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

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Periodic points of algebraic functions

related to a continued fraction of

Ramanujan

Sushmanth J. Akkarapakam and Patrick Morton

Sept. 23, 2022

Abstract

A continued fraction v(τ) of Ramanujan is evaluated at certain argu-

ments in the ﬁeld K=Q(√−d), with −d≡1 (mod 8), in which the ideal

(2) = ℘2℘0

2is a product of two prime ideals. These values of v(τ) are

shown to generate the inertia ﬁeld of ℘2or ℘0

2in an extended ring class

ﬁeld over the ﬁeld K. The conjugates over Qof these same values, to-

gether with 0,−1±√2, are shown to form the exact set of periodic points

of a ﬁxed algebraic function ˆ

F(x), independent of d. These are analogues

of similar results for the Rogers-Ramanujan continued fraction.

1 Introduction

This paper is concerned with values of Ramanujan’s continued fraction

v(τ) = q1/2

1 + q+

1 + q3+

1 + q5+

1 + q7+. . . , q =e2πiτ ,

which is also given by the inﬁnite product

v(τ) = q1/2∞

n=1 1−qn(2

n), q =e2πiτ ,

for τin the upper half-plane. Here, 2

nis the Kronecker symbol. See [8], [5,

p. 153]. The continued fraction v(τ) is analogous to the Rogers-Ramanujan

continued fraction

r(τ) = q1/5∞

n=1 1−qn(5

n)q=e2πiτ ,

whose properties were considered in the papers [13], [14]. In [13] it was shown

that certain values of r(τ), for τin the imaginary quadratic ﬁeld K=Q(√−d)

with discriminant −d≡ ±1 (mod 5), are periodic points of a ﬁxed algebraic

arXiv:2210.00659v2 [math.NT] 12 Feb 2023

function, independent of d, and generate certain class ﬁelds ΣfΩfover K. Here

Σfis the ray class ﬁeld of conductor f=℘5or ℘0

5over K, where (5) = ℘5℘0

in the ring of integers RKof K; and Ωfis the ring class ﬁeld of conductor f

corresponding to the order R−dof discriminant −d=dKf2in K(dKis the

discriminant of K).

Here we will show that a similar situation holds for certain values of the

continued fraction v(τ). We consider discriminants of the form −d≡1 (mod

8) and arguments in the ﬁeld K=Q(√−d). Let RKbe the ring of integers in

this ﬁeld and let the prime ideal factorization of (2) in RKbe (2) = ℘2℘0

2. We

deﬁne the algebraic integer wby

w=a+√−d

2, a2+d≡0 (mod 25),(N(w), f) = 1,(1.1)

where ℘2= (2, w). Also, the positive (and odd) integer fis deﬁned by −d=

dKf2, where dKis the discriminant of K/Q.

We will show that

v(w/8) = ±1±√1 + π2

π,

where πis a generator in Ωfof the ideal ℘2(or rather, its extension ℘2RΩfin

Ωf). The algebraic integer πand its conjugate ξin Ωfwere studied in [10] and

shown to satisfy

π4+ξ4= 1,(π) = ℘2,(ξ) = ℘0

2, ξ =πτ2+ 1

πτ2−1,(1.2)

where τ=Ωf/K

℘2is the Artin symbol (Frobenius automorphism) for the prime

ideal ℘2and the ring class ﬁeld Ωfover Kwhose conductor is f. It follows from

results of [10] that

π= (−1)cp(w),

where cis an integer satisfying the congruence

c≡1−a2+d

32 (mod 2)

and p(τ) is the modular function p(τ) = f2

2(τ/2)

f2(τ/2) , deﬁned in terms of the Weber-

Schl¨aﬂi functions f2(τ),f(τ). (See [16], [4], [15].) The above formula for v(w/8)

follows from the identity

p(8τ)=1−v2(τ)

v(τ)=1

v(τ)−v(τ),

for τin the upper half-plane, which we prove in Proposition 5.

As in [13], we consider a diophantine equation, namely

C2:X2+Y2=σ2(1 + X2Y2), σ =−1 + √2.

An identity for the continued fraction v(τ) implies that (X, Y )=(v(w/8), v(−1/w))

is a point on C2. We prove the following theorem relating the coordinates of

this point.

Theorem A. Let wbe given by (1.1) with ℘2= (2, w)in RK, and let −d=

dKf2≡1(mod 8).

(a) The ﬁeld F1=Q(v(w/8)) = Q(v2(w/8)) equals the ﬁeld Σ℘03

2Ωf, where Σ℘03

is the ray class ﬁeld of conductor f=℘03

2and Ωfis the ring class ﬁeld of

conductor fover the ﬁeld K. The ﬁeld F1is the inertia ﬁeld for ℘2in the

extended ring class ﬁeld LO,8= Σ8Ωfover K, where O=R−dis the order

of discriminant −din K.

(b) We have F2=Q(v(−1/w)) = Σ℘3

2Ωf, the inertia ﬁeld of ℘0

2in LO,8/K.

℘2, then

v(−1/w) = v(w/8)τ2+ (−1)cσ

σv(w/8)τ2−(−1)c.(1.3)

See Theorems 1, 3 and 4 and their corollaries.

From part (c) of this theorem we deduce the following.

