1 Introduction
Let p be a prime number and let
${\mathbb {Z}}_{p}$
denote the ring of p-adic integers. The analogy between knots and primes, or 3-manifolds and the rings of integers of number fields, has played an important role since Gauss’s era. In modern times, Barry Mazur initially pointed out the analogy between Iwasawa theory of
${\mathbb {Z}}_{p}$
-extensions and Alexander–Fox theory of
${\mathbb {Z}}$
-covers [Reference Mazur16], and such analogies have been systematically arranged by Kapranov [Reference Kapranov9], Reznikov [Reference Reznikov27, Reference Reznikov28], Morishita [Reference Morishita20–Reference Morishita22], Kim [Reference Kim, Neumann and Schroll10], and others. The theories of
${\mathbb {Z}}_{p}$
-extensions and
${\mathbb {Z}}$
-covers may be seen as deformation theories of
$\mathrm { GL}_1$
-representations, and this viewpoint extends to the analogy between Hida–Mazur’s Galois deformation theory and Thurston’s hyperbolic deformation theory [Reference Morishita and Terashima24].
In the context of Hida–Mazur theory and explorations of its more precise analog in low-dimensional topology [Reference Bénard, Tange, Tran and Ueki1, Reference Kitayama, Morishita, Tange and Terashima11, Reference Kitayama, Morishita, Tange and Terashima12, Reference Morishita, Takakura, Terashima and Ueki23, Reference Tange, Morishita, Nakamura and Ueki32–Reference Tange and Ueki34], there are special interests in irreducible
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-representations whose residual
$\mathrm { SL}_2{\mathbb {F}}_p$
-representations are reducible. In this article, following Mazur’s suggestion [Reference Mazur17, Section 19], we aim to “go the other way,” by considering liminal representations and characters.
Let
$\pi $
be a group. A function
$\chi :\pi \to {\mathbb {Z}}_{p}$
is called an
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-character if there exists an
$\mathrm {SL}_2$
-representation
$\rho $
over an extension of
${\mathbb {Z}}_{p}$
such that
$\chi =\mathrm {tr}\,\rho $
holds. An
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-representation (resp. character) is said to be liminal if it is reducible and its every open neighborhood contains an irreducible
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-representation (resp. character). Our main result in this article will be the following observation.
Theorem 1.1 Let K be a genus one two-bridge knot in
$S^3$
. If an odd prime number p (resp.
$2^3$
) divides the size of the 1st homology group of some odd-th cyclic branched cover of K, then its group
$\pi _1(S^3-K)$
admits a liminal
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-character (resp.
$\mathrm {SL}_2{\mathbb {Z}}_2$
-character).
One may find this result interesting, partially for the following reason. It has been classically known by Burde and de Rham [Reference Burde2, Reference de Rham3] that such deformative characters correspond to the zeros of the Alexander polynomial (see also [Reference Heusener, Porti and Peiró8]), and Fox–Weber’s formula asserts that the size of
$H_1$
of a finite cyclic cover is calculated by using the Alexander polynomial, so our result has a deep context in its background. However, there is no obvious implication, and this result would rather suggest a new interaction between
$\mathrm {GL}_1$
-deformation theory and
$\mathrm {SL}_2$
-deformation theory. We will describe further queries and perspectives for readers with specific interests in Remark 6.5. We also remark that a p-adic analog of Burde–de Rham theorem and its application to number theoretical settings are due to the second author and others [Reference Mizusawa, Tange and Terashima19].
To prove Theorem 1.1, we utilize the character varieties, the cyclic resultants of the Alexander polynomials, the Legendre symbols, Hensel’s lemma, and establish a property of certain Fibonacci/Lucas-type sequences as well.
The argument will be given as follows. In Section 2, we recollect properties of genus one two-bridge knots, namely, double twist knots of type
$J(2k,2l)$
with
$(0,0)\neq (k,l)\in {\mathbb {Z}}^2$
and calculate the intersection of the varieties of irreducible and reducible
$\mathrm { SL}_2{\mathbb {C}}$
-characters (Proposition 2.1). In Section 3, by using Hensel’s lemma several times, we prove that
$J(2k,2l)$
admits a liminal
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-character iff (i)
$p=2$
and
$r\equiv 1$
mod 8 or (ii)
$p\neq 2$
and the Legendre symbol satisfies
$(\frac {r}{p})=1$
, where r denotes the square-free part of
$4k^2l^2-kl$
(Theorem 3.1), and calculate several examples. In Section 4, we further discuss the existence of liminal
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-representations (Proposition 4.2), and give a remark on the case of a general two-bridge knot (Remark 4.3).
