The Iwasawa Main Conjecture for semistable abelian varieties over function fields

Lai, King Fai; Longhi, Ignazio; Tan, Ki-Seng; Trihan, Fabien

doi:10.1007/s00209-015-1550-4

Cited by 6 publications

(5 citation statements)

References 22 publications

(8 reference statements)

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“…Let us mention also that the Iwasawa Main Conjecture for semistable abelian varieties over the arithmetic Z p -extension was proven in [LLTT2]. In §4.5, we check that our result is compatible with the latter one.…”

Section: Introductionsupporting

confidence: 73%

“…Let A/K be a constant ordinary abelian variety. Then Theorem 4.9 is equivalent to the Iwasawa Main Conjecture proved in [9].…”

Section: Comparison With [9]mentioning

confidence: 89%

“…Nowadays, there are many authors working on the Iwasawa Main Conjecture for the Selmer Groups over p-adic extensions of number fields, but results over function fields are rare. One reason might be the essential difficulty of dealing with p in characteristic p: even in the case of the arithmetic Z p -extension, one needs to use techniques beyond Kato-Trihan [7] to overcome this difficulty; see [9]. In general, if L/K is a Z p -extension of global function fields that does ramify at some place, then the techniques from syntomic cohomology do not work.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Iwasawa Main Conjecture for constant ordinary abelian varieties over function fields

Lai¹,

Longhi²,

Tan³

et al. 2016

Proc. London Math. Soc.

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Section: Introductionsupporting

confidence: 73%

“…Let A/K be a constant ordinary abelian variety. Then Theorem 4.9 is equivalent to the Iwasawa Main Conjecture proved in [9].…”

Section: Comparison With [9]mentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Iwasawa Main Conjecture for constant ordinary abelian varieties over function fields

Lai¹,

Longhi²,

Tan³

et al. 2016

Proc. London Math. Soc.

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“…In this case Trihan has obtained a proof of the Iwasawa Main Conjecture, by techniques of syntomic cohomology. No assumption on the abelian variety A/F is needed: the relevant p-adic L-function is defined by means of cohomology and it interpolates the Hasse-Weil L-function (see [40]).…”

Section: 3mentioning

confidence: 99%

Aspects of Iwasawa theory over function fields

Bandini¹,

Bars²,

Longhi³

2020

$T$-Motives: Hodge Structures, Transcendence and Other Motivic Aspects

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We consider Z N p -extensions F of a global function field F and study various aspects of Iwasawa theory with emphasis on the two main themes already (and still) developed in the number fields case as well. When dealing with the Selmer group of an abelian variety A defined over F , we provide all the ingredients to formulate an Iwasawa Main Conjecture relating the Fitting ideal and the p-adic L-function associated to A and F. We do the same, with characteristic ideals and p-adic L-functions, in the case of class groups (using known results on characteristic ideals and Stickelberger elements for Z d p -extensions). The final section provides more details for the cyclotomic Z N p -extension arising from the torsion of the Carlitz module: in particular, we relate cyclotomic units with Bernoulli-Carlitz numbers by a Coates-Wiles homomorphism.

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“…Application to constant ordinary abelian varieties. This final section briefly presents the main results of [10] (for all the details see the original paper and its companion [9]), in order to provide a simple but (in our opinion) meaningful application of our formula to the setting of constant ordinary abelian varieties. We keep notations as close as possible to the ones in [10].…”

Section: Akashi Series For Z Dmentioning

confidence: 99%

Euler characteristic and Akashi series for Selmer groups over global function fields

Bandini

Valentino

2018

Journal of Number Theory

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Let A be an abelian variety defined over a global function field F of positive characteristic p and let K/F be a p-adic Lie extension with Galois group G. We provide a formula for the Euler characteristic χ(G, Sel A (K)p) of the p-part of the Selmer group of A over K. In the special case G = Z d p and A a constant ordinary variety, using Akashi series, we show how the Euler characteristic of the dual of Sel A (K)p is related to special values of a p-adic L-function.

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The Iwasawa Main Conjecture for semistable abelian varieties over function fields

Abstract: Abstract. We prove the Iwasawa Main Conjecture over the arithmetic Zp-extension for semistable abelian varieties over function fields of characteristic p > 0.

Cited by 6 publications

References 22 publications

The Iwasawa Main Conjecture for constant ordinary abelian varieties over function fields

The Iwasawa Main Conjecture for constant ordinary abelian varieties over function fields

Aspects of Iwasawa theory over function fields

Euler characteristic and Akashi series for Selmer groups over global function fields

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