Dirichlet $L$-series at $s=0$ and the scarcity of Euler systems

Abstract

We study Euler systems for ${\mathbb{G}}_m$ over a number field $k$. Motivated by a distribution-theoretic idea of Coleman, we formulate a conjecture regarding the existence of such systems that is elementary to state and yet strictly finer than Kato's equivariant Tamagawa number conjecture for Dirichlet $L$-series at $s=0$. To investigate the conjecture, we develop an abstract theory of `Euler limits' and, in particular, prove the existence of canonical `restriction' and `localisation' sequences in this theory. By using this approach we obtain a variety of new results, ranging from a proof, modulo standard $\mu$-vanishing hypotheses, of our central conjecture in the case $k$ is $\mathbb{Q}$ or imaginary quadratic to a proof of the `minus part' of Kato's conjecture in the case $k$ is totally real. In proving these results, we also show that higher-rank Euler systems for a wide class of $p$-adic representations control the structure of Iwasawa-theoretic Selmer groups in the manner predicted by `main conjectures'.

Corrected a mistake in previous version, treatment of Coleman's distributions-theoretic conjecture moved to a separate article

References in corpus (4)

• On the theory of higher rank Euler, Kolyvagin and Stark systems, III: applications
• A higher rank Euler system for the multiplicative group over a totally real field
• On Euler systems for the multiplicative group over general number fields
• On a conjecture of Coleman concerning Euler systems