The theory of Néron models has been extensively developed over the last fifty years. The motivation behind their original introduction in [A. Neron, Publ. Math., Inst. Hautes Étud. Sci. 21, 128 (1964; Zbl 0132.41403)] was to control the behaviour of heights on abelian varieties, and this later proved decisive in, for example, Falting’s proof of the Mordell Conjecture, but the focus here is not on diophantine problems at all. The Néron model \({\mathcal A}\) of an abelian variety \(A\) over the quotient field \(K\) of a DVR \(R\) is essentially the minimal smooth \(R\)-model of \(A\) as a group scheme. As such it is naturally associated to \(A\) and has good functorial properties, but its association with the base is more problematic. It does not commute with ramified base change, and a central problem is to understand and control this failure.
Considerable work in this direction has been done, by the authors of this book among others, in the case of tame ramification. An important invariant is the component group \(\Phi(A)\), which is the group of components of the reduction, i.e.of the group scheme \({\mathcal A}\times_R k\), where \(k\) is the residue field of \(R\). The order of this group is a generalisation of the notion of Tamagawa number: in the case where \(A\) is the localisation of an abelian variety over a number field at a prime \(p\), the order of the component group is precisely the Tamagawa number at \(p\) that appears in the Birch–Swinnerton-Dyer conjecture.
In earlier work [Math. Ann. 348, No. 3, 749–778 (2010; Zbl 1245.11072)] and [Adv. Math. 227, No. 1, 610–653 (2011; Zbl 1230.11076)] the authors introduced two formal power series that are generating functions for the Néron models of tamely ramified extensions of \(K\). Fix a separable closure \(K^s\) of \(K\) and for \(d\) coprime to \(\text{Char}\, k\) denote by \(K(d)\) the unique degree \(d\) extension of \(K\) inside \(K^s\) and by \(A(d)=A\times_K K(d)\) the corresponding abelian variety, with Néron model \({\mathcal A}(d)\). The motivic zeta function \[ Z_A(T)=\sum_d \left[{\mathcal A}(d)_k\right]{\mathbb L}^{\text{ord}_A(d)} T^d \in K_0({\mathbf{Var}}_k)[[T]] \] has coefficients in the Grothendieck ring of \(k\)-varieties, and is closely related to the in principle cruder component series \[ S^\Phi_A(T)=\sum_d\left|\Phi(A(d))\right|T^d \in {\mathbb Z}[[T]]. \] In particular the earlier work of the authors shows that both these series are rational functions if \(A\) is tamely ramified. Moreover, \(Z_A({\mathbb L}^{-s})\) has a unique pole where \(s\) is equal to Chai’s base change conductor \(c(A)\).
The main aim of the present book is to extend these results and ideas to other group schemes and to some case of wildly ramification. A fully general theory still appears out of reach at the moment, but by focussing on semi-abelian varieties and on wildly ramified Jacobians the authors are able to identify the main features and provide some technical tools, as well as obtaining good results for those cases.
The general principle used here as well as in the authors’ earlier work is that although Néron models do change under tame base change they do not do so more than they can help. If \(K'/K\) is a tame extension of degree \(d\) then the size of the component group gets multiplied by \(d^{t(A)}\) (where \(t(A)\) is the rank of the torus part of \({\mathcal A}_k^0\)) in the good case where \({\mathcal A}_k^0\) has no unipotent part (\(A\) is said to have semi-abelian reduction in this case). This will not always be the case for arbitrary tame \(K'/K\) and arbitrary \(A\), but it does hold if, essentially, what we have to adjoin to \(K\) to get a field \(L\) over which \(A\) has semi-abelian reduction has nothing to do with \(K'\) (the authors say that \(K'\) is “sufficiently orthogonal” to \(L\)). Under these conditions one also expects the torus rank, unipotent rank and abelian part of the central fibres to be unchanged by the base change to \(K'\), and that amounts to saying that they give the same element of the Grothendieck ring \(K_0({\text{Var}}_k)\).
In the tame case it is easy to say what “sufficiently orthogonal” should mean: the degrees of \(L/K\) and \(K'/K\) being coprime is enough. If \(A\) is wildly ramified the situation is much less clear. However, for Jacobians \(A={\text{Jac}}\, C\), one can pass to the curves and extract a new invariant, introduced here, called the stabilisation index \(e(C)\): then one requires \(d\) to be coprime to \(e(C)\), and the desirable consequences above for the Néron model follow.
The definition of \(e(C)\) and the details of the argument above, leading to a proof that the Néron component series \(S^\Phi_A(T)\) is a rational function in the Jacobian case, occupy Chapter 4 of the book: Chapters 1–3 are in various ways introductory. The invariant \(e(C)\) is in the tame case simply the degree of the extension necessary to obtain a normal crossings curve, but the general definition is more complicated and depends on the multiplicities of components in a suitable model of \(C\).
Chapter 5 deals with the semi-abelian case. The difficulty here is that one cannot remain in the algebraic setting. Instead the authors use the rigid analytic geometry approach of [S. Bosch and X. Xarles, Math. Ann. 306, No. 3, 459–486 (1996; Zbl 0869.14020)] (making a correction to that paper as they do so) to control the torsion part of the component group – the component group itself is in this context no longer finite.
The next part of the book examines some other invariants of Néron models, starting with an analogue of Chai’s base-change conductor called the tame base-change conductor, defined for any semi-abelian \(K\)-variety \(G\) and agreeing with Chai’s conductor if (and only if) \(G\) is tamely ramified. It is the sum of the jumps in Edixhoven’s filtration of the central fibre [B. Edixhoven, Compos. Math. 81, No. 3, 291–306 (1992; Zbl 0759.14033)]: Chapter 6 establishes its basic properties. In Chapter 7 the authors attempt to relate it to other arithmetic invariants such as the Artin conductor, obtaining results for Jacobians of curves of genus 1 and genus 2 but in a rather ad hoc way, making for instance essential use of the hyperellipticity.
Finally we come to motivic zeta functions. Chapter 8 makes the promised use of the rationality results for the component series to deduce results about rationality and poles for \(Z_A\) from those for \(S^\Phi_A\), in the cases where the latter have now been proved (tamely ramified semi-abelian varieties, or Jacobians, with a mention also of Pryms). Chapter 9 adds a cohomological interpretation of the motivic zeta function via a trace formula, and deduces some consequences. Chapter 10 lists some open problems and future directions, some of which have been implicitly mentioned earlier in the book.