Pith. sign in

REVIEW 2 major objections 5 minor 35 references

Anticyclotomic Iwasawa theory of CM elliptic curves at ramified primes

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves an integral Iwasawa main conjecture for CM elliptic curves at ramified primes: the characteristic ideal of the signed Selmer group $X^{-\varepsilon}(E)$ equals the ideal of the $p$-adic $L$-function…

desk verdict A real main conjecture at ramified primes, but the load-bearing local input is an unpublished prequel; referee both together. read the letter →

arxiv 2608.06879 v1 pith:IND7FFI6 submitted 2026-08-07 math.NT

Ashay Burungale , Shinichi Kobayashi , Kentaro Nakamura , Kazuto Ota This is my paper · ORCID
classification math.NT MSC 11R2311G0511G1511S40
keywords IwasawamainconjectureCMellipticcurvesanticyclotomicZ_p-extensionramifiedprimessignedSelmergroupslocalsigndecompositionGaussiancyclotomicpolynomialsp-adicL-functions
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 aims to establish an integral Iwasawa main conjecture for CM elliptic curves at primes $p$ ramified in the imaginary quadratic field, a regime where earlier frameworks stop working. The central result equates the characteristic ideal of a signed anticyclotomic Selmer group, built from one half of a canonical splitting of local Iwasawa cohomology, with the ideal generated by the $p$-adic $L$-function. The root numbers of twists split evenly at every layer of the anticyclotomic tower, so no single local condition interpolates the Bloch–Kato subgroups; the signed framework handles both signs at once. If correct, this gives the first main conjecture for a $p$-adic deformation none of whose geometric specialisations is trianguline at $p$, and it ties the $p$-adic $L$-function both to central $L$-values and to Selmer classes responsible for Mordell–Weil growth.

What carries the argument

The load-bearing object is the local sign decomposition from the prequel: for the conjugate symplectic self-dual deformation over the ramified local field, the local Iwasawa cohomology $H^1(\Psi, T_\psi)$ splits as a direct sum of two free rank-one Lagrangian $\Lambda$-submodules $H^1_+ \oplus H^1_-$ whose specialisations at de Rham twists reproduce the Bloch–Kato subgroups exactly when the completed local $\varepsilon$-constant has the matching sign. On top of this, the paper introduces Gaussian plus/minus cyclotomic polynomials $\Phi^\pm_k(\gamma)$, products over all anticyclotomic characters whose $\varepsilon$-constant has a given sign; these polynomials annihilate the opposite-sign specialisations and generate the signed Bloch–Kato subgroups at finite layers. The signed Selmer groups are then defined by the corresponding Lagrangian local conditions, and the proof of the main conjecture globalises them via elliptic units: an anticyclotomic elliptic-unit main conjecture, descended from the classical elliptic-unit main conjecture, is transformed by Poitou–Tate duality into the equality of characteristic ideals.

What would settle it

Compute a single finite layer: take a ramified quadratic extension $\Psi/\mathbb{Q}_p$ and a character $\psi$ satisfying the conjugate symplectic self-dual condition, and check whether the plus/minus Bloch–Kato subgroups really give the claimed $p$-torsion-free direct-sum decomposition of Theorem 2.18; a failure at any $n$ would contradict the construction. Alternatively, find one anticyclotomic character $\chi$ whose local epsilon constant does not follow the quadratic-residue rule $\varepsilon(\phi\chi^b) = (b/p)\varepsilon(\phi\chi)$, which would break the equidistribution underlying the signed framework.

