Dynamics of quadratic polynomials and rational points on a curve of genus 4

Let f t ( z stretchy="false">) = 2 + f_t(z)=z^2+t . For any alttext="z element-of double-struck upper Q"> ∈ Q encoding="application/x-tex">z\in \mathbb {Q} , let alttext="upper S z"> S encoding="application/x-tex">S_z be the collection of alttext="t encoding="application/x-tex">t\in such that alttext="z"> encoding="application/x-tex">z is preperiodic for encoding="application/x-tex">f_t In this article, assuming a well-known conjecture Flynn, Poonen, and Schaefer [Duke Math. J. 90 (1997), pp. 435–463], we prove uniform result regarding size over order to it, need determine set rational points on specific non-hyperelliptic curve C"> C encoding="application/x-tex">C genus alttext="4"> 4 encoding="application/x-tex">4 defined alttext="double-struck encoding="application/x-tex">\mathbb We use Chabauty’s method, which requires us Mordell-Weil rank Jacobian J"> J encoding="application/x-tex">J give two proofs alttext="1"> 1 encoding="application/x-tex">1 : an analytic proof, conditional BSD some standard conjectures L-series, algebraic unconditional, but relies computation class groups number fields degree alttext="12"> 12 encoding="application/x-tex">12 alttext="24"> 24 encoding="application/x-tex">24 respectively. finally combine information obtained from both provide numerical verification strong