Browsing by Author "Morton, Patrick"
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Item Arithmetic properties of 3-cycles of quadratic maps over Q(Elsevier, 2022-11) Morton, Patrick; Raianu, Serban; Mathematical Sciences, School of ScienceIt 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 fc(x) = x2 + 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 fc(x) are proved. A graph on the numerators of the rational 3-periodic points of maps fc is considered which reflects connections between solutions of norm equations from the cubic field of discriminant -23.Item Genera of integer representations and the Lyndon-Hochschild-Serre spectral sequence(2021-08) Neuffer, Christopher; Ramras, Daniel; Ji, Ronghui; Morton, Patrick; Buse, OlgutaThere has been in the past ten to fifteen years a surge of activity concerning the cohomology of semi-direct product groups of the form $\mathbb{Z}^{n}\rtimes$G with G finite. A problem first stated by Adem-Ge-Pan-Petrosyan asks for suitable conditions for the Lyndon-Hochschild-Serre Spectral Sequence associated to this group extension to collapse at second page of the Lyndon-Hochschild-Serre spectral sequence. In this thesis we use facts from integer representation theory to reduce this problem to only considering representatives from each genus of representations, and establish techniques for constructing new examples in which the spectral sequence collapses.Item The Hasse invariant of the Tate normal form E5 and the class number of Q(−5l)(Elsevier, 2021-10) Morton, Patrick; Mathematical Sciences, School of ScienceIt is shown that the number of irreducible quartic factors of the form g(x)=x4+ax3+(11a+2)x2−ax+1 which divide the Hasse invariant of the Tate normal form E5 in characteristic l is a simple linear function of the class number h(−5l) of the field Q(−5l), when l≡2,3 modulo 5. A similar result holds for irreducible quadratic factors of g(x), when l≡1,4 modulo 5. This implies a formula for the number of linear factors over Fp of the supersingular polynomial ssp(5⁎)(x) corresponding to the Fricke group Γ0⁎(5).Item On the Hasse invariants of the Tate normal forms E5 and E7(Elsevier, 2021) Morton, Patrick; Mathematical Sciences, School of ScienceA formula is proved for the number of linear factors over Fl of the Hasse invariant of the Tate normal form E5(b) for a point of order 5, as a polynomial in the parameter b, in terms of the class number of the imaginary quadratic eld K = Q(pl), 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 speci c form in terms of the class number of Q(p 5l). These results are shown to imply a recent conjecture of Nakaya on the number of linear factors over Fl of the supersingular polynomial ss(5 ) l (X) corresponding to the Fricke group 0 (5). The degrees and forms of the irreducible factors of the Hasse invariant of the Tate normal form E7 for a point of order 7 are determined, which is used to show that the polynomial ss(N ) l (X) for the group 0 (N) has roots in Fl2 , for any prime l 6= N, when N 2 f2; 3; 5; 7g.Item Periodic points of algebraic functions and Deuring’s class number formula(Springer, 2019) Morton, Patrick; Mathematical Sciences, School of ScienceThe exact set of periodic points in Q of the algebraic function ˆ F(z) = (−1±p1 − z4)/z2 is shown to consist of the coordinates of certain solutions (x, y) = ( , ) of the Fermat equation x4+y4 = 1 in ring class fields f over imaginary quadratic fields K = Q(p−d) of odd conductor f, where −d = dKf2 1 (mod 8). This is shown to result from the fact that the 2-adic function F(z) = (−1 + p1 − z4)/z2 is a lift of the Frobenius automorphism on the coordinates for which | |2 < 1, for any d 7 (mod 8), when considered as elements of the maximal unramified extension K2 of the 2-adic field Q2. 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 ˆ F(z) in Q are also determined.Item Product formulas for the 5-division points on the Tate normal form and the Rogers–Ramanujan continued fraction(Elsevier, 2019) Morton, Patrick; Mathematical Sciences, School of ScienceExplicit formulas are proved for the 5-torsion points on the Tate normal form E5 of an elliptic curve having (X,Y)=(0,0) as a point of order 5. These formulas express the coordinates of points in E5[5]−⟨(0,0)⟩ 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 E5 is given as b=(ε5u5−ε−5)/(u5+1) and ε=(−1+5–√)/2. If r(τ) is the Rogers-Ramanujan continued fraction and b=r5(τ), then the coordinates of points of order 5 in E5[5]−⟨(0,0)⟩ are shown to be products of linear fractional expressions in r(5τ) with coefficients in Q(ζ5).Item The quartic Fermat equation in Hilbert class fields of imaginary quadratic fields(World Scientific, 2015-09) Lynch, Rodney; Morton, Patrick; Department of Mathematical Sciences, School of ScienceIt is shown that the quartic Fermat equation x4 + y4 = 1 has nontrivial integral solutions in the Hilbert class field Σ of any quadratic field whose discriminant satisfies -d ≡ 1 (mod 8). A corollary is that the quartic Fermat equation has no nontrivial solution in , for p (> 7) a prime congruent to 7 (mod 8), but does have a nontrivial solution in the odd degree extension Σ of K. These solutions arise from explicit formulas for the points of order 4 on elliptic curves in Tate normal form. The solutions are studied in detail and the results are applied to prove several properties of the Weber singular moduli introduced by Yui and Zagier.Item Solutions of diophantine equations as periodic points of p-adic algebraic functions, I(2016) Morton, Patrick; Department of Mathematical Sciences, School of ScienceSolutions of the quartic Fermat equation in ring class fields of odd conductor over quadratic fields K = Q(√-d) with -d Ξ 1 (mod 8) are shown to be periodic points of a fixed algebraic function T(z) defined on the punctured disk 0 < |z|2 ≥ 1/2 of the maximal unramified, algebraic extension K2 of the 2-adic field Q2. 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.Item Solutions of diophantine equations as periodic points of p-adic algebraic functions, II: The Rogers-Ramanujan continued fraction(2019) Morton, Patrick; Mathematical Sciences, School of ScienceIn this part we show that the diophantine equation X5+Y5=ε5(1−X5Y5) , where ε=−1+5√2 , has solutions in specific abelian extensions of quadratic fields K=Q(−d−−−√) in which −d≡±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.Item Solutions of diophantine equations as periodic points of p-adic algebraic functions, III(arXiv, 2020) Morton, Patrick; Mathematical Sciences, School of Science