# Local and global canonical height functions for affine space regular automorphisms

@article{Kawaguchi2009LocalAG, title={Local and global canonical height functions for affine space regular automorphisms}, author={Shu Kawaguchi}, journal={arXiv: Algebraic Geometry}, year={2009} }

Let f: A^N \to A^N be a regular polynomial automorphism defined over a number field K. For each place v of K, we construct the v-adic Green functions G_{f,v} and G_{f^{-1},v} (i.e., the v-adic canonical height functions) for f and f^{-1}. Next we introduce for f the notion of good reduction at v, and using this notion, we show that the sum of v-adic Green functions over all v gives rise to a canonical height function for f that satisfies the Northcott-type finiteness property. Using previous…

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## References

SHOWING 1-10 OF 25 REFERENCES

### Canonical height functions for affine plane automorphisms

- Mathematics
- 2006

Let be a polynomial automorphism of dynamical degree δ≥2 over a number field K. We construct height functions defined on that transform well relative to f, which we call canonical height functions…

### Filtrations, hyperbolicity and dimension for polynomial automorphisms of Cn

- Mathematics
- 2002

In this paper we study the dynamics of regular polynomial automorphisms of C^n. These maps provide a natural generalization of complex Henon maps in C^2 to higher dimensions. For a given regular…

### Filtrations, hyperbolicity and dimension for polynomial automorphisms

- Mathematics

In this paper we study the dynamics of regular polynomial automorphisms of C n. These maps provide a natural generalization of complex Hénon maps in C 2 to higher dimensions. For a given regular…

### Height Bounds and Preperiodic Points for Families of Jointly Regular Affine Maps

- Mathematics
- 2006

h ( φ(P ) ) = d · h(P ) + O(1) for all P ∈ P (K̄) combined with the fact that there are only finitely many K-rational points of bounded height leads immediately to a proof of Northcott’s Theorem [18]…

### An upper bound for the height for regular affine automorphisms of A^n

- Mathematics
- 2009

In 2006, Kawaguchi proved a lower bound for height of h(f(P)) when f is a regular affine automorphism of A^2, and he conjectured that a similar estimate is also true for regular affine automorphisms…

### Fundamentals of Diophantine Geometry

- Mathematics
- 1983

1 Absolute Values.- 2 Proper Sets of Absolute Values. Divisors and Units.- 3 Heights.- 4 Geometric Properties of Heights.- 5 Heights on Abelian Varieties.- 6 The Mordell-Weil Theorem.- 7 The…

### Geometric and arithmetic properties of the Hénon map

- Mathematics
- 1994

The real and complex dynamics of the H6non map 4) : Ik2--*~ 2, 4 ) ( x , y ) = ( y , y 2 + b + a x ) (1) have been extensively studied since H6non [6] showed that such maps may have strange…

### Nonarchimedean Green functions and dynamics on projective space

- Mathematics
- 2007

Let $${\varphi: \mathbb{P}^N_K\to\mathbb{P}^N_K}$$ be a morphism of degree d ≥ 2 defined over a field K that is algebraically closed field and complete with respect to a nonarchimedean absolute…

### The Arithmetic of Dynamical Systems

- Mathematics
- 2007

* Provides an entry for graduate students into an active field of research
* Each chapter includes exercises, examples, and figures
* Will become a standard reference for researchers in the field
…

### Heights in Diophantine Geometry

- Mathematics
- 2006

I. Heights II. Weil heights III. Linear tori IV. Small points V. The unit equation VI. Roth's theorem VII. The subspace theorem VIII. Abelian varieties IX. Neron-Tate heights X. The Mordell-Weil…