[Paper Review] Solutions of diophantine equations as periodic points of $p$-adic algebraic functions, I

Patrick Morton|arXiv (Cornell University)|Oct 17, 2014

Advanced Mathematical Identities11 references4 citations

TL;DR

This paper establishes a dynamical correspondence between periodic points of a specific 2-adic algebraic function $ T(z) $ and solutions to the quartic Fermat equation in ring class fields of imaginary quadratic fields $ K = \mathbb{Q}(\sqrt{-d}) $ with $ -d \equiv 1 \pmod{8} $. It proves that all but two periodic points of $ T(z) $ in $ \overline{\mathbb{Q}}_2 $ generate ring class fields over $ p $-admissible imaginary quadratic fields, providing a dynamical proof of a class number relation originally due to Deuring.

ABSTRACT

Solutions of the quartic Fermat equation in ring class fields of odd conductor over quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ with $-d \equiv 1$ (mod $8$) are shown to be periodic points of a fixed algebraic function $T(z)$ defined on the punctured disk $0< |z|_2 \le \frac{1}{2}$ of the maximal unramified, algebraic extension $ extsf{K}_2$ of the $2$-adic field $\mathbb{Q}_2$. All ring class fields of odd conductor over imaginary quadratic fields in which the prime $p=2$ splits are shown to be generated by complex periodic points of the algebraic function $T$, and conversely, all but two of the periodic points of $T$ generate ring class fields over suitable imaginary quadratic fields. This gives a dynamical proof of a class number relation originally proved by Deuring. It is conjectured that a similar situation holds for an arbitrary prime $p$ in place of $p=2$, where the case $p=3$ has been previously proved by the author, and the case $p=5$ will be handled in Part II.

Motivation & Objective

• To establish a dynamical correspondence between periodic points of a $ p $-adic algebraic function and ring class fields over imaginary quadratic fields.
• To provide a dynamical proof of a class number relation originally proved by Deuring using periodic points of $ T(z) $ in the $ 2 $-adic setting.
• To extend the framework of complex multiplication and class field theory by embedding ring class fields in the $ p $-adic dynamical system defined by $ T(z) $.
• To support a broader conjecture that similar dynamical structures exist for all primes $ p $, with $ p=2 $ as a foundational case.

Proposed method

• Define a $ 2 $-adic algebraic function $ T(z) $ on the punctured disk $ 0 < |z|_2 \leq \frac{1}{2} $ in the maximal unramified extension $ \mathsf{K}_2 $ of $ \mathbb{Q}_2 $, using binomial expansions.
• Show that solutions to the quartic Fermat equation in ring class fields over $ K = \mathbb{Q}(\sqrt{-d}) $ with $ -d \equiv 1 \pmod{8} $ are periodic points of $ T(z) $.
• Use iterated resultants to compute minimal polynomials $ b_d(x) $ of periodic points, linking them to class field generation.
• Prove that all but two periodic points of $ T(z) $ in $ \overline{\mathbb{Q}}_2 $ generate ring class fields over $ p $-admissible imaginary quadratic fields.
• Derive a class number relation by counting periodic points of minimal period $ n $, showing $ \sum_{-d \in \mathfrak{D}_n} h(-d) = n N_4(n) = \sum_{k|n} \mu(n/k) 2^{2k} $ for $ n > 1 $.
• Verify factorizations of resultant polynomials $ R_n(x) $ into products of $ b_d(x) $, confirming that periodic points correspond to ring class field generators.

Experimental results

Research questions

• RQ1 How can periodic points of a $ p $-adic algebraic function be used to generate ring class fields over imaginary quadratic fields?
• RQ2 What is the precise dynamical system (algebraic function) that realizes the class field theory of ring class fields for $ p=2 $?
• RQ3 Can the periodic points of this function be shown to parametrize solutions to Diophantine equations such as the quartic Fermat equation?
• RQ4 To what extent do the periodic points of $ T(z) $ generate all ring class fields over $ p $-admissible imaginary quadratic fields, and how many exceptions exist?
• RQ5 Does the counting of periodic points of a given minimal period yield a class number relation, and if so, can it be proven via this dynamical approach?

Key findings

• All ring class fields of odd conductor over $ p $-admissible imaginary quadratic fields $ K = \mathbb{Q}(\sqrt{-d}) $ with $ -d \equiv 1 \pmod{8} $ are generated by periodic points of the $ 2 $-adic function $ T(z) $.
• All but two periodic points of $ T(z) $ in $ \overline{\mathbb{Q}}_2 $ generate ring class fields over some $ p $-admissible imaginary quadratic field.
• The number of periodic points of minimal period $ n $ in the domain $ \textsf{D}_2 = \{ y : 0 < |y|_2 \leq \frac{1}{2} \} $ is given by $ n N_4(n) = \sum_{k|n} \mu(n/k) 2^{2k} $, which equals the sum of class numbers $ h(-d) $ over discriminants $ -d \in \mathfrak{D}_n $.
• The minimal polynomials $ b_d(x) $ of periodic points are verified via factorization of iterated resultants $ R_n(x) $, with explicit factorizations into $ b_d(x) $ for various $ d $, such as $ b_{23}(x), b_{31}(x), \dots, b_{255}(x) $, confirming their role in class field generation.
• The dynamical system provides a new proof of a class number relation originally due to Deuring, now derived from the periodic orbit structure of $ T(z) $.
• All periodic points of $ T(z) $ with $ n > 1 $ are prime elements in the local field $ \mathsf{K}_2 $, indicating their arithmetic significance in the $ 2 $-adic setting.