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Patrick Morton

Publications and source records attributed to Patrick Morton.

At least 19 recordsLinked to original sources
arXiv2026

Legendre polynomials and complex multiplication, II: class numbers of quadratic fields and genus 2 supersingular polynomials

The factorizations over $\mathbb{F}_p$ of two supersingular polynomials $h_p(x)$ and $g_p(x)$ for genus $2$ curves, discussed by Ibukiyama, Katsura and Oort in their 1986 paper, are investigated. These polynomials are congruent modulo $p$ to the Jacobi polynomials $P_n^{(\alpha,0)}(1-2x)$, for $\alpha = \pm 1/4, \pm 1/6$, respectively. The number of their linear factors (mod $p$) is determined in terms of class numbers of the imaginary quadratic fields $\mathbb{Q}(\sqrt{-dp})$, where $d \in \{1,2,3\}$. The proofs use a quadratic transformation relating these polynomials to the Legendre polynomials $P_n(x)$; previous results on linear and binomial quadratic factors of $P_{(p-e)/4}(x)$ and $P_{(p-\bar e)/3}(x)$ proved by Brillhart and Morton; and properties of the irreducible quadratic factors of the class equations $H_{-3p}(X)$ and $H_{-12p}(X)$ (mod $p$). The quadratic transformation and linear factor results for $h_p(x) \equiv P_n^{(\pm1/4,0)}(1-2x)$ were first discovered using artificial intelligence.

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arXiv2025

Algebraic functions and class number formulas

A class number formula is proved for extended ring class fields $L_{\mathcal{O},9}$ over imaginary quadratic fields $K_d = \mathbb{Q}(\sqrt{-d})$, in which the prime $p = 3$ splits, by determining the fields generated by the periodic points of a well-chosen algebraic function. The number of periodic points of a given period $n \ge 2$ for this algebraic function equals six times the sum of class numbers of imaginary quadratic orders $\textsf{R}_{-d}$, for which the Artin symbol for a prime ideal divisor $\wp_3$ in $K_d$ of $3$ has order $n$ in the Galois group of $F_d/K_d$, where $F_d$ is the inertia field of $\wp_3$ in $L_{\mathcal{O},9}/K_d$.

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arXiv2025

A quadrilateral half-turn theorem

If $ABC$ is a given triangle in the plane, $P$ is any point not on the extended sides of $ABC$ or its anticomplementary triangle, $Q$ is the complement of the isotomic conjugate of $P$ with respect to $ABC$, $DEF$ is the cevian triangle of $P$, and $D_0$ and $A_0$ are the midpoints of segments $BC$ and $EF$, respectively, a synthetic proof is given for the fact that the complete quadrilateral defined by the lines $AP, AQ, D_0Q, D_0A_0$ is perspective by a Euclidean half-turn to the similarly defined complete quadrilateral for the isotomic conjugate $P'$ of $P$ . This fact is used to define and prove the existence of a generalized circumcenter and generalized orthocenter for any such point $P$.

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arXiv2024

A proof of Sugawara's conjecture on Hasse-Weber ray class invariants

In this paper a proof is given of Sugawara's conjecture from 1936, that the ray class field of conductor $\mathfrak{f}$ over an imaginary quadratic field $K$ is generated over $K$ by a single primitive $\mathfrak{f}$-division value of the $\tau$-function, first defined by Weber and then modified by Hasse in his 1927 paper giving a new foundation of complex multiplication.

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arXiv2023

On Ramanujan's cubic continued fraction

The periodic points of the algebraic function defined by the equation $g(x,y) = x^3(4y^2+2y+1)-y(y^2-y+1)$ are shown to be expressible in terms of Ramanujan's cubic continued fraction $c(\tau)$ with arguments in an imaginary quadratic field in which the prime $3$ splits. If $w = (a+\sqrt{-d})/2$ lies in an order of conductor $f$ in $K$ and $9 \mid N_{K/\mathbb{Q}}(w)$, then one of these periodic points is $c(w/3)$, which is shown to generate the ring class field of conductor $2f$ over $K$.

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arXiv2023

New proofs of two identities of Ramanujan

A proof of several identities of Ramanujan involving theta functions of level $7$ is given which uses a specific modular function for $\Gamma_1(7)$ and Klein's projective representation of $PSL(2,7)$ into $PSL(3, \mathbb{C})$. Four identities of Berndt and Zhang are derived as algebraic corollaries of the main proof.

