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We study motivic zeta functions of degenerating families of Calabi-Yau varieties. Our main result says that they satisfy an analog of Igusa's monodromy conjecture if the family has a so-called Galois-equivariant Kulikov model; we provide several classes of examples where this condition is verified. We also establish a close relation between the zeta function and the skeleton that appeared in Kontsevich and Soibelman's non-archimedean interpretation of the SYZ conjecture in mirror symmetry., Comment: New result on existence of Kulikov models for abelian varieties added in section 5.1

We analyze the geometry of rational p-division points in degenerating families of elliptic curves in characteristic p. We classify the possible Kodaira symbols and determine for the Igusa moduli problem the reduction type of the universal curve. Special attention is paid to characteristic 2 and 3 where wild ramification and stacky phenomena show up., 44 pages, minor changes

Let [math] be the Néron model of an abelian variety [math] over the fraction field [math] of a discrete valuation ring [math] . By work of Mazur and Messing, there is a functorial way to prolong the universal extension of [math] by a vector group to a smooth and separated group scheme over [math] , called the canonical extension of [math] . Here we study the canonical extension when [math] is the Jacobian of a smooth, proper and geometrically connected curve [math] over [math] . Assuming that [math] admits a proper flat regular model [math] over [math] that has generically smooth closed fiber, our main result identifies the identity component of the canonical extension with a certain functor [math] classifying line bundles on [math] that have partial degree zero on all components of geometric fibers and are equipped with a regular connection. This result is a natural extension of a theorem of Raynaud, which identifies the identity component of the Néron model [math] of [math] with the functor [math] . As an application of our result, we prove a comparison isomorphism between two canonical integral structures on the de Rham cohomology of [math] .

Let $p$ be a prime. Let $(R,\ideal{m})$ be a regular local ring of mixed characteristic $(0,p)$ and absolute index of ramification $e$. We provide general criteria of when each abelian scheme over $\Spec R\setminus\{\ideal{m}\}$ extends to an abelian scheme over $\Spec R$. We show that such extensions always exist if $e\le p-1$, exist in most cases if $p\le e\le 2p-3$, and do not exist in general if $e\ge 2p-2$. The case $e\le p-1$ implies the uniqueness of integral canonical models of Shimura varieties over a discrete valuation ring $O$ of mixed characteristic $(0,p)$ and index of ramification at most $p-1$. This leads to large classes of examples of N��ron models over $O$. If $p>2$ and index $p-1$, the examples are new., 28 pages. Final version identical (modulo style) to the galley proofs. To appear in Doc. Math