Watch

Extended reading notes

Core claim

The authors' central claim, stated as Theorem 1.7, is that for a CM elliptic curve $E$ over $\mathbb{Q}$, an odd prime $p \geq 5$ ramified in the CM field, and the sign $\varepsilon$ determined by the ratio of global and local epsilon constants, the signed Selmer group $X^{-\varepsilon}(E)$ is a torsion module over the anticyclotomic Iwasawa algebra $\Lambda$, and its characteristic ideal is exactly the principal ideal generated by the integral $p$-adic $L$-function: $\mathrm{Ch}_\Lambda(X^{-\varepsilon}(E)) = (\mathscr{L}_{p,v_\varepsilon}(E))$ as ideals of $\Lambda$. This is complemented by an interpolation theorem showing that $\mathscr{L}_p(E)$ recovers the central Hecke $L$-values of the twists with epsilon-constant $+1$, for characters of arbitrary infinity type, and by a theorem expressing its values at twists with epsilon-constant $-1$ as a product of a dual exponential value and a Bloch–Kato logarithm of a Selmer class. The paper also proves that the Mordell–Weil rank over the $n$-th layer satisfies $\operatorname{rank}_{\mathcal{O}_K} E(K^{\mathrm{ac}}_n) = (p^n-1)/2 + c$ for all sufficiently large $n$.

Load-bearing premise

The entire construction rests on the prequel's local sign decomposition, the claim that over each layer of the ramified anticyclotomic tower the local deformation splits into two rank-one Lagrangian pieces that exactly track the Bloch–Kato subgroups by sign; if that decomposition fails, the signed Selmer groups, the $p$-adic $L$-function, and the main conjecture all collapse.

Editorial extensions

If this is right

  • The $p$-adic $L$-function $\mathscr{L}_{p,v_\varepsilon}(E)$ determines the full characteristic ideal of the signed Selmer group $X^{-\varepsilon}(E)$, so the algebraic structure of that Selmer group is governed by an analytic object.
  • The interpolation formula recovers central Hecke $L$-values at all de Rham twists with $\varepsilon = +1$, including twists of arbitrary infinity type, making $\mathscr{L}_p(E)$ a genuine $p$-adic $L$-function for the ramified anticyclotomic deformation.
  • Values of $\mathscr{L}_p(E)$ at twists with $\varepsilon = -1$ are identified with Bloch–Kato logarithms and dual exponentials of Selmer classes, linking the $L$-function outside its interpolation range to non-torsion cohomology classes.
  • The control theorem yields the asymptotic formula $\operatorname{rank}_{\mathcal{O}_K} E(K^{\mathrm{ac}}_n) = (p^n-1)/2 + c$ for $n \gg 0$, matching the equidistribution of root numbers.
  • The theorem supplies a test case in which a main conjecture is provable even though no geometric specialisation of the deformation is trianguline at $p$.

Reading between the lines

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

  • If the same two-Lagrangian template is applied to other deformations whose local epsilon constants equidistribute, one would predict analogous signed main conjectures, with the Gaussian polynomials replaced by finer sign filtrations.
  • The characteristic-ideal equality, combined with standard control theorems, should determine the asymptotic order of the $p$-primary Tate–Shafarevich groups at finite layers in terms of special values of $\mathscr{L}_p$; this is a concrete numerical prediction.
  • The formula at $\varepsilon = -1$ twists suggests a derivative-type relation: the first Taylor coefficient of $\mathscr{L}_p$ at such a twist should be the height or logarithm of the corresponding Selmer class, analogous to BDP-type formulas in the inert-prime setting.
  • The plus/minus local points constructed here are natural candidates for the local input in a ramified-prime Euler system, which would give an independent route to the main conjecture.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. This paper develops an integral Iwasawa theory for anticyclotomic Z_p-extensions of CM fields at primes p ramified in K. For a conjugate symplectic self-dual Hecke character φ of infinity type (1,0), the authors use the local sign decomposition of [5] to define signed Selmer groups Sel^{±}, a Rubin-type p-adic L-function L_{p,v_ε}(E) as the coordinate of the anticyclotomic elliptic unit in the ε-Lagrangian submodule, and prove an integral Iwasawa main conjecture Ch_Λ(X^{-ε}(E)) = (L_{p,v_ε}(E)) (Theorem 1.7). They also prove an interpolation formula for L_p(E) at de Rham specialisations with ε(φχ)=+1 (Theorem 1.5), and a formula at twists with ε(φχ)=−1 relating L_p(E) to logarithms of Selmer elements (Theorem 1.8). The main proofs rely on a descent from Rubin's elliptic unit main conjecture (Theorem 3.14), an explicit reciprocity law (Prop. 4.10), and the sign containment of the elliptic unit (Prop. 4.6).

