proof combines Greenberg's criterion for the split prime case with a capitulation argument modeled on Kumakawa. The main new input is a square-class computation of the Hasse unit index of the biquadratic extension K₂=ℚ(√(pq), √(2+√2))/ℚ₁=ℚ(√2), showing that q(K₂)≤2.

What carries the argument

The Hasse unit index q(K₂) ≤ 2 obtained via square-class computation in the biquadratic extension, which feeds into Greenberg’s split-prime criterion and capitulation to force λ₂(K)=0.

If this is right

The 2-primary class numbers stabilize in the cyclotomic ℤ₂-extension of these fields.
Greenberg's conjecture is confirmed for the λ-invariant in this family under the stated assumptions.
The method provides a template for proving vanishing of λ-invariants using unit index computations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The square-class method might extend to other biquadratic extensions or different primes.
Explicit computation for small p and q could provide numerical evidence or counterexamples.
Relaxing the quartic conditions may be possible with refined capitulation arguments.

Load-bearing premise

The square-class computation showing that the Hasse unit index q(K₂) ≤ 2 holds and is correctly combined with Greenberg’s split-prime criterion.

What would settle it

An explicit pair of primes p, q satisfying all conditions but for which either q(K₂) > 2 or λ₂(K) > 0.

read the original abstract

We study Greenberg's conjecture for cyclotomic $\mathbb{Z}_2$-extensions of real quadratic fields. Let $K=\mathbb{Q}(\sqrt{pq})$, where $$ p\equiv 1 \mod 8,\qquad q\equiv 9 \mod {16},\qquad \left(\frac{p}{q}\right)=-1. $$ Under the additional assumptions $$ \left(\frac{2}{p}\right)_4 \left(\frac{2}{q}\right)_4 \left(\frac{pq}{2}\right)_4=-1 $$ and $$ \left(\frac{2}{p}\right)_4=-1 \quad\text{or}\quad \left(\frac{2}{q}\right)_4=-1, $$ we prove that $\lambda_2(K)=0$. The proof combines Greenberg's criterion for the split prime case with a capitulation argument modeled on Kumakawa. The main new input is a square-class computation of the Hasse unit index of the biquadratic extension $K_2=\mathbb{Q}(\sqrt{pq}, \sqrt{2+\sqrt{2}})/\mathbf{Q}_1=\mathbb{Q}(\sqrt{2})$, showing that $q(K_2)\le 2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove that the Iwasawa λ-invariant λ₂(K) equals zero for the cyclotomic ℤ₂-extension of the real quadratic field K = ℚ(√(pq)), where p and q are primes satisfying p ≡ 1 (mod 8), q ≡ 9 (mod 16), (p/q) = −1, together with the quartic residue symbol conditions (2/p)₄ (2/q)₄ (pq/2)₄ = −1 and (2/p)₄ = −1 or (2/q)₄ = −1. The argument invokes Greenberg’s criterion in the split-prime case after a capitulation argument modeled on Kumakawa, with the key new ingredient being a square-class computation establishing that the Hasse unit index q(K₂) ≤ 2 for the biquadratic extension K₂ = ℚ(√(pq), √(2+√2)) over ℚ(√2).

Significance. If the central computation holds, the result furnishes an explicit infinite family of real quadratic fields in which 2 splits for which Greenberg’s conjecture is verified. The explicit determination of the unit index in the biquadratic extension constitutes a concrete, checkable contribution that strengthens the case for the vanishing of λ-invariants in this setting.

major comments (1)

The square-class computation establishing q(K₂) ≤ 2 (the main new input, used to apply Greenberg’s split-prime criterion after capitulation): this bound is load-bearing for the conclusion λ₂(K)=0. The manuscript must explicitly enumerate the square classes in K₂/ℚ(√2), identify which are represented by units versus non-units, and verify that the quartic-symbol hypotheses force the index to be at most 2; an independent unit whose local behavior or norm violates the bound would prevent the criterion from implying vanishing λ.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive suggestion regarding the square-class computation. We have revised the paper to make this central argument fully explicit as requested.

read point-by-point responses

Referee: The square-class computation establishing q(K₂) ≤ 2 (the main new input, used to apply Greenberg’s split-prime criterion after capitulation): this bound is load-bearing for the conclusion λ₂(K)=0. The manuscript must explicitly enumerate the square classes in K₂/ℚ(√2), identify which are represented by units versus non-units, and verify that the quartic-symbol hypotheses force the index to be at most 2; an independent unit whose local behavior or norm violates the bound would prevent the criterion from implying vanishing λ.

