Publications
Publications

A refinement of the Artin conductor and the base change conductor (with C.L. Chai) (ArXiv)For a local field K with positive residue characteristic p, we introduce, in the first part of this paper, a refinement bAr_K of the classical Artin distribution Ar_K. It takes values in cyclotomic extensions of Q which are unramified at p, and it bisects Ar_K in the sense that Ar_K is equal to the sum of bAr_K and its conjugate distribution. Compared with 1/2 Ar_K, the bisection bAr_K provides a higher resolution on the level of tame ramification. In the second part of this article, we prove that the base change conductor c(T) of an analytic Ktorus T is equal to the value of bAr_{K} on the Q_prational Galois representation X^*(T)_{Q_p} that is given by the character module X^*(T) of T. We hereby generalize a formula for the base change conductor of an algebraic Ktorus, and we obtain a formula for the base change conductor of a semiabelian Kvariety with potentially ordinary reduction.

Abelian varieties over number fields, tame ramification and big Galois image (with S. AriasdeReyna, to appear in Mathematical Research Letters) (ArXiv)Given a natural number n \geq 1 and a number field K, we show the existence of an integer l_0 such that for any prime number l \geq l_0, there exists a finite extension F/K, unramified in all places above l, together with a principally polarized abelian variety A of dimension n over F such that the resulting ltorsion representation is surjective and everywhere tamely ramified. In particular, we realize GSp2n(F_l) as the Galois group of a finite tame extension of number fields F′/F such that F is unramified above l.

Néron models of formally finite type (International Mathematics Research Notices 2012; doi: 10.1093/imrn/rns201) (ArXiv) (pdf)We introduce Néron models of formally finite type for uniformly rigid spaces, and we prove that they generalize the notion of formal Néron models for rigidanalytic groups as it was defined by Bosch and Schlöter. Using this compatibility result, we give examples of uniformly rigid groups whose Néron models are not of topologically finite type.

We define a new category of nonarchimedean analytic spaces over a complete discretely valued field, which we call uniformly rigid. It extends the category of rigid spaces, and it can be described in terms of bounded functions on products of open and closed polydiscs. We relate uniformly rigid spaces to their associated classical rigid spaces, and we transfer various constructions and results from rigid geometry to the uniformly rigid setting. In particular, we prove an analog of Kiehl’s patching theorem for coherent ideals, and we define the uniformly rigid generic fiber of a formal scheme of formally finite type. This uniformly rigid generic fiber is more intimately linked to its model than the classical rigid generic fiber obtained via Berthelot’s construction.
Further papers

We introduce a new category of nonarchimedean analytic spaces over a complete discretely valued field. These spaces, which we call uniformly rigid, may be viewed as classical rigidanalytic spaces together with an additional uniform structure: On the level of points, a uniformly rigid space coincides with its underlying rigid space, while its Gtopology is coarser and its sheaf of holomorphic functions is smaller. A uniformly rigid structure is, locally, induced by an integral formal model of formally finite type. In fact, Berthelot's generic fiber functor factors naturally through the category of uniformly rigid spaces. The uniformly rigid generic fiber of a quasicompact formal scheme of formally finite type is quasicompact, and its global functions are bounded. We construct our new category by starting out from semiaffinoid algebras and nested rational coverings; we prove a uniformly rigid acyclicity theorem, and we study coherent modules. We then define formal N\'eron models for uniformly rigid spaces to be smooth formal schemes of formally finite type satisfying the universal N\'eron property. We establish the existence of formal N\'eron models for certain classes of uniformly rigid groups, and we discuss applications to Chai's program concerning the computation of the base change conductor for abelian varieties with potentially multiplicative reduction. This paper is a faithful copy of the author's doctoral thesis.

Diploma thesis: "Kongruenzen von NéronModellen für Tori" (pdf)My diploma thesis is an elaboration of the article Congruences of Néron models for tori and the Artin conductor by Chai, Yu and de Shalit.