The Iwasawa Main Conjecture for Hilbert Modular Forms

Wan, Xin

doi:10.1017/fms.2015.16

Cited by 34 publications

(41 citation statements)

References 62 publications

(177 reference statements)

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“…When is split above , Theorem A can essentially be deduced from a general theorem of Hida [Hid91] (cf. [Wan15, §7.3]), except for the location of the possible poles. Theorems C (4) and D in the classical context are due to Howard [How05] (in fact, Theorems B and C (4) were first envisioned by Mazur [Maz83] in that context, whereas Perrin-Riou [Per87] had conjectured the equality in (1.5.2)).…”

Section: Introductionmentioning

confidence: 99%

The -adic Gross–Zagier formula on Shimura curves

Disegni¹

2017

Compositio Math.

117

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Abstract. -We prove a general formula for the p-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above p. The formula is in terms of the cyclotomic derivative of a Rankin-Selberg p-adic L-function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author, to the context of the work of Yuan-Zhang-Zhang on the archimedean Gross-Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is +1 rather than −1, by an anticyclotomic version of the Waldspurger formula.When combined with work of Fouquet, the anticyclotomic Gross-Zagier formula implies one divisibility in a p-adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.

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Section: Introductionmentioning

confidence: 99%

The -adic Gross–Zagier formula on Shimura curves

Disegni¹

2017

Compositio Math.

117

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“…Since all the elliptic curves here have no semistable prime in their conductors, Skinner-Urban's work [SU14] does not apply to these examples. X. Wan's work [Wan15] could apply only if one can find suitable real quadratic fields. From now on, λ means Iwasawa λ-invariants, not a place dividing p. All four elliptic curves E i (i = 1, · · · , 4) share the following properties:…”

Section: Examplesmentioning

confidence: 99%

“…Kur14b, Theorem 4.(2)]. In[Wan15, Theorem 4], it is required to find a suitable real quadratic field. It seems difficult to find it (at least algorithmically).…”

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confidence: 99%

On the indivisibility of derived Kato’s Euler systems and the main conjecture for modular forms

Kim

Sun

2020

Sel. Math. New Ser.

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We provide a simple and efficient numerical criterion to verify the Iwasawa main conjecture and the indivisibility of derived Kato's Euler systems for modular forms of weight two at any good prime under mild assumptions. In the ordinary case, the criterion works for all members of a Hida family once and for all. The criterion also shows a relation between Kato's Euler systems and Euler systems of Gauss sum typeà la Kurihara. The key ingredient is the explicit computation of the integral image of the derived Kato's Euler systems under the dual exponential map. We provide some explicit new examples at the end. This work does not appeal to the Eisenstein congruence method at all.

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“…The Iwasawa main conjecture for classical modular forms of integral weight is formulated over

G L_{2}

and this has been a recent, active research area with its connections to the Birch and Swinnerton‐Dyer conjecture, see [8, 9]. Provided one has both the analytic and algebraic machinery, the Iwasawa main conjecture can be formulated for higher dimensional modular forms and groups, for example, [17].…”

Section: Introductionmentioning

confidence: 99%

p‐adic L‐functions on metaplectic groups

Mercuri

2020

J. London Math. Soc.

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With respect to the analytic‐algebraic dichotomy, the theory of Siegel modular forms of half‐integral weight is lopsided; the analytic theory is strong, whereas the algebraic lags behind. In this paper, we capitalise on this to establish the fundamental object needed for the analytic side of the Iwasawa main conjecture — the p‐adic L‐function obtained by interpolating the complex L‐function at special values. This is achieved through the Rankin–Selberg method and the explicit Fourier expansion of non‐holomorphic Siegel Eisenstein series. The construction of the p‐stabilisation in this setting is also of independent interest.

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The Iwasawa Main Conjecture for Hilbert Modular Forms

Cited by 34 publications

References 62 publications

The -adic Gross–Zagier formula on Shimura curves

The -adic Gross–Zagier formula on Shimura curves

On the indivisibility of derived Kato’s Euler systems and the main conjecture for modular forms

p‐adic L‐functions on metaplectic groups

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