# Manchester Number Theory Seminar - Francesco Ballini

Dates: | 15 November 2022 |
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Times: | 15:00 - 16:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Current University students |

Speaker: Francesco Ballini (University of Oxford)

Title: Hyperelliptic continued fractions

Abstract: We can define a continued fraction for formal series f(t)=\sum_{i=-\infty}^d c_it^i by repeatedly removing the polynomial part, \sum_{i=0}^d c_it^i, (the equivalent of the integral part) and inverting the remaining part, as in the real case. This way, the partial quotients are polynomials. Both the usual real continued fractions and the polynomial continued fractions carry properties of best approximation. However, while for square roots of rationals the real continued fraction is eventually periodic, such periodicity does not always occur for \sqrt{D(t)}. The correct analogy was found by Abel in 1826: the continued fraction of \sqrt{D(t)} is eventually periodic if and only if there exist nontrivial polynomials x(t),y(t) such that x(t)^2-D(t)y(t)^2=1 (the polynomial Pell equation). Notice that the same holds also for square root of integers in the real case. In 2014 Zannier found that some periodicity survives for all the \sqrt{D(t)}: the degrees of their partial quotients are eventually periodic. All these statements are strongly related to geometry and they are based on the study of the Jacobian of the curve u^2=D(t). We give a brief survey of the theory of polynomial continued fractions and their interplay with Jacobians.

Room: Frank Adams 1