# On Shafarevich-Tate groups and analytic ranks in Hida families of modular forms

13 Jan 2020 Vigni Stefano

Let \$f\$ be a newform of weight \$2\$, square-free level and trivial character, let \$A_f\$ be the abelian variety attached to \$f\$ and for every good ordinary prime \$p\$ for \$f\$ let \$\boldsymbol f^{(p)}\$ be the \$p\$-adic Hida family through \$f\$. We prove that, for all but finitely many primes \$p\$ as above, if \$A_f\$ is an elliptic curve such that \$A_f(\mathbb Q)\$ has rank \$1\$ and the \$p\$-primary part of the Shafarevich-Tate group of \$A_f\$ over \$\mathbb Q\$ is finite then all specializations of \$\boldsymbol f^{(p)}\$ of weight congruent to \$2\$ modulo \$2(p-1)\$ and trivial character have finite (\$p\$-primary) Shafarevich-Tate group and \$1\$-dimensional image of the relevant \$p\$-adic \'etale Abel-Jacobi map... (read more)

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• NUMBER THEORY
• ALGEBRAIC GEOMETRY