Algebra Seminar

Speaker: Séverin Philip (Institut Fourier Université Grenoble Alpes )

"On the degree of semi-stable reduction for abelian varieties"

Time: 14:30

Room: ZOOM

For an abelian variety $A$ over a number field $K$ a seminal theorem of Grothendieck asserts the existence of a finite extension $L/K$ such that $A$ acquires semi-stable reduction over $L$. My work is centered around studying this phenomenon of acquiring 'semi-stability' where arithmetic and geometry meet. I will showcase this with elliptic curves. The main object of study will be the minimal degree $d(A)$ of the extensions $L/K$ that appear this way and I will give almost optimal bounds depending only on the dimension of $A$ for its maximum (for varying $A$ and $K$ of a given dimension). These bounds will come from a study of the so-called finite monodromy groups of $A$, groups that I will introduce and show that they give a theoretical computation of $d(A)$. This computation and the work of Silverberg and Zarhin will give our upper bound. The lower bound will come from the study of a moduli space for abelian varieties and an explicit construction using the theory CM of abelian varieties.