Rational points on algebraic curves in infinite towers of number fields

Ray, Anwesh

doi:10.1007/s11139-022-00583-3

Cited by 1 publication

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“…Mazur [12] initiated the Iwasawa theory of elliptic curves over number fields, which had applications to the growth of Mordell-Weil ranks of elliptic curves in certain infinite towers of number fields. One hopes to extend such lines of investigation to curves of higher genus (see [14]) and, more generally, to study the stability and growth of solutions to any Diophantine equation in an infinite tower of global fields. We study certain function field analogues of such questions.…”

Section: Introductionmentioning

confidence: 99%

Diophantine Equations of the Form Over Function Fields

Ray

2023

Bull. Aust. Math. Soc.

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Let $\ell $ and p be (not necessarily distinct) prime numbers and F be a global function field of characteristic $\ell $ with field of constants $\kappa $ . Assume that there exists a prime $P_\infty $ of F which has degree $1$ and let $\mathcal {O}_F$ be the subring of F consisting of functions with no poles away from $P_\infty $ . Let $f(X)$ be a polynomial in X with coefficients in $\kappa $ . We study solutions to Diophantine equations of the form $Y^{n}=f(X)$ which lie in $\mathcal {O}_F$ and, in particular, show that if m and $f(X)$ satisfy additional conditions, then there are no nonconstant solutions. The results apply to the study of solutions to $Y^{n}=f(X)$ in certain rings of integers in $\mathbb {Z}_{p}$ -extensions of F known as constant $\mathbb {Z}_p$ -extensions. We prove similar results for solutions in the polynomial ring $K[T_1, \ldots , T_r]$ , where K is any field of characteristic $\ell $ , showing that the only solutions must lie in K. We apply our methods to study solutions of Diophantine equations of the form $Y^n=\sum _{i=1}^d (X+ir)^m$ , where $m,n, d\geq 2$ are integers.

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Section: Introductionmentioning

confidence: 99%