Frobenius and settled elements in iterated Galois extensions

Abstract: Understanding Frobenius elements in iterated Galois extensions is a major goal in arithmetic dynamics. In 2012 Boston and Jones conjectured that any quadratic polynomial f over a finite field that is different from x^2 is settled, namely the weighted proportion of f-stable factors in the factorization of the n-th iterate of f tends to 1 as n tends to infinity. This can be rephrased in terms of Frobenius elements: given a quadratic polynomial f over a number field K, an element \alpha in K and the extension K_\infty generated by all the f^n-preimages of \alpha, the Frobenius elements of unramified primes in K_\infty are settled. In this talk, we will explain how to construct uncountably many non-conjugate settled elements that cannot be the Frobenius of any ramified or unramified prime, for any quadratic polynomial. The key result is a description of the critical orbit modulo squares for quadratic polynomials over local fields. This is joint work with Carlo Pagano.

algebraic geometrynumber theory

Audience: researchers in the topic

FGC-HRI-IPM Number Theory Webinars

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