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**Stark systems and equivariant main conjectures.**
*(English)*
Zbl 1519.11061

This is a further contribution to the theory of Stark systems with an application to equivariant main conjectures of elliptic curves. The main result is one divisibility of the main conjecture under certain hypotheses. This was already known by former work of the author [Math. Z. 298, No. 3–4, 1653–1725 (2021; Zbl 07373932)] under the additional assumption that a certain \(\mu\)-invariant vanishes.

Recently, the theory of Euler systems and of the related Stark and Kolyvagin systems has been further developed by several authors including D. Burns and T. Sano [Int. Math. Res. Not. 2021, No. 13, 10118–10206 (2021; Zbl 1496.11132)] and R. Sakamoto [Algebra Number Theory 12, No. 10, 2295–2326 (2018; Zbl 1425.11175)]. A novelty was the notion of exterior power biduals, which turned out to be essential for higher rank theory. In this paper the author extracts more refined consequences from the existence of a Stark system. He introduces the notion of (primitive) basic elements for perfect complexes and proves that each (primitive) Stark system gives rise to a (primitive) basic element of an arithmetic complex (we point out that this notion implicitly already appeared in the literature; for instance in [D. Burns et al., Doc. Math. 21, 555–626 (2016; Zbl 1407.11133)]). The main application is as follows.

Let \(E/\mathbb Q\) be an elliptic curve with good reduction at a prime \(p \geq 5\). Let \(F\) be a finite abelian extension of \(\mathbb Q\) and denote its cyclotomic \(\mathbb Z_p\)-extension by \(F_{\infty}\). Let \(\mathcal{R}_F := \mathbb{Z}_p[[\mathrm{Gal}(F_{\infty}/\mathbb Q)]]\) be the Iwasawa algebra. Let us assume here for simplicity that \(E\) has ordinary reduction at \(p\) (this is not assumed in the paper!). It is known that the dual Selmer group has projective dimension at most \(1\) over \(\mathcal{R}_F\) and thus its Fitting ideal is principal. The main conjecture then simply asserts that \[ \mathrm{Fitt}_{\mathcal{R}_F}((\mathrm{Sel}_S(E/F_{\infty})^{\vee})) = (\mathcal{L}_S(E/F_{\infty})), \] where \(S\) is a certain finite set of places and \(\mathcal{L}_S(E/F_{\infty}) \in \mathcal{R}_F \otimes_{\mathbb Z_p} \mathbb Q_p\) is a \(p\)-adic \(L\)-function. Now under certain standard hypotheses the author proves the inclusion “\(\supseteq\)” of the main conjecture. Crucially, the hypotheses do not include the vanishing of any \(\mu\)-invariant as in the author’s previous work.

Recently, the theory of Euler systems and of the related Stark and Kolyvagin systems has been further developed by several authors including D. Burns and T. Sano [Int. Math. Res. Not. 2021, No. 13, 10118–10206 (2021; Zbl 1496.11132)] and R. Sakamoto [Algebra Number Theory 12, No. 10, 2295–2326 (2018; Zbl 1425.11175)]. A novelty was the notion of exterior power biduals, which turned out to be essential for higher rank theory. In this paper the author extracts more refined consequences from the existence of a Stark system. He introduces the notion of (primitive) basic elements for perfect complexes and proves that each (primitive) Stark system gives rise to a (primitive) basic element of an arithmetic complex (we point out that this notion implicitly already appeared in the literature; for instance in [D. Burns et al., Doc. Math. 21, 555–626 (2016; Zbl 1407.11133)]). The main application is as follows.

Let \(E/\mathbb Q\) be an elliptic curve with good reduction at a prime \(p \geq 5\). Let \(F\) be a finite abelian extension of \(\mathbb Q\) and denote its cyclotomic \(\mathbb Z_p\)-extension by \(F_{\infty}\). Let \(\mathcal{R}_F := \mathbb{Z}_p[[\mathrm{Gal}(F_{\infty}/\mathbb Q)]]\) be the Iwasawa algebra. Let us assume here for simplicity that \(E\) has ordinary reduction at \(p\) (this is not assumed in the paper!). It is known that the dual Selmer group has projective dimension at most \(1\) over \(\mathcal{R}_F\) and thus its Fitting ideal is principal. The main conjecture then simply asserts that \[ \mathrm{Fitt}_{\mathcal{R}_F}((\mathrm{Sel}_S(E/F_{\infty})^{\vee})) = (\mathcal{L}_S(E/F_{\infty})), \] where \(S\) is a certain finite set of places and \(\mathcal{L}_S(E/F_{\infty}) \in \mathcal{R}_F \otimes_{\mathbb Z_p} \mathbb Q_p\) is a \(p\)-adic \(L\)-function. Now under certain standard hypotheses the author proves the inclusion “\(\supseteq\)” of the main conjecture. Crucially, the hypotheses do not include the vanishing of any \(\mu\)-invariant as in the author’s previous work.

Reviewer: Andreas Nickel (München)

### MSC:

11R23 | Iwasawa theory |

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### References:

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