Néron models are mathematical objects which play a very important role in contemporary arithmetic geometry. However, they usually behave badly, particularly in respect of exact sequences and base change, which makes most problems regarding their behaviour very delicate. Chai introduced the base change conductor, a rational number associated with a semiabelian variety $G$ which measures the failure of the Néron model of $G$ to commute with (ramified) base change. Moreover, Chai conjectured that this invariant is additive in certain exact sequences. We shall introduce a new method to study the Néron models of Jacobians of proper (possibly singular) curves, and sketch a proof of Chai's conjecture for semiabelian varieties which are also Jacobians. 

Let K be a number field and let L/K be an abelian extension. The genus field of L/K is the largest extension of L which is unramified at all places of L and abelian as an extension of K. The genus group is its Galois group over L, which is a quotient of the class group of L, and the genus number is the size of the genus group. We study the quantitative behaviour of genus numbers as one varies over abelian extensions L/K with fixed Galois group. We give an asymptotic formula for the average value of the genus number and show that any given genus number appears only 0% of the time. This is joint work with Christopher Frei and Daniel Loughran.