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arXiv2022

Periodic points of algebraic functions related to a continued fraction of Ramanujan

A continued fraction $v(\tau)$ of Ramanujan is evaluated at certain arguments in the field $K = \mathbb{Q}(\sqrt{-d})$, with $-d \equiv 1$ (mod $8$), in which the ideal $(2) = \wp_2 \wp_2'$ is a product of two prime ideals. These values of $v(\tau)$ are shown to generate the inertia field of $\wp_2$ or $\wp_2'$ in an extended ring class field over the field $K$. The conjugates over $\mathbb{Q}$ of these same values, together with $0, -1 \pm \sqrt{2}$, are shown to form the exact set of periodic points of a fixed algebraic function $\hat F(x)$, independent of $d$. These are analogues of similar results for the Rogers-Ramanujan continued fraction.

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arXiv2022

The Hasse invariant of the Tate normal form $E_7$ and the supersingular polynomial for the Fricke group $\Gamma_0^*(7)$

A formula is proved for the number of linear factors and irreducible cubic factors over $\mathbb{F}_l$ of the Hasse invariant $\hat H_{7,l}(a)$ of the Tate normal form $E_7(a)$ for a point of order $7$, as a polynomial in the parameter $a$, in terms of the class number of the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-l})$. Conjectural formulas are stated for the numbers of quadratic and sextic factors of $\hat H_{7,l}(a)$ of certain specific forms in terms of the class number of $\mathbb{Q}(\sqrt{-7l})$, which are shown to imply a recent conjecture of Nakaya on the number of linear factors over $\mathbb{F}_l$ of the supersingular polynomial $ss_l^{(7*)}(X)$ corresponding to the Fricke group $\Gamma_0^*(7)$.

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arXiv2021

Supersingular conjectures for the Fricke group

A proof is given of several conjectures from a recent paper of Nakaya concerning the supersingular polynomial $ss_p^{(N*)}(X)$ for the Fricke group $\Gamma_0^*(N)$, for $N \in \{2, 3, 5, 7\}$. One of these conjectures gives a formula for the square of $ss_p^{(N*)}(X)$ (mod $p$) in terms of a certain resultant, and the other relates the primes $p$ for which $ss_p^{(N*)}(X)$ splits into linear factors (mod $p$) to the orders of certain sporadic simple groups.

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arXiv2021

Arithmetic properties of $3$-cycles of quadratic maps over $\mathbb{Q}$

It is shown that $c=-29/16$ is the unique rational number of smallest denominator, and the unique rational number of smallest numerator, for which the map $f_c(x) = x^2+c$ has a rational periodic point of period $3$. Several arithmetic conditions on the set of all such rational numbers $c$ and the rational orbits of $f_c(x)$ are proved. A graph on the numerators of the rational $3$-periodic points of maps $f_c$ is considered which reflects connections between solutions of norm equations from the cubic field of discriminant $-23$.

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arXiv2021

The Hasse invariant of the Tate normal form $E_5$ and the class number of $\mathbb{Q}(\sqrt{-5l})$

It is shown that the number of irreducible quartic factors of the form $g(x) = x^4+ax^3+(11a+2)x^2-ax+1$ which divide the Hasse invariant of the Tate normal form $E_5$ in characteristic $l$ is a simple linear function of the class number $h(-5l)$ of the field $\mathbb{Q}(\sqrt{-5l})$, when $l \equiv 2,3$ modulo $5$. A similar result holds for irreducible quadratic factors of $g(x)$, when $l \equiv 1, 4$ modulo $5$. This implies a formula for the number of linear factors over $\mathbb{F}_p$ of the supersingular polynomial $ss_p^{(5*)}(x)$ corresponding to the Fricke group $Γ_0^*(5)$.

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arXiv2020

On the Hasse invariants of the Tate normal forms $E_5$ and $E_7$

A formula is proved for the number of linear factors over $\mathbb{F}_l$ of the Hasse invariant of the Tate normal form $E_5(b)$ for a point of order $5$, as a polynomial in the parameter $b$, in terms of the class number of the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-l})$, proving a conjecture of the author from 2005. A similar theorem is proved for quadratic factors with constant term $-1$, and a theorem is stated for the number of quartic factors of a specific form in terms of the class number of $\mathbb{Q}(\sqrt{-5l})$. These results are shown to imply a recent conjecture of Nakaya on the number of linear factors over $\mathbb{F}_l$ of the supersingular polynomial $ss_l^{(5*)}(X)$ corresponding to the Fricke group $Γ_0^*(5)$. The degrees and forms of the irreducible factors of the Hasse invariant of the Tate normal form $E_7$ for a point of order $7$ are determined, which is used to show that the polynomial $ss_l^{(N*)}(X)$ for the group $Γ_0^*(N)$ has roots in $\mathbb{F}_{l^2}$, for any prime $l \neq N$, when $N \in \{2,3,5,7\}$.