In Section 5, for a given
$m\in {\mathbb {Z}}$
, we write
$t^2-t+m=(t-a)(t-b)$
and define the Lucas-type sequence
$(L_n)_n\in {\mathbb {Z}}^{\mathbb {N}}$
by
$L_n=a^n+b^n$
. We prove that the condition
$p\mid L_{2n+1}$
for some
$n\in {\mathbb {Z}}_{\geq 0}$
implies
$(\frac {4m^2-m}{p})=1$
(Theorem 5.1) and exhibit examples. This part is tailored toward readers with independent interests also. Note that the classical one with
$m=-1$
appears in the case of the figure-eight knot
$J(2,-2)=4_1$
.
In Section 6, we recall some properties of the cyclic covers
$M_n\to S^3$
of a knot K, prove for a knot K with its Alexander polynomial
$\Delta _K(t)=mt^2-(2m-1)t+m$
that
$\#H_1(M_{2n+1})=L_{2n+1}^{\,2}$
holds, yielding that
$p\mid \#H_1(M_{2n+1})$
implies
$(\frac {4m^2-m}{p})=1$
(Theorem 6.1), and deduce Theorem 1.1. We also attach further remarks (Remark 6.5).
2 Genus one two-bridge knots
It is known that every genus one two-bridge knot is realized as a double twist knot of type
$J(2k,2l)$
with
$(0,0)\neq (k,l)\in {\mathbb {Z}}^2$
defined by the following diagram. A basic reference is [Reference Tran35].
Especially,
$J(2,2l)$
is known as a twist knot;
$J(2,0)$
is the unknot
$0_1$
,
$J(2,2)$
is the trefoil
$3_1$
, and
$J(2,-2)$
is the figure-eight knot
$4_1$
.
The group of such a knot admits a presentation
$$\begin{align*}\pi:=\pi_1(S^3-J(2k,2l))=\langle a,b\mid w^la=bw^l\rangle\end{align*}$$
with
$a,b$
being meridians and
$w=(ba^{-1})^k(b^{-1}a)^k$
.
A Seifert matrix is given by
$V={\small \begin {pmatrix}k&1\\0&l\end {pmatrix}}$
, so its Alexander polynomial becomes
$$\begin{align*}\Delta_{J(2k,2l)}(t)=\mathrm{det}\,(tV-V^{\perp})=klt^2+(1-2kl)t+kl.\end{align*}$$
For each
$g\in \pi $
, let
$\mathrm {tr}\, g$
denote the map
$\mathrm {Hom}(\pi ,\mathrm {SL}_2{\mathbb {C}})\to {\mathbb {C}}; \rho \mapsto \mathrm {tr}\,\rho (g)$
. Then, the conjugacy classes of
$\mathrm {SL}_2{\mathbb {C}}$
-representations are parameterized by
$$\begin{align*}x:=\mathrm{tr}\,a, \ \ y:=\mathrm{tr}\,ab^{-1}.\end{align*}$$
(This y is different from that in [Reference Tange and Ueki34] and has a strong advantage in our calculation. The old y coincides with
$\mathrm {tr}\,ab=\mathrm {tr}\,a\,\mathrm {tr}\,b-\mathrm {tr}\,ab^{-1}=x^2-y$
.) We further put
$z:=\mathrm {tr}\,w$
.
Each conjugacy class of non-abelian
$\mathrm {SL}_2{\mathbb {C}}$
-representation is presented by Riley’s universal representation
$\rho ^{\mathrm {R}}:\pi \to \mathrm {SL}_2{\mathbb {C}}$
defined by
$$\begin{align*}\rho^{\mathrm{R}}(a)={\small \begin{pmatrix}s&1\\0&s^{-1}\end{pmatrix}}, \ \ \rho^{\mathrm{R}}(b)={\small \begin{pmatrix}s&0\\ 2-y&s^{-1}\end{pmatrix}},\end{align*}$$
where
$x=s+s^{-1}$
and
$x,y$
satisfy Riley’s polynomial equation
$f_{k,l}(x,y)=\Phi _{k,l}(x, y-2)=0$
with
$f_{k,l}(x,y)\in {\mathbb {Z}}[x,y]$
(see also [Reference Le14, Theorem 3.3.1] and [Reference Riley29, Reference Riley30]).
