On the conjectures of Birch and Swinnerton-Dyer in characteristic p &gt;0

Katô, Kazuya; Trihan, Fabien

doi:10.1007/s00222-003-0299-2

Cited by 64 publications

(91 citation statements)

References 21 publications

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“…The proof shall be given in Corollaries 2.1.5 and 2.1.11. Note that, by [KT03], X p (A/K (p) ∞ ) as a submodule of (N 1 ∞ ) ∨ is also Λ-torsion. Formula (17) in section 2.2 will define f A/K Finally, we can state our analogue of the Iwasawa Main Conjecture: Theorem 1.3.…”

Section: Introductionmentioning

confidence: 99%

“…The proof is based on a generalization of a lemma of σ-linear algebra that was used to prove the cohomological formula of the Birch and Swinnerton-Dyer conjecture (see [KT03,Lemma 3.6]). Section 3.2 investigates the consequences of Theorem 1.3 in the direction of a p-adic Birch and Swinnerton-Dyer conjecture.…”

Section: Introductionmentioning

confidence: 99%

“…We endow C with the log structure induced by the smooth divisor Z and denote C # this log-scheme. Let D be the (covariant) log Dieudonné crystal over C # /W (F) associated with A/K as constructed in [KT03,IV]. Recall the following theorem of [KT03]:…”

Section: Introductionmentioning

confidence: 99%

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The Iwasawa Main Conjecture for semistable abelian varieties over function fields

et al. 2015

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Iwasawa Main Conjecture for semistable abelian varieties over function fields

et al. 2015

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“…Write K = F q (t). The main result of [KT03] (specialized to elliptic curves) states that (10) holds whenever X(K, E) is finite (in fact, it suffices that…”

Section: The Proof Of Theorem 18mentioning

confidence: 99%

There are genus one curves of every index over every infinite, finitely generated field

Clark

Lacy

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. Every nontrivial abelian variety over a Hilbertian field in which the weak Mordell-Weil theorem holds admits infinitely many torsors with period any n > 1 which is not divisible by the characteristic. The corresponding statement with "period" replaced by "index" is plausible but much more challenging. We show that for every infinite, finitely generated field K, there is an elliptic curve E /K which admits infinitely many torsors with index any n > 1.

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“…After these preliminary considerations Theorem 1.6 follows immediately from the comparison of formulas in the previous two chapters. We deduce Corollary 1.7 by comparing this theorem with results of Milne [14] and Kato-Trihan [11]. For every f ∈ C 0 (X, M) and µ ∈ M(X, R) we define the modulus µ( f ) of f with respect to µ as the Z-submodule of R generated by the elements µ( f −1 (g)), where 0 = g ∈ M. We also define the integral of f on X with respect to µ as the sum:…”

Section: (Contents)mentioning

confidence: 99%

Proof of an exceptional zero conjecture for elliptic curves over function fields

Pál¹

2006

Math. Z.

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Based on the analogy between number fields and function fields of one variable over finite fields, we formulate and prove an analogue of the exceptional zero conjecture of Mazur, Tate and Teitelbaum for elliptic curves defined over function fields. The proof uses modular parametrization by Drinfeld modular curves and the theory of non-archimedean integration. As an application we prove a refinement of the Birch-Swinnerton-Dyer conjecture if the analytic rank of the elliptic curve is zero. Mathematics Subject Classification (2000)11G18(primary) · 11G40 (secondary) Historical introduction and announcement of results (Historical background)In the more recent history of number theory one of the major themes is to formulate and prove so-called p-adic analogues of class number formulas. The first result of this sort was perhaps Leopoldt's class number formula. In general these theorems and conjectures relate p-adic L-functions, whose characteristic property is that they interpolate suitable normalized special values of classical complex L-functions to p-adic analytic invariants of global number theoretical objects, as the p-adic logarithm of global units in number fields or p-adic heights of global points in elliptic curves defined over number fields.The strong analogy between number fields and function fields of one variable over finite fields has been known already to mathematicians in the classical period of the 19th century. This analogy suggests that the aforementioned results and conjectures should have appropriate analogues over function fields as well, although their precise form is not at all clear if someone looks at classical results such as Leopoldt's class number formula. It was first B. Gross who made progress on this A. Pál

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On the conjectures of Birch and Swinnerton-Dyer in characteristic p >0

Cited by 64 publications

References 21 publications

The Iwasawa Main Conjecture for semistable abelian varieties over function fields

The Iwasawa Main Conjecture for semistable abelian varieties over function fields

There are genus one curves of every index over every infinite, finitely generated field

Proof of an exceptional zero conjecture for elliptic curves over function fields

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