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arXiv · 2111.14689

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

Dominik Bullach·David Burns·Alexandre Daoud·Soogil Seo
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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'.

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Dominik Bullach, David Burns, Alexandre Daoud, Soogil Seo. 2021-11-29. Dirichlet $L$-series at $s=0$ and the scarcity of Euler systems. https://arxiv.org/abs/2111.14689

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