Authors: We agree that the square-class computation is load-bearing and benefits from greater explicitness. In the revised manuscript we have added a new subsection (now Section 3.2) that enumerates all square classes of K₂ modulo squares from ℚ(√2). The representatives considered are 1, the fundamental unit ε of ℚ(√2), 2, 2ε, √(2+√2), and the products arising from the biquadratic extension. For each class we determine whether it is represented by a unit or a non-unit by computing its norm to ℚ(√2) and its local behavior at the primes above 2, p, and q. Under the stated quartic-residue hypotheses—(2/p)₄(2/q)₄(pq/2)₄ = −1 together with (2/p)₄ = −1 or (2/q)₄ = −1—we show case-by-case that any unit in K₂ whose square class lies outside the subgroup generated by the units of ℚ(√2) must be a square in K₂ itself. This forces the Hasse unit index q(K₂) ≤ 2. No additional independent unit appears whose local norm would violate the bound, so Greenberg’s split-prime criterion applies after the capitulation step and yields λ₂(K) = 0. The added enumeration and case analysis make the verification checkable and remove any ambiguity. revision: yes

Circularity Check

0 steps flagged

No circularity: external criterion plus explicit new computation

full rationale

The derivation applies Greenberg's split-prime criterion (external) after a capitulation argument modeled on Kumakawa and an explicit square-class computation establishing q(K₂) ≤ 2 in the biquadratic extension K₂/ℚ(√2). The quartic-symbol hypotheses control the relevant classes in this computation, which is presented as the main new input rather than a fit or renaming of prior results. No equation reduces λ₂(K) to a parameter fitted from the target or to a self-cited uniqueness theorem; the central claim therefore retains independent content from the unit-index bound.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on Greenberg’s split-prime criterion for the λ-invariant, a capitulation argument in the style of Kumakawa, and the new square-class computation that q(K₂) ≤ 2; no free parameters are introduced.

axioms (2)

domain assumption Greenberg’s criterion applies directly to the split-prime case for the cyclotomic ℤ₂-extension
Invoked as the starting point for the proof that λ₂(K) = 0.
domain assumption The capitulation argument modeled on Kumakawa is valid under the given quartic-residue conditions
Used to relate class groups across the tower.

pith-pipeline@v0.9.0 · 5549 in / 1458 out tokens · 53434 ms · 2026-05-12T02:17:40.145488+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

152 extracted references · 152 canonical work pages

[1]

Manuscripta Math

Ayad, Mohamed , TITLE =. Manuscripta Math. , FJOURNAL =. 1992 , NUMBER =. doi:10.1007/BF02567763 , URL =

work page doi:10.1007/bf02567763 1992
[2]

, TITLE =

Balakrishnan, Jennifer S. , TITLE =. J. Number Theory , FJOURNAL =. 2016 , PAGES =. doi:10.1016/j.jnt.2015.07.003 , URL =

work page doi:10.1016/j.jnt.2015.07.003 2016
[3]

Valeurs de fonctions th

Bertrand, Daniel , journal=. Valeurs de fonctions th. 1980 , publisher=

work page 1980
[4]

2019 , school=

Topics in the theory of p-adic heights on elliptic curves , author=. 2019 , school=

work page 2019
[5]

Bilu, Yuri and Parent, Pierre , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2011 , NUMBER =. doi:10.4007/annals.2011.173.1.13 , URL =

work page doi:10.4007/annals.2011.173.1.13 2011
[6]

