On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields

Longo, Matteo

doi:10.5802/aif.2197

Cited by 15 publications

(10 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This morphism is smooth outside of the finite set of supersingular points. Moreover, if x is a geometric point in the special fibre M n,nrd(H) ⊗ κ, then the fibre over x in (43) is given by a union of smooth, irreducible curves indexed by P 1 (O Fv /v n ) that intersect transversally at each supersingular point, and nowhere else.…”

Section: Integral Models Fix a Compact Open Subgroup H ⊂ B × Let Umentioning

confidence: 99%

On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms

Order¹

2013

Can. j. math.

View full text Add to dashboard Cite

Abstract. We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston and others. The construction has direct applications to Iwasawa main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated p-adic L-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique of Skinner and Urban.

show abstract

Section: Integral Models Fix a Compact Open Subgroup H ⊂ B × Let Umentioning

confidence: 99%

On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms

Order¹

2013

Can. j. math.

View full text Add to dashboard Cite

show abstract

“…To begin with, we need the following For a prime number p ∈ Σ and an integer m ≥ 1 define (25) S p m := ℓ prime number | ℓ is inert in K, ℓ ∤ N and p m | ℓ + 1 .…”

Section: ±-Eigenspacesmentioning

confidence: 99%

A refined Beilinson–Bloch conjecture for motives of modular forms

Longo

Vigni

2017

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. We propose a refined version of the Beilinson-Bloch conjecture for the motive associated with a modular form of even weight. This conjecture relates the dimension of the image of the relevant p-adic Abel-Jacobi map to certain combinations of Heegner cycles on Kuga-Sato varieties. We prove theorems in the direction of the conjecture and, in doing so, obtain higher weight analogues of results for elliptic curves due to Darmon.

show abstract

“…Denote by I f the kernel of the map T n → O f,π /(π n ) associated to the modular form f . The results contained in [L2,Theorem 4.13] when f has rational coefficients can be easily extended to this general case (see [L1,Section 4.8]) proving that there exists a canonical submodule D ⊆ T p (J ( ) )/I f , such that D T f,n (as Galois modules); moreover, T p (J ( ) ) decomposes (as Galois module) in a direct sum D ⊕ D .…”

Section: Congruences Between Modular Forms and The Euler Systemmentioning

confidence: 99%

“…In is enough to show that∂ (P * m ) ≡L f,cp ∞ . The Cěrednik-Drinfeld description of the special fiber X ( ) at of the integral model of the Shimura curve X ( ) (which is recalled in [L2,Sections 4.2,4.3] or [Zh2,Section 5]) combined with the -adic description of the image ofP m in X ( ) (see [L2,Section 5.2]) imply thatP m can be identified with a pair (g,…”

Section: Explicit Reciprocity Lawsmentioning

confidence: 99%