Significance. If the local results imported from [5] are valid, the paper achieves a genuine milestone: the first Iwasawa main conjecture of p-adic-L-function type for a deformation with no trianguline geometric specialisation, together with a precise control theorem and asymptotic rank formula. The paper is carefully structured and proves substantial results in the text: the Gaussian plus/minus polynomials and their class-number interpretation (Lemma 2.11), the descent from Rubin's theorem to the anticyclotomic elliptic unit main conjecture (Theorem 3.14), and the explicit reciprocity law (Prop. 4.10). The central deficit is that Theorem 2.5 and Proposition 4.6, on which the entire framework is built, are imported from an unpublished preprint and asserted without self-contained proof.

major comments (2)
  1. [Section 2, Theorem 2.5] The decomposition H^1(Ψ, T_ψ) = H^1_+ ⊕ H^1_- into free rank-one Lagrangian Λ-submodules is stated as a consequence of [5, Thm. 7.25] and Lemma 2.3, but no proof is given in this paper. This decomposition is the foundation for the signed Selmer groups (Definition 5.1), the local points (Theorem 2.18), the p-adic L-function (Definition 4.7), and the characteristic-ideal equality in Theorem 5.6. As [5] is an unpublished preprint by the same authors, the main theorem is conditional on an external result. Please either include a full proof of Theorem 2.5 or state the main conjecture explicitly as conditional on [5] being made publicly available and refereed.
  2. [Section 4, Proposition 4.6] The containment loc_p(z^{ac}_{p^∞ f}) ∈ H^1_ε is the sign input that selects the correct Selmer group in Theorem 1.7. The proof is deferred with 'one may proceed just as in the proof of [29, Cor. 3.3] or [5, Prop. 8.11]'. This step is directly load-bearing: if the sign were reversed, the roles of ε and −ε in the exact sequence (5.3) would be interchanged and the stated equality Ch_Λ(X^{-ε}) = (L_{p,v_ε}) would be false. The proof must be written out.
minor comments (5)
  1. [Section 1.1] The assertion that no geometric specialisation is trianguline at p is stated without proof or reference. Since this claim appears in the abstract and frames the novelty but is not used in the proof of Theorem 1.7, please add a proof or reference, or explicitly mark it as a conjecture or expectation.
  2. [Definition 2.9 and Remark 4.3] The sign convention for ϵ(Ψ) differs from the notation in [5], as noted in Remark 4.3 for the global case; a short table comparing the two conventions would help the reader avoid sign errors when checking the local inputs.
  3. [Section 4.4.1, equation (4.11)] The factor (−2πp/|d_K|)^k involves the notation 2πp which is only implicitly defined; please spell out that this is the p-adic period associated to the choices of e_{Q_p} and the Haar measures.
  4. [References, [32]] Reference [32] is given as 'Ph.D. thesis, Princeton University' with no title, year, or other identifying information; please provide the full citation.
  5. [Title and front matter] The running title in the text contains spacing artifacts ('IW ASA W A THEOR Y'); please ensure the compiled manuscript is free of such artifacts.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the signed main conjecture is proved from Rubin's independent elliptic-unit main conjecture via a cited local decomposition theorem, not by construction.

