nLab Euler systems



An Euler system is a collection of classes in Galois cohomology satisfying certain compatibility conditions. The existence of a nontrivial Euler system can imply bounds on the Selmer group and thus the theory of Euler systems finds applications in the study of special values of L-functions, for instance in Victor Kolyvagin’s work on the Birch and Swinnerton-Dyer conjecture.


The reference for this section is Section 1.4 of #LoefflerZerbes18.


Let G G_{\mathbb{Q}} be the absolute Galois group of \mathbb{Q}. Let VV be a Galois representation valued in p\mathbb{Q}_{p}, and let TVT\subset V be a G KG_{K}-stable p\mathbb{Z}_{p}-lattice. Let Σ\Sigma be a finite set of primes containing pp and all ramified primes.

Let P (V,t)P_{\ell}(V,t) be the local Euler factor at \ell, i.e.

P (V,t)=det(1tρ(Frob v 1)).P_{\ell}(V,t)=\det(1-t\cdot\rho(\Frob_{v}^{-1})).

An Euler system for (T,Σ)(T,\Sigma) is a collection c=(c m) m\mathbf{c}=(c_{m})_{m}, where c mH 1((μ m),T)c_{m}\in H^{1}(\mathbb{Q}(\mu_{m}),T) satisfying the following compatibility conditions:

norm (μ m) (μ m)(c m)={c mifΣorm| P (V *(1),σ 1)c motherwise\mathrm{norm}_{\mathbb{Q}(\mu_{m})}^{\mathbb{Q}(\mu_{m\ell})}(c_{m\ell})= \begin{cases} c_{m}\;\if \ell\in\Sigma\;\or\;m\vert \ell\\ P_{\ell}(V^{*}(1),\sigma_{\ell}^{-1})\cdot c_{m}\;\otherwise \end{cases}

where σ \sigma_{\ell} is the image of Frob \mathrm{Frob}_{\ell} in Gal((μ m)/)\mathrm{Gal}(\mathbb{Q}(\mu_{m})/\mathbb{Q}).

Application to bounding the Selmer group


(Theorem 2 of #LoefflerZerbes18) Suppose c\mathbf{c} is an Euler system for (T,Σ)(T,\Sigma) with c 1c_{1} nonzero, and suppose that VV satisfies certain technical conditions. Then the strict Selmer group Sel strict(,V *(1))\Sel_{\strict}(\mathbb{Q},V^{*}(1)) is zero.

Relation to motivic cohomology

One of the ways to construct examples of Euler systems is via motivic cohomology (see chapter 3 of #LoefflerZerbes18).


David Loeffler and Sarah Zerbes, Euler Systems, Arizona Winter School 2018 Notes (pdf)

Last revised on August 5, 2023 at 19:39:24. See the history of this page for a list of all contributions to it.