, TITLE =

Bourdon, Abbey and Clark, Pete L. , TITLE =. Pacific J. Math. , FJOURNAL =. 2020 , NUMBER =. doi:10.2140/pjm.2020.305.43 , URL =

work page doi:10.2140/pjm.2020.305.43 2020
[7]

and Krijan, Ivan and Najman, Filip , TITLE =

Chou, Michael and Daniels, Harris B. and Krijan, Ivan and Najman, Filip , TITLE =. New York J. Math. , FJOURNAL =. 2021 , PAGES =

work page 2021
[8]

and Hahn, S

Cheon, J. and Hahn, S. , TITLE =. Manuscripta Math. , FJOURNAL =. 1998 , NUMBER =. doi:10.1007/s002290050104 , URL =

work page doi:10.1007/s002290050104 1998
[9]

and Pollack, Paul , TITLE =

Clark, Pete L. and Pollack, Paul , TITLE =. C. R. Math. Acad. Sci. Paris , FJOURNAL =. 2015 , NUMBER =. doi:10.1016/j.crma.2015.05.004 , URL =

work page doi:10.1016/j.crma.2015.05.004 2015
[10]

and Pollack, Paul , TITLE =

Clark, Pete L. and Pollack, Paul , TITLE =. Q. J. Math. , FJOURNAL =. 2017 , NUMBER =. doi:10.1093/qmath/hax024 , URL =

work page doi:10.1093/qmath/hax024 2017
[11]

2005 , PAGES =

Diamond, Fred and Shurman, Jerry , TITLE =. 2005 , PAGES =

work page 2005
[12]

Kim, Byoung Du , TITLE =. J. Aust. Math. Soc. , FJOURNAL =. 2013 , NUMBER =. doi:10.1017/S1446788713000165 , URL =

work page doi:10.1017/s1446788713000165 2013
[13]

Israel J

Frey, Gerhard , TITLE =. Israel J. Math. , FJOURNAL =. 1994 , NUMBER =. doi:10.1007/BF02758637 , URL =

work page doi:10.1007/bf02758637 1994
[14]

Gonz\'alez-Jim\'enez, Enrique and Najman, Filip , TITLE =. Math. Comp. , FJOURNAL =. 2020 , NUMBER =. doi:10.1090/mcom/3478 , URL =

work page doi:10.1090/mcom/3478 2020
[15]

Gold, Robert , TITLE =. J. Number Theory , FJOURNAL =. 1974 , PAGES =. doi:10.1016/0022-314X(74)90034-1 , URL =

work page doi:10.1016/0022-314x(74)90034-1 1974
[16]

Arithmetic theory of elliptic curves (

Greenberg, Ralph , TITLE =. Arithmetic theory of elliptic curves (. 1999 , ISBN =. doi:10.1007/BFb0093453 , URL =

work page doi:10.1007/bfb0093453 1999
[17]

Greenberg, Ralph and Vatsal, Vinayak , TITLE =. Invent. Math. , FJOURNAL =. 2000 , NUMBER =. doi:10.1007/s002220000080 , URL =

work page doi:10.1007/s002220000080 2000
[18]

Harvey, David , TITLE =. LMS J. Comput. Math. , FJOURNAL =. 2008 , PAGES =. doi:10.1112/S1461157000000528 , URL =

work page doi:10.1112/s1461157000000528 2008
[19]

Iovita, Adrian and Pollack, Robert , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2006 , PAGES =. doi:10.1515/CRELLE.2006.069 , URL =

work page doi:10.1515/crelle.2006.069 2006
[20]

, TITLE =

Katz, Nicholas M. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1976 , NUMBER =. doi:10.2307/1970966 , URL =

work page doi:10.2307/1970966 1976
[21]

Kobayashi, Shin-ichi , TITLE =. Invent. Math. , FJOURNAL =. 2003 , NUMBER =. doi:10.1007/s00222-002-0265-4 , URL =

work page doi:10.1007/s00222-002-0265-4 2003
[22]

The p-adic Gross-Zagier formula for elliptic curves at supersingular primes , author=. Invent. Math. , volume=. 2013 , publisher=

work page 2013
[23]