full rationale

The central derivation is not definitionally circular. In Definition 4.7, L_{p,v_epsilon,t}(phi) is defined as the coordinate of the anticyclotomic elliptic unit loc_p(z) in the free rank-one epsilon-Lagrangian H^1_epsilon, so the equality Ch(H^1_epsilon / Lambda loc_p(z)) = (L_p) is immediate. But the target characteristic ideal Ch(X^{-epsilon}) is an independent global object. The proof of Theorem 5.6 uses the exact sequence (5.3) and the characteristic-ideal computation (5.4) to express Ch(X^{-epsilon}) as a product of Ch(X^ac_str) and Ch(H^1_epsilon / S^ac_rel). The first factor is matched to Ch(S^epsilon / Lambda z) by the elliptic-unit main conjecture Theorem 3.14, which is deduced from Rubin's external theorem [30], not from the signed Selmer machinery. The second factor is then combined with the exact sequence 0 -> S^epsilon/Lambda z -> H^1_epsilon/Lambda loc_p(z) -> H^1_epsilon/S^epsilon -> 0 to recover (L_p). Every input in this chain has independent content: Proposition 4.6 (the sign containment loc_p(z) in H^1_epsilon) is a proved statement about elliptic units, not a definitional choice; Theorem 2.5 (the local sign decomposition) is imported from the authors' preprint [5] as a separate local theorem with its own stated hypotheses, and it does not assume the main conjecture. Under the review rules, such a parameter-free cited result counts as real evidence rather than circularity. The interpolation Theorem 4.12 likewise rests on Kato's explicit reciprocity law (Proposition 4.10) plus the non-vanishing of the period from [5, Cor. 3.2], again independent of the main conjecture. The unproved assertion in Section 1.1 that no geometric specialisation is trianguline is a correctness/verification risk, but it is not used in the proof of Theorem 1.7 and does not constitute a circular step. Accordingly, no load-bearing step reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on two main inputs: the local sign decomposition and construction of L_p from the authors' preprint [5], and Rubin's elliptic unit main conjecture. Both are domain theorems rather than fitted parameters. No free parameters or invented entities (in the sense of new particles, forces, or dimensions) appear; the Gaussian plus/minus cyclotomic polynomials are new mathematical objects but are constructed from existing characters and are not postulated entities.

assumptions (4)
  • domain assumption Correctness of the local sign decomposition and freeness of H^1_± from [5, Thm 1.18 and 7.25]
    The paper's global results (Theorems 1.3, 1.7) and the p-adic L-function all depend on this decomposition into two free rank-one Lagrangian submodules interpolating Bloch-Kato subgroups; it is cited from the authors' unpublished preprint [5], not proved here.
  • domain assumption Assumption 3.1(2): existence of a finite prime-to-p extension H over which the Hecke character φ is associated to an abelian variety
    Used to apply Rubin's main conjecture and the control theorems. The authors note it holds for canonical Hecke characters with H the Hilbert class field, and p≥5 with p∤h_K in the main theorems.
  • standard math Rubin's elliptic unit main conjecture for imaginary quadratic fields [30]
    External theorem used to deduce the anticyclotomic elliptic unit main conjecture (Theorem 3.14), which is the key input for Theorem 1.7.
  • standard math Kato's explicit reciprocity law [18, Thm 12.5 and related results]
    External theorem used in the proof of Proposition 4.10, which underpins the interpolation formula in Theorem 1.5.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Anticyclotomic Iwasawa theory of CM elliptic curves at ramified primes." pith.science (2026). https://pith.science/paper/IND7FFI6

@misc{pith2026260806879,
  author       = {Pith},
  title        = {Pith review of: Anticyclotomic Iwasawa theory of CM elliptic curves at ramified primes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IND7FFI6}},
  note         = {Machine review of arXiv:2608.06879}
}
abstract

We propose an integral framework for the anticyclotomic Iwasawa theory of CM elliptic curves $E$ at primes $p$ ramified in the CM field. The $\varepsilon$-constants of the geometric specialisations of the associated $p$-adic conjugate symplectic self-dual deformation equidistribute between $\pm 1$ within every layer of the anticyclotomic tower, and none of the geometric specialisations are trianguline at $p$. We define signed Selmer groups via the Lagrangian local conditions arising from the local sign decomposition established in the prequel \cite{BKNO}, and our central result is the formulation and proof of an integral Iwasawa main conjecture relating one of them to the $p$-adic $L$-function $\mathscr{L}_p(E)$ constructed there. We further show that $\mathscr{L}_p(E)$ interpolates the central Hecke $L$-values of the twists with $\varepsilon$-constant $+1$, including twists of arbitrary infinity type, and relate its values at twists with $\varepsilon$-constant $-1$ to the $p$-adic logarithm of certain Selmer elements. This provides the first Iwasawa main conjecture in terms of a $p$-adic $L$-function and Selmer groups for a $p$-adic deformation admitting no trianguline geometric specialisation. The proofs rest on our resolution of a Rubin-type conjecture for the underlying local deformation, together with a theory of plus/minus local points along the anticyclotomic tower, based on the Gaussian plus/minus cyclotomic polynomials rooted in Gauss' Disquisitiones Arithmeticae.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

