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Register a new userOn the variation of the rank of Jacobian varieties on unramified abelian towers over number fields
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Authors
Amilcar Pacheco
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Let $C$ be a smooth projective curve defined over a number field $k$, $X/k(C)$ a smooth projective curve of positive genus, $J_X$ the Jacobian variety of $X$ and $(tau,B)$ the $k(C)/k$-trace of $J_X$. We estimate how the rank of $J_X(k(C))/tau B(k)$ varies when we take an unramified abelian cover $pi:Cto C$ defined over $k$.
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Given a natural number n and a number field K, we show the existence of an integer ell_0 such that for any prime number ellgeq ell_0, there exists a finite extension F/K, unramified in all places above ell, together with a principally polarized abelian variety A of dimension n over F such that the resulting ell-torsion representation rho_{A,ell} from G_F to GSp(A[ell](bar{F})) is surjective and everywhere tamely ramified. In particular, we realize GSp_{2n}(mathbb{F}_ell) as the Galois group of a finite tame extension of number fields F/F such that F is unramified above ell.
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