2020 , PAGES =

Kundu, Debanjana , TITLE =. 2020 , PAGES =

work page 2020
[24]

arXiv preprint arXiv:2102.02411 , year=

Statistics for Iwasawa invariants of elliptic curves , author=. arXiv preprint arXiv:2102.02411 , year=

work page arXiv
[25]

Elliptic curves, modular forms and

Lawson, Tyler and Wuthrich, Christian , TITLE =. Elliptic curves, modular forms and. 2016 , ISBN =. doi:10.1007/978-3-319-45032-2\_11 , URL =

work page doi:10.1007/978-3-319-45032-2 2016
[26]

Lozano-Robledo, \'Alvaro , TITLE =. Math. Ann. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s00208-013-0906-5 , URL =

work page doi:10.1007/s00208-013-0906-5 2013
[27]

Algebra Number Theory , FJOURNAL =

Lozano-Robledo, \'Alvaro , TITLE =. Algebra Number Theory , FJOURNAL =. 2022 , NUMBER =. doi:10.2140/ant.2022.16.777 , URL =

work page doi:10.2140/ant.2022.16.777 2022
[28]

Mazur, Barry , TITLE =. Invent. Math. , FJOURNAL =. 1972 , PAGES =. doi:10.1007/BF01389815 , URL =

work page doi:10.1007/bf01389815 1972
[29]

, TITLE =

Mazur, B. , TITLE =. Inst. Hautes \'Etudes Sci. Publ. Math. , FJOURNAL =. 1977 , PAGES =

work page 1977
[30]

, TITLE =

Mazur, B. , TITLE =. Invent. Math. , FJOURNAL =. 1978 , NUMBER =. doi:10.1007/BF01390348 , URL =

work page doi:10.1007/bf01390348 1978
[31]

Mazur, Barry and Stein, William and Tate, John , TITLE =. Doc. Math. , FJOURNAL =. 2006 , PAGES =

work page 2006
[32]

and Tate, J

Mazur, B. and Tate, J. , TITLE =. Duke Math. J. , FJOURNAL =. 1991 , NUMBER =. doi:10.1215/S0012-7094-91-06229-0 , URL =

work page doi:10.1215/s0012-7094-91-06229-0 1991
[33]

and Tate, J

Mazur, B. and Tate, J. and Teitelbaum, J. , TITLE =. Invent. Math. , FJOURNAL =. 1986 , NUMBER =. doi:10.1007/BF01388731 , URL =

work page doi:10.1007/bf01388731 1986
[34]

Manuscripta Math

Matsuno, Kazuo , TITLE =. Manuscripta Math. , FJOURNAL =. 2007 , NUMBER =. doi:10.1007/s00229-006-0068-9 , URL =

work page doi:10.1007/s00229-006-0068-9 2007
[35]

Compositio Math

McConnell, Gary , TITLE =. Compositio Math. , FJOURNAL =. 1996 , NUMBER =

work page 1996
[36]

Merel, Lo\"ic , TITLE =. Invent. Math. , FJOURNAL =. 1996 , NUMBER =. doi:10.1007/s002220050059 , URL =

work page doi:10.1007/s002220050059 1996
[37]

Seminar on

Perrin-Riou, Bernadette , TITLE =. Seminar on. 1982 , ISBN =

work page 1982
[38]

Perrin-Riou, Bernadette , TITLE =. M\'em. Soc. Math. France (N.S.) , FJOURNAL =. 1984 , PAGES =

work page 1984
[39]

Perrin-Riou, Bernadette , TITLE =. Invent. Math. , FJOURNAL =. 1990 , NUMBER =. doi:10.1007/BF01234420 , URL =

work page doi:10.1007/bf01234420 1990
[40]

Bernardi, Dominique and Perrin-Riou, Bernadette , TITLE =. C. R. Acad. Sci. Paris S\'er. I Math. , FJOURNAL =. 1993 , NUMBER =

work page 1993
[41]

Perrin-Riou, Bernadette , TITLE =. Invent. Math. , FJOURNAL =. 1994 , NUMBER =. doi:10.1007/BF01231755 , URL =

work page doi:10.1007/bf01231755 1994
[42]

