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Register a new userMinimally critical endomorphisms of P^N
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Authors
Patrick Ingram
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We study the dynamics of the map endomorphism of N-dimensional projective space defined by f(X)=AX^d, where A is a matrix and d is at least 2. When d>N^2+N+1, we show that the critical height of such a morphism is comparable to its height in moduli space, confirming a case of a natural generalization of a conjecture of Silverman.
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We study the dynamics of a class of endomorphisms of A^N which restricts, when N = 1, to the class of unicritical polynomials. Over the complex numbers, we obtain lower bounds on the sum of Lyapunov exponents, and a statement which generalizes the compactness of the Mandelbrot set. Over the algebraic numbers, we obtain estimates on the critical height, and over general algebraically closed fields we obtain some rigidity results for post-critically finite morphisms of this form.
We give an effective algorithm to determine the endomorphism ring of a Drinfeld module, both over its field of definition and over a separable or algebraic closure thereof. Using previous results we deduce an effective description of the image of the adelic Galois representation associated to the Drinfeld module, up to commensurability. We also give an effective algorithm to decide whether two Drinfeld modules are isogenous, again both over their field of definition and over a separable or algebraic closure thereof.
For a natural number $Ngeq 2$ and a real $alpha$ such that $0 < alpha leq sqrt{N}-1$, we define $I_alpha:=[alpha,alpha+1]$ and $I_alpha^-:=[alpha,alpha+1)$ and investigate the continued fraction map $T_alpha:I_alpha to I_alpha^-$, which is defined as $T_alpha(x):= N/x-d(x),$ where $d(x):=left lfloor N/x -alpharight rfloor$. For all natural $N geq 7$, for certain values of $alpha$, open intervals $(a,b) subset I_alpha$ exist such that for almost every $x in I_{alpha}$ there is an natural number $n_0$ for which $T_alpha^n(x) otin (a,b)$ for all $ngeq n_0$. These emph{gaps} $(a,b)$ are investigated in the square $Upsilon_alpha:=I_alpha times I_alpha^-$, where the emph{orbits} $T_alpha^k(x), k=0,1,2,ldots$ of numbers $x in I_alpha$ are represented as cobwebs. The squares $Upsilon_alpha$ are the union of emph{fundamental regions}, which are related to the cylinder sets of the map $T_alpha$, according to the finitely many values of $d$ in $T_alpha$. In this paper some clear conditions are found under which $I_alpha$ is gapless. When $I_alpha$ consists of at least five cylinder sets, it is always gapless. In the case of four cylinder sets there are usually no gaps, except for the rare cases that there is one, very wide gap. Gaplessness in the case of two or three cylinder sets depends on the position of the endpoints of $I_alpha$ with regard to the fixed points of $I_alpha$ under $T_alpha$.
Let f be a degree d polynomial defined over the nonarchimedean field C_p, normalized so f is monic and f(0)=0. We say f is post-critically bounded, or PCB, if all of its critical points have bounded orbit under iteration of f. It is known that if p is greater than or equal to d and f is PCB, then all critical points of f have p-adic absolute value less than or equal to 1. We give a similar result for primes between d/2 and d. We also explore a one-parameter family of cubic polynomials over the 2-adic numbers to illustrate that the p-adic Mandelbrot set can be quite complicated when p is less than d, in contrast with the simple and well-understood p > d case.
We show that for primes $N, p geq 5$ with $N equiv -1 bmod p$, the class number of $mathbb{Q}(N^{1/p})$ is divisible by $p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when $N equiv -1 bmod p$, there is always a cusp form of weight $2$ and level $Gamma_0(N^2)$ whose $ell$-th Fourier coefficient is congruent to $ell + 1$ modulo a prime above $p$, for all primes $ell$. We use the Galois representation of such a cusp form to explicitly construct an unramified degree $p$ extension of $mathbb{Q}(N^{1/p})$.
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