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arXiv2020

Solutions of diophantine equations as periodic points of $p$-adic algebraic functions, III

All the periodic points of a certain algebraic function related to the Rogers-Ramanujan continued fraction $r(τ)$ are determined. They turn out to be $0, \frac{-1 \pm \sqrt{5}}{2}$, and the conjugates over $\mathbb{Q}$ of the values $r(w_d/5)$, where $w_d$ is one of a specific set of algebraic integers, divisible by the square of a prime divisor of 5, in the field $K_d=\mathbb{Q}(\sqrt{-d})$, as $-d$ ranges over all negative quadratic discriminants for which $\left(\frac{-d}{5}\right) = +1$. This yields new insights on class numbers of orders in the fields $K_d$. Conjecture 1 of Part I is proved for the prime $p=5$, showing that the ring class fields over fields of type $K_d$ whose conductors are relatively prime to $5$ coincide with the fields generated over $\mathbb{Q}$ by the periodic points (excluding -1) of a fixed $5$-adic algebraic function.

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arXiv2019

Genus theory and the factorization of class equations over $\mathbb{F}_p$

A new proof, depending only on genus theory, is given of a theorem of Stankewicz, which characterizes the primes $p$ for which the class equation $H_D(X)$ of the maximal order of the imaginary quadratic field $K=\mathbb{Q}(\sqrt{D})$ has a linear factor (mod $p$). This yields a prime decomposition law for the primes $p$ with $p \nmid D$ in the real subfield of the Hilbert class field of $K$.

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arXiv2018

Solutions of diophantine equations as periodic points of $p$-adic algebraic functions, II: The Rogers-Ramanujan continued fraction

In this part we show that the diophantine equation $X^5+Y^5=\varepsilon^5(1-X^5Y^5)$, where $\varepsilon=\frac{-1+\sqrt{5}}{2}$, has solutions in specific abelian extensions of quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ in which $-d \equiv \pm 1$ (mod $5$). The coordinates of these solutions are values of the Rogers-Ramanujan continued fraction $r(τ)$, and are shown to be periodic points of an algebraic function.

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arXiv2018

Product formulas for the $5$-division points on the Tate normal form and the Rogers-Ramanujan continued fraction

Explicit formulas are proved for the $5$-torsion points on the Tate normal form $E_5$ of an elliptic curve having $(X,Y)=(0,0)$ as a point of order $5$. These formulas express the coordinates of points in $E_5[5] - \langle(0,0)\rangle$ as products of linear fractional quantities in terms of $5$-th roots of unity and a parameter $u$, where the parameter $b$ which defines the curve $E_5$ is given as $b=(\varepsilon^5 u^5- \varepsilon^{-5})/(u^5+1)$ and $\varepsilon = (-1+\sqrt{5})/2$. If $r(τ)$ is the Rogers-Ramanujan continued fraction and $b=r^5(τ)$, then the coordinates of points of order $5$ in $E_5[5] - \langle(0,0)\rangle$ are shown to be products of linear fractional expressions in $r(5τ)$ with coefficients in $\mathbb{Q}(ζ_5)$.

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arXiv2018

Periodic points of algebraic functions and Deuring's class number formula

The exact set of periodic points in $\overline{\mathbb{Q}}$ of the algebraic function $\widehat{F}(z)=(-1\pm \sqrt{1-z^4})/z^2$ is shown to consist of the coordinates of certain solutions $(x,y)=(π, ξ)$ of the Fermat equation $x^4+y^4=1$ in ring class fields $Ω_f$ over imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ of odd conductor $f$, where $-d \equiv 1$ (mod $8$). This is shown to result from the fact that the $2$-adic function $F(z)=(-1+ \sqrt{1-z^4})/z^2$ is a lift of the Frobenius automorphism on the coordinates $π$ for which $|π|_2<1$, for any $d \equiv 7$ (mod $8$), when considered as elements of the maximal unramified extension $\textsf{K}_2$ of the $2$-adic field $\mathbb{Q}_2$. This gives an interpretation of the case $p=2$ of a class number formula of Deuring. An algebraic method of computing these periodic points and the corresponding class equations $H_{-d}(x)$ is given that is applicable for small periods. The pre-periodic points of $\widehat{F}(z)$ in $\overline{\mathbb{Q}}$ are also determined.

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