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Variation of the Mordell-Weil rank in families of abelian varieties Hindry, Marc


We consider a family of abelian varieties over a number field $K$ , i.e. a variety $X$ with a map to a curve $B$ whose fibres are abelian varieties (the interesting cases are when $B$ is the projective line or an elliptic curve with positive rank). The generic fibre is an abelian variety over the function field $K(B)$ and the group of $K(B)$-rational points has a rank $r$. For almost all points $t$ in $B(K)$ the fibre is an abelian variety $X_t$ over $K$ and the group of $K$-rational point has rank $r(t)$. A specialisation theorem of Silverman says that for or almost all points $t$ in $B(K)$ the rank $r(t)$ is greater or equal to $r$. We want to understand the distribution of $r(t)$, in particular we ask wether there are infinitely many $t$'s 1) with $r(t)=r$, 2) with $r(t)>r$. The problem looks very hard in general, but, under specific geometric conditions, we will settle the second question, and provide interesting example where much more can be proven. This is a joint work with Cecília Salgado.

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