Theorem B. If wand care as above, then the generator (−1)1+cv(w/8) of

the ﬁeld Σ℘03

2Ωfover Qis a periodic point of the multivalued algebraic function

F(x)given by

F(x) = −x2−1

2±1

2px4−6x2+ 1; (1.4)

and v2(w/8) is a periodic point of the algebraic function ˆ

T(x)deﬁned by

T(x) = 1

2(x2−4x+ 1) ±x−1

2px2−6x+ 1.(1.5)

The minimal period of (−1)1+cv(w/8) (and of v2(w/8)) is equal to the order of

the automorphism τ2in Gal(F1/K). Together with the numbers 0,−1±√2, the

values (−1)1+cv(w/8) and their conjugates over Qare the only periodic points

of the algebraic function ˆ

F(x)in Qor C. The only periodic points of ˆ

T(x)in

Qor Care 0,(−1±√2)2, and the conjugates of the values v2(w/8) over Q.

We understand by a periodic point of the multivalued algebraic function

F(x) the following. Let f(x, y) = x2y+x2+y2−ybe the minimal polynomial

of ˆ

F(x) over Q(x). A periodic point of ˆ

F(x) is an algebraic number afor which

there exist a1, a2, . . . , an−1∈Qsatisfying

f(a, a1) = f(a1, a2) = ··· =f(an−1, a) = 0.

See [11], [12]. Thus, if a∈Qis a periodic point of ˆ

F(x), so are its conjugates

over Q, because f(x, y) has coeﬃcients in Q. We show in Section 8 that v2(w/8)

is actually a periodic point in the usual sense of the single-valued 2-adic function

T(x) = x2−4x+ 2 −(x−1)(x−3) ∞

k=1

Ck−1

(x−3)2k,

deﬁned on a subset of the maximal unramiﬁed, algebraic extension K2of the

2-adic ﬁeld Q2. (Ckis the k-th Catalan number.) This follows from the fact

that

v(w/8)2τ2=T(v(w/8)2),

in the completion F1,q⊂K2of F1= Σ℘03

2Ωfwith respect to a prime divisor q

of ℘2in F1. This implies that the minimal period of v2(w/8) with respect to

the function T(x) is n= ord(τ2).

These facts are all analogues of the corresponding facts for the Rogers-

Ramanujan continued fraction r(τ) which were proved in [13] and [14].

An important corollary of the fact that the conjugates of the values v(w/8)

in Theorem B are, together with the three ﬁxed points, all the periodic points

of the algebraic function ˆ

F(x), is the following class number formula. In this

formula, h(−d) denotes the class number of the order R−dof discriminant −din

the quadratic ﬁeld K=Kd, and Dn,2is the ﬁnite set of negative discriminants

−d≡1 (mod 8) for which the Frobenius automorphism τ2in Theorem A has

order nin Gal(F1/Kd) (F1=F1,d also depends on d):

−d∈Dn,2

h(−d) = 1

k|n

µ(n/k)2k, n > 1.(1.6)

(µ(n) is the M¨obius function.) See Theorem 9. This fact is the analogue for the

prime p= 2 of Theorem 1.3 in [14] for the prime p= 5, or of Conjecture 1 of

that paper for a prime p > 5.

The layout of the paper is as follows. Section 2 contains a number of q-

identities (following Ramanujan) and theta function identities which we use to

prove identities for various modular functions in Sections 3-5. In Sections 6 and

7 we prove Theorem A. The proofs of Theorem B and (1.6) are given in Sections

8 and 9.

2 Preliminaries.

As is customary, let us set

(a;q)0:= 1,(a;q)n:=

n−1

k=0

(1 −aqk), n ≥1

and

(a;q)∞:= ∞

k=0

(1 −aqk),|q| ≤ 1.

Ramanujan’s general theta function f(a, b) is deﬁned as

f(a, b) := ∞

n=−∞

an(n+1)/2bn(n−1)/2.(2.1)

Three special cases are deﬁned, in Ramanujan’s notation, as

ϕ(q) := f(q, q) = ∞

n=−∞

qn2,(2.2)

ψ(q) := f(q, q3) = ∞

n=0

qn(n+1)/2,(2.3)

f(−q) := f(−q, −q2) = ∞

n=−∞

(−1)nqn(3n−1)/2.(2.4)

The Jacobi’s triple product identity, in Ramanujan’s notation, takes the form

f(a, b)=(−a;ab)∞(−b;ab)∞(ab;ab)∞.(2.5)

Using this, the above three functions can be written as

ϕ(q)=(−q;q2)2

∞(q2;q2)∞,(2.6)

ψ(q)=(−q;q)∞(q2;q2)∞=(q2;q2)∞

(q;q2)∞

,(2.7)

f(−q) = (q;q)∞.(2.8)

The equality that relates the right hand sides of both the equations for f(−q)

in (2.4) and (2.8) is Euler’s pentagonal number theorem.

Another important function that plays a prominent role is given by

χ(q) := (−q;q2)∞.(2.9)

All the above four functions satisfy a myriad of relations, most of which are

listed and proved in Berndt’s books on Ramanujan’s notebooks, and we will

refer to them as needed.

Last but not least, the Dedekind-eta function is deﬁned as

η(τ) = q1/24 f(−q), q =e2πiτ ,Im τ > 0.(2.10)

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摘要：

PeriodicpointsofalgebraicfunctionsrelatedtoacontinuedfractionofRamanujanSushmanthJ.AkkarapakamandPatrickMortonSept.23,2022AbstractAcontinuedfractionv()ofRamanujanisevaluatedatcertainargu-mentsintheeldK=Q(pd),withd1(mod8),inwhichtheideal(2)=}2}02isaproductoftwoprimeideals.Thesevaluesofv()areshown...

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

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