35 extracted references · 34 canonical work pages

  1. [5]

    A local sign decomposition for symplectic self-dual Galois representations of rank two

    A. Burungale, S. Kobayashi, K. Nakamura, and K. Ota,A local sign decomposition of Galois cohomology for symplectic self-dual representations, arXiv:2508.17776

    work page Pith review arXiv

  2. [1]

    Agboola and B

    A. Agboola and B. Howard,Anticyclotomic Iwasawa theory of CM elliptic curves. II, Math. Res. Lett.12(2005), no. 5-6, 611–621

    work page 2005

  3. [2]

    Andreatta and A

    F. Andreatta and A. Iovita,Katz typep-adicL-functions for primespnon-split in theCMfield, Comment. Math. Helv.99(2024), no. 4, 641–716

    work page 2024

  4. [3]

    Bertolini, H

    M. Bertolini, H. Darmon, and K. Prasanna,Generalized Heegner cycles andp-adic RankinL-series, Duke Math. J.162(2013), no. 6, 1033–1148. With an appendix by Brian Conrad

    work page 2013

  5. [4]

    Burungale, W

    A. Burungale, W. He, S. Kobayashi, and K. Ota,HeckeL-values, definite Shimura sets and modℓnon-vanishing, arXiv:2408.13932

    arXiv

  6. [6]

    Burungale, S

    A. Burungale, S. Kobayashi, and K. Ota,Rubin ’s conjecture on local units in the anticyclotomic tower at inert primes, Ann. of Math. (2)194(2021), no. 3, 943–966

    work page 2021

  7. [7]

    ,p-adicL-functions and rational points on CM elliptic curves at inert primes, J. Inst. Math. Jussieu23 (2024), no. 3, 1417–1460

    work page 2024

  8. [8]

    ,Thep-adic valuation of local resolvents, generalized Gauss sums and anticyclotomic HeckeL-values of imaginary quadratic fields at inert primes, Amer. J. Math., to appear

    work page

Show all 35 references
  1. [9]

    ,On the Tate–Shafarevich groups of CM elliptic curves over anticyclotomicZ p-extensions at inert primes, proceedings of the Bertolinifest, to appear

  2. [10]

    Burungale, S

    A. Burungale, S. Kobayashi, K. Ota, and S. Yasuda,Kato’s epsilon conjecture for anticyclotomic CM deforma- tions at inert primes, J. Number Theory270(2025), 17–67. 41

    2025

  3. [11]

    B¨ uy¨ ukboduk,On the anticyclotomic Iwasawa theory of CM forms at supersingular primes, Rev

    K. B¨ uy¨ ukboduk,On the anticyclotomic Iwasawa theory of CM forms at supersingular primes, Rev. Mat. Iberoam. 31(2015), no. 1, 109–126

    2015

  4. [12]

    Coates and A

    J. Coates and A. Wiles,On the conjecture of Birch and Swinnerton-Dyer, Invent. Math.39(1977), no. 3, 223–251

    1977

  5. [13]

    Greenberg,On the Birch and Swinnerton-Dyer conjecture, Invent

    R. Greenberg,On the Birch and Swinnerton-Dyer conjecture, Invent. Math.72(1983), no. 2, 241–265

    1983

  6. [14]

    ,Iwasawa theory andp-adic deformations of motives, Motives (Seattle, W A, 1991), Proc. Sympos. Pure Math., vol. 55, Part 2, Amer. Math. Soc., Providence, RI, 1994, pp. 193–223

    1991

  7. [15]

    Ser., vol

    ,Introduction to Iwasawa theory for elliptic curves, Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Math. Ser., vol. 9, Amer. Math. Soc., Providence, RI, 2001, pp. 407–464

    1999

  8. [16]

    Jia,OnL-functions of Hecke characters and anticyclotomic towers, arXiv:2412.05867

    H. Jia,OnL-functions of Hecke characters and anticyclotomic towers, arXiv:2412.05867

    arXiv

  9. [17]

