arXiv subjects

Dominik Bullach

Publications and source records attributed to Dominik Bullach.

7 recordsLinked to original sources

arXiv2026

On Euler systems and Nekovář-Selmer complexes

We develop a theory of Euler and Kolyvagin systems relative to the Nekovář--Selmer complexes of $p$-adic representations over local complete Gorenstein rings. This theory is both finer and requires fewer hypotheses than those of Mazur and Rubin over discrete valuation rings and of Sakamoto et al. over Gorenstein rings. In particular, given appropriate Euler systems, it allows one to study Selmer groups defined relative to Greenberg local conditions. As initial applications, we prove new cases of Kato's generalised Iwasawa main conjecture for both $\mathbb{Z}_p(a)$ and the $p$-adic Tate modules of rational elliptic curves, new cases of the Quillen--Lichtenbaum Conjecture, and a strengthening of existing results on the Birch--Swinnerton-Dyer Conjecture for CM elliptic curves.

math.NT↗

arXiv2025

On the refined `Birch--Swinnerton-Dyer type' conjectures of Mazur and Tate

We prove a substantial part of conjectures of Mazur and Tate that refine the conjecture of Birch and Swinnerton-Dyer. Our approach, which also leads to some results even finer than the predictions of Mazur and Tate, is via the `rank-zero component' of the relevant case of the equivariant Tamagawa Number conjecture.

math.NT↗

arXiv2023

Annihilating class groups in $p$-elementary extensions

We derive new cases of conjectures of Rubin and of Burns--Kurihara--Sano concerning derivatives of Dirichlet $L$-series at $s = 0$ in $p$-elementary extensions of number fields for arbitrary prime numbers $p$. In naturally arising examples of such extensions one therefore obtains annihilators of class groups from $S$-truncated Dirichlet $L$-series for `large-enough' sets of places $S$.

math.NT↗

arXiv2023

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

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 $μ$-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'.

math.NT↗

arXiv2021

The equivariant Tamagawa Number Conjecture for abelian extensions of imaginary quadratic fields

We prove the Iwasawa-theoretic version of a Conjecture of Mazur--Rubin and Sano in the case of elliptic units. This allows us to derive the $p$-part of the equivariant Tamagawa number conjecture at $s = 0$ for abelian extensions of imaginary quadratic fields in the semi-simple case and, provided that a standard $μ$-vanishing hypothesis is satisfied, also in the general case.

math.NT↗

arXiv2021

On $p$-adic families of special elements for rank-one motives

We conjecture that special elements associated with rank-one motives are obtained $p$-adically from Rubin-Stark elements by means of a precise `higher-rank Soulé twist' construction. We show this conjecture incorporates a variety of known results and existing predictions and also gives rise to a concrete strategy for proving the equivariant Tamagawa Number Conjecture for rank-one motives. We then use this approach to obtain new evidence in support of the equivariant Tamagawa Number Conjecture in the setting of CM abelian varieties.

math.NT↗

arXiv2021

On Universal Norms for $p$-adic Representations in Higher Rank Iwasawa Theory

We begin a systematic investigation of universal norms for $p$-adic representations in higher rank Iwasawa theory. After establishing the basic properties of the module of higher rank universal norms we construct an Iwasawa-theoretic pairing that is relevant to this setting. This allows us, for example, to refine the classical Iwasawa Main Conjecture for cyclotomic fields, and also to give applications to various well-known conjectures in arithmetic concerning Iwasawa invariants and leading terms of $L$-functions.

math.NT↗