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Seminars

Number Theory Seminar

Periodic Curves and Preimage Trees in Arithmetic Dynamics

2026/10/02 (Fri.) 10:30~11:45
  • Date : 2026/10/02 (Fri.) 10:30~11:45
  • Location : Seminar Room 617, Institute of Mathematics (NTU Campus)
  • Speaker : Wayne Peng (National Center University)
  • Organizer : Julie Tzu-Yueh Wang (AS)
  • Abstract :
    In arithmetic dynamics, the tree of iterated preimages of a point under a polynomial carries a natural action of the absolute Galois group and may be viewed as a dynamical analog of the Tate module of an elliptic curve. This leads to a natural question: when do two points, possibly for two different polynomial dynamical systems, have isomorphic Galois-equivariant preimage trees?

    In this talk, I will describe a geometric approach to this question using periodic curves in the product dynamical system \((f,g)\) on \(\mathbb A^1\times\mathbb A^1\). For non-special polynomials, we introduce a class of balanced periodic curves and show that, away from a distinguished center, points lying on such a curve give rise to isomorphic preimage trees. For indecomposable polynomials, periodic curves can, in turn, be described in terms of these balanced curves.

    I will also discuss the converse direction. Using ramification in iterated preimage fields, together with results related to dynamical gcd estimates, one obtains—conditionally on the \(abc\) and Vojta conjectures—that points not lying on an appropriate periodic curve cannot have isomorphic preimage trees. Thus, in a broad class of polynomial dynamical systems, the arithmetic relation between two infinite preimage trees is governed by the geometry of periodic curves.
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