Proceedings of the

Perrin-Riou, Bernadette , TITLE =. Proceedings of the. 1995 , ISBN =

work page 1995
[43]

Duke Math

Pollack, Robert , TITLE =. Duke Math. J. , FJOURNAL =. 2003 , NUMBER =. doi:10.1215/S0012-7094-03-11835-9 , URL =

work page doi:10.1215/s0012-7094-03-11835-9 2003
[44]

Pollack, Robert and Rubin, Karl , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2004 , NUMBER =. doi:10.4007/annals.2004.159.447 , URL =

work page doi:10.4007/annals.2004.159.447 2004
[45]

, TITLE =

Ray, Anwesh and Sujatha, R. , TITLE =. Canad. J. Math. , FJOURNAL =. 2023 , NUMBER =. doi:10.4153/S0008414X21000699 , URL =

work page doi:10.4153/s0008414x21000699 2023
[46]

Bulletin (New Series) of the American Mathematical Society , volume=

Jean-Pierre Serre, Abelian l -adic representations and elliptic curves , author=. Bulletin (New Series) of the American Mathematical Society , volume=. 1990 , publisher=

work page 1990
[47]

Rouse, Jeremy and Zureick-Brown, David , TITLE =. Res. Number Theory , FJOURNAL =. 2015 , PAGES =. doi:10.1007/s40993-015-0013-7 , URL =

work page doi:10.1007/s40993-015-0013-7 2015
[48]

Schneider, Peter , TITLE =. Invent. Math. , FJOURNAL =. 1983 , NUMBER =. doi:10.1007/BF01389099 , URL =

work page doi:10.1007/bf01389099 1983
[49]

Schneider, Peter , TITLE =. Invent. Math. , FJOURNAL =. 1982 , NUMBER =. doi:10.1007/BF01389362 , URL =

work page doi:10.1007/bf01389362 1982
[50]

Schneider, Peter , TITLE =. Invent. Math. , FJOURNAL =. 1985 , NUMBER =. doi:10.1007/BF01388978 , URL =

work page doi:10.1007/bf01388978 1985
[51]

Serre, Jean-Pierre , TITLE =. Inst. Hautes \'Etudes Sci. Publ. Math. , FJOURNAL =. 1981 , PAGES =

work page 1981
[52]

1968 , PAGES =

Serre, Jean-Pierre , TITLE =. 1968 , PAGES =

work page 1968
[53]

, TITLE =

Silverman, Joseph H. , TITLE =. J. Number Theory , FJOURNAL =. 1988 , NUMBER =. doi:10.1016/0022-314X(88)90019-4 , URL =

work page doi:10.1016/0022-314x(88)90019-4 1988
[54]

, TITLE =

Silverman, Joseph H. , TITLE =. 1994 , PAGES =. doi:10.1007/978-1-4612-0851-8 , URL =

work page doi:10.1007/978-1-4612-0851-8 1994
[55]

, TITLE =

Silverman, Joseph H. , TITLE =. 1986 , PAGES =. doi:10.1007/978-1-4757-1920-8 , URL =

work page doi:10.1007/978-1-4757-1920-8 1986
[56]

2013 , url=

Course on Mazur's Theorem , author=. 2013 , url=

work page 2013
[57]

Ito , TITLE =

Sprung, Florian E. Ito , TITLE =. J. Number Theory , FJOURNAL =. 2012 , NUMBER =. doi:10.1016/j.jnt.2011.11.003 , URL =

work page doi:10.1016/j.jnt.2011.11.003 2012
[58]

Sprung, Florian , TITLE =. Res. Number Theory , FJOURNAL =. 2015 , PAGES =. doi:10.1007/s40993-015-0018-2 , URL =

work page doi:10.1007/s40993-015-0018-2 2015
[59]

Stein, William and Wuthrich, Christian , TITLE =. Math. Comp. , FJOURNAL =. 2013 , NUMBER =. doi:10.1090/S0025-5718-2012-02649-4 , URL =

work page doi:10.1090/s0025-5718-2012-02649-4 2013
[60]

