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The finiteness theorems for generalized Jacobians with nontrivial torsion

V. S. Zhgoonab

a Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region
b National Research University Higher School of Economics, Moscow


https://vkvideo.ru/video-222947497_456239110
https://youtu.be/tPxEJvc_IuI

Аннотация: The questions on the finiteness of the set of rational points on an algebraic curve over algebraic number fields are fundamental questions of arithmetic algebraic geometry. In my talk I shall discuss new questions about the finiteness of the set of generalized Jacobians of a curve $C$, defined over algebraic number field, that are associated to the modules $m$ such that a fixed class of finite order in the Jacobian of $C$ lifts to the torsion class in the generalized Jacobian $J_m$. On one hand, such set of generalized Jacobians with the property mentioned above is infinite. On the other hand, we obtained finiteness results under additional assumptions on the support of $m$ or on the structure of the subgroup $J_m$. These results were applied to the problem of quasi-periodicity of continued rational fractions constructed in the power series $k((1/x))$, for some special class of elements of the field of rational functions on the hyperelliptic curve $y^2 = f(x)$ over an algebraic number field. In particular, for any $n$ we established the finiteness of the set of polynomials $g(x)$ of degree bounded by $n$, for which the expansion of the elements $g(x) \sqrt{f(x)}$ in the continued fraction is periodic.

Язык доклада: английский


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