There is a bijective correspondence between conjugacy classes of irreducible representations and points on
$f_{k,l}(x,y)=0$
with
$y-2\neq 0$
. Each point of
$y-2=0$
corresponds to a pair of the class of reducible representations and that of abelian representations. We claim the following.
Proposition 2.1 The intersection of
$f_{k,l}(x,y)=0$
and
$y-2=0$
is
$(x,y)=\left(\pm \sqrt {4-\frac {1}{kl}}, 2\right)$
.
For each
$n\in {\mathbb {Z}}$
, let
$S_n(z)\in {\mathbb {Z}}[z]$
denote the n-th Chebyshev polynomial of the second type defined by
$S_{n-1}(2\cos \theta )=\frac {\sin n\theta }{\sin \theta }$
with
$\theta \in {\mathbb {R}}$
,
$\sin \theta \neq 0$
. This is equivalent to say that
$S_{-1}(z)=0$
,
$S_0(z)=1$
, and
$S_{n+1}(z)-zS_n(z)+S_{n-1}(z)=0$
for every
$n\in {\mathbb {Z}}$
. These polynomials satisfy
$S_{-1-n}(z)=-S_{-1+n}(z)$
and
$S_{n-1}(\pm 2)=(\pm 1)^{n-1}n$
.
By using the
$\mathrm {SL}_2$
trace relations, we may obtain
$$\begin{align*}z=\mathrm{tr}\,w=2+(y-2)(-x^2+y+2)S_{m-1}^2(y).\end{align*}$$
By [Reference Tran35, Subsection 2.2], the variety of irreducible characters is given by
$$\begin{align*}f_{k,l}(x,y)= S_l(z)-(1+(-x^2+y+2)S_{k-1}(y)(S_k(y)-S_{k-1}(y))S_{l-1}(z).\end{align*}$$
Proof of Proposition 2.1
If
$y-2=0$
, then we have
$z=2$
,
$0=f_{k,l}(x,2)=l-(1+(-x^2+2+2)k((k+1)-k))l=1-(4-x^2)kl$
, and hence
$x=\pm \sqrt {4-\frac {1}{kl}}$
.
Remark 2.2 In the old parameters, the intersection points become
$(x,\eta )=\left(\pm \sqrt {4-\frac {1}{kl}}, -\frac {1}{kl}\right)$
. Note that Proposition 2.1 persists over any algebraically closed field
${\mathbb {F}}$
with
$\mathrm {char}\,{\mathbb {F}}\nmid kl$
.
3 Liminal
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-characters
Theorem 3.1 Suppose
$kl\neq 0$
. Then, the group of
$J(2k,2l)$
admits a liminal
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-character iff
i)
$p=2$
and
$4k^2l^2-kl \equiv 1\ \mathrm {mod}\ 8$
or
ii)
$p\neq 2$
,
$p\nmid kl$
, and the Legendre symbol satisfies
$\left (\dfrac {r}{p}\right )=1$
, where r denotes the square-free part of
$4k^2l^2-kl$
.
Here, we define the square-free part of
$0\neq a\in {\mathbb {Z}}$
by
$a/b^2$
, where b denotes the maximal integer with
$b^2\mid a$
. (Note that some people use this term for other notions.) The Legendre symbol
$(\frac {\,a\,}{p})\in \{0,\pm 1\}$
of
$a\in {\mathbb {Z}}$
over p is defined as follows: If
$p\mid a$
, we put
$(\frac {\,a\,}{p})=0$
. Suppose
$p\nmid a$
. If
$a\equiv x^2$
mod p for some
$x\in {\mathbb {Z}}$
, then we put
$(\frac {\,a\,}{p})=1$
. Otherwise, we put
$(\frac {\,a\,}{p})=-1$
. The following lemmas are consequences of Hensel’s lemma (cf.[