nLab Euler systems

Idea

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.

Definition

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

Definition

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

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

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

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

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

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

Application to bounding the Selmer group

Theorem

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

References

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