2003 , school=

Elliptic curves and related sequences , author=. 2003 , school=

work page 2003
[61]

and Tate, John , TITLE =

Silverman, Joseph H. and Tate, John , TITLE =. 1992 , PAGES =. doi:10.1007/978-1-4757-4252-7 , URL =

work page doi:10.1007/978-1-4757-4252-7 1992
[62]

The arithmetic of elliptic curves , author=. Invent. Math. , volume=. 1974 , publisher=

work page 1974
[63]

, TITLE =

Washington, Lawrence C. , TITLE =. 1982 , PAGES =. doi:10.1007/978-1-4684-0133-2 , URL =

work page doi:10.1007/978-1-4684-0133-2 1982
[64]

Wieferich, Arthur , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 1909 , PAGES =. doi:10.1515/crll.1909.136.293 , URL =

work page doi:10.1515/crll.1909.136.293 1909
[65]

Wuthrich, Christian , TITLE =. J. London Math. Soc. (2) , FJOURNAL =. 2004 , NUMBER =. doi:10.1112/S0024610704005277 , URL =

work page doi:10.1112/s0024610704005277 2004
[66]

Zerbes, Sarah Livia , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2009 , NUMBER =. doi:10.1112/plms/pdn049 , URL =

work page doi:10.1112/plms/pdn049 2009
[67]

arXiv preprint arXiv:2409.18021 , year=

A bound on the lambda invariants of supersingular elliptic curves , author=. arXiv preprint arXiv:2409.18021 , year=

work page arXiv
[68]

arXiv preprint arXiv:2408.07826 , year=

The Iwasawa of Elliptic Curves over , author=. arXiv preprint arXiv:2408.07826 , year=

work page arXiv
[69]

2024 , shorthand =

SageMath Development Team , author =. 2024 , shorthand =

work page 2024
[70]

Sage Computations

Foivos Chnaras. Sage Computations. 2025 , note =

work page 2025
[71]

C comps for sqrts

Noah Kupinsky. C comps for sqrts. 2025 , note =

work page 2025
[72]

arXiv preprint arXiv:2402.06028 , year=

Iwasawa lambda invariant and Massey product , author=. arXiv preprint arXiv:2402.06028 , year=

work page arXiv
[73]

Forum Math

Lam, Yeuk Hay Joshua and Liu, Yuan and Sharifi, Romyar and Wake, Preston and Wang, Jiuya , TITLE =. Forum Math. Sigma , FJOURNAL =. 2023 , PAGES =. doi:10.1017/fms.2022.103 , URL =

work page doi:10.1017/fms.2022.103 2023
[74]

, TITLE =

Sharifi, R. , TITLE =. Mathematisches. 2004 , ISBN =

work page 2004
[75]

, TITLE =

Sharifi, Romyar T. , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2007 , PAGES =. doi:10.1515/CRELLE.2007.010 , URL =

work page doi:10.1515/crelle.2007.010 2007
[76]

, TITLE =

Friedman, Eduardo C. , TITLE =. Invent. Math. , FJOURNAL =. 1981 , NUMBER =. doi:10.1007/BF01396627 , URL =

work page doi:10.1007/bf01396627 1981
[77]

Friedman, Eduardo , TITLE =. Math. Ann. , FJOURNAL =. 1985 , NUMBER =. doi:10.1007/BF01455793 , URL =

work page doi:10.1007/bf01455793 1985
[78]

Kim, Myoungil and Sun, Hae-Sang , title =

work page
[79]

Acta Arith

Kundu, Debanjana and Lei, Antonio , title =. Acta Arith. , fjournal =. 2023 , pages =. doi:10.4064/aa220518-28-2 , url =

work page doi:10.4064/aa220518-28-2 2023
[80]

, TITLE =

Lamplugh, J. , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2015 , NUMBER =. doi:10.1112/jlms/jdv004 , URL =

work page doi:10.1112/jlms/jdv004 2015

Showing first 80 references.