On the numerically and cohomologically trivial automorphisms of elliptic surfaces II: $χ(S)>0$
Fabrizio Catanese, Wenfei Liu, Matthias Schuett
TL;DR
The paper advances the understanding of numerically and cohomologically trivial automorphisms on properly elliptic surfaces S with χ(S) > 0 by showing that Aut_Q(S) can be large yet controlled via the torsion of the Mordell–Weil group of the relative Jacobian J(S). A central construction links Aut_Q(S) to MW(J(S))_tor through a base-change/log-transform framework, allowing explicit realization of prescribed finite abelian groups (notably 2-generated ones) and providing sharp bounds in terms of χ(S), q(S), and P_2(S). The work also resolves the less-explored case of cohomologically trivial automorphisms, proving sharp bounds in pg = 0 and isotrivial settings and constructing infinite families with order-2 and order-3 automorphisms, while proving abelianness of Aut_Z(S) in all χ(S) > 0 cases. The results have strong implications for the structure of automorphism groups of elliptic surfaces, linking topological invariants, Mordell–Weil theory, and explicit geometric constructions to yield a near-complete picture in the χ(S) > 0 regime and highlighting key open questions in the non-isotrivial, pg > 0 setting.
Abstract
In this second part we study first the group $Aut_{\mathbb{Q}}(S)$ of numerically trivial automorphisms of a properly elliptic surface $S$, that is, of a minimal surface with Kodaira dimension $κ(S)=1$, in the case $χ(S) \geq 1$. Our first surprising result is that, against what has been believed for over 40 years, we have nontrivial such groups for $p_g(S) >0$. Indeed, we show even that there is no absolute upper bound for their cardinalities $|Aut_{\mathbb{Q}}(S)|$. At any rate, we give explicit and nearly optimal upper bounds for $|Aut_{\mathbb{Q}}(S)|$ in terms of the numerical invariants of $S$, as $χ(S)$, or the irregularity $q(S)$, or the bigenus $P_2(S)$. Moreover, we come quite close to a complete description of the possible groups $Aut_{\mathbb{Q}}(S)$ as 2-generated finite abelian groups, and we give an effective criterion for surfaces to have trivial $Aut_{\mathbb{Q}}(S)$. Our second surprising results concern the group $Aut_{\mathbb{Z}}(S)$ of cohomologically trivial automorphisms; we are able to give the explicit upper bounds for $|Aut_{\mathbb{Z}}(S)|$ in special cases: $9$ when $p_g(S) =0$, and the sharp upper bound $3$ when $S$ (i.e., the pluricanonical elliptic fibration) is isotrivial. We produce also non isotrivial examples where $Aut_{\mathbb{Z}}(S)$ is a cyclic group of order $2$ or $3$.