    Johnson-Leung and G

    J. Johnson-Leung and G. Kings,On the equivariant main conjecture for imaginary quadratic fields, J. Reine Angew. Math.653(2011), 75–114

    2011

  10. [18]

    Kato,p-adic Hodge theory and values of zeta functions of modular forms, Ast´ erisque295(2004), 117–290

    K. Kato,p-adic Hodge theory and values of zeta functions of modular forms, Ast´ erisque295(2004), 117–290

    2004

  11. [19]

    Kobayashi,Iwasawa theory for elliptic curves at supersingular primes, Invent

    S. Kobayashi,Iwasawa theory for elliptic curves at supersingular primes, Invent. Math.152(2003), no. 1, 1–36

    2003

  12. [20]

    Lerch,Essais sur le calcul du nombre des classes de formes quadratiques binaires aux coefficients entiers, Acta Math.29(1905), no

    M. Lerch,Essais sur le calcul du nombre des classes de formes quadratiques binaires aux coefficients entiers, Acta Math.29(1905), no. 1, 333–424

    1905

  13. [21]

    1, 203–293

    ,Essais sur le calcul du nombre des classes de formes quadratiques binaires aux coefficients entiers, Acta Math.30(1906), no. 1, 203–293

    1906

  14. [22]

    Mazur,Rational points of abelian varieties with values in towers of number fields, Invent

    B. Mazur,Rational points of abelian varieties with values in towers of number fields, Invent. Math.18(1972), 183–266

    1972

  15. [23]

    Perrin-Riou,FonctionsL p-adiques des repr´ esentationsp-adiques, Ast´ erisque229(1995), 198 pp

    B. Perrin-Riou,FonctionsL p-adiques des repr´ esentationsp-adiques, Ast´ erisque229(1995), 198 pp

    1995

  16. [24]

    ,Th´ eorie d’Iwasawa des repr´ esentationsp-adiques semi-stables, M´ em. Soc. Math. Fr. (N.S.)84(2001), vi+111 pp

    2001

  17. [25]

    Pollack,On thep-adicL-function of a modular form at a supersingular prime, Duke Math

    R. Pollack,On thep-adicL-function of a modular form at a supersingular prime, Duke Math. J.118(2003), no. 3, 523–558

    2003

  18. [26]

    Pottharst,Analytic families of finite-slope Selmer groups, Algebra Number Theory7(2013), no

    J. Pottharst,Analytic families of finite-slope Selmer groups, Algebra Number Theory7(2013), no. 7, 1571–1612

    2013

  19. [27]

    K. A. Ribet,Galois representations attached to eigenforms with Nebentypus, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Math., vol. 601, Springer, Berlin-New York, 1977, pp. 17–51

    1976

  20. [28]

    D. E. Rohrlich,OnL-functions of elliptic curves and anticyclotomic towers, Invent. Math.75(1984), 383–408

    1984

  21. [29]

    Rubin,Local units, elliptic units, Heegner points and elliptic curves, Invent

    K. Rubin,Local units, elliptic units, Heegner points and elliptic curves, Invent. Math.88(1987), no. 2, 405–422

    1987

  22. [30]

    main conjectures

    ,The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math.103(1991), no. 1, 25–68

    1991

  23. [31]

    ,Euler systems, Ann. of Math. Stud., vol. 147, Princeton University Press, Princeton, NJ, 2000

    2000

  24. [32]

    Sangiovanni Vincentelli

    M. Sangiovanni Vincentelli. Ph.D. thesis, Princeton University

  25. [33]

    Schneider and J

    P. Schneider and J. Teitelbaum,p-adic Fourier theory, Doc. Math.6(2001), 447–481

    2001

  26. [34]

    Serre and J

    J.-P. Serre and J. Tate,Good reduction of abelian varieties, Ann. of Math. (2)88(1968), no. 3, 492–517

    1968

  27. [35]

    Tate,Number theoretic background, Automorphic forms, representations andL-functions (Proc

    J. Tate,Number theoretic background, Automorphic forms, representations andL-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Proc. Sympos. Pure Math., vol. XXXIII, Amer. Math. Soc., Providence, RI, 1979, pp. 3–26. Ashay A. Burungale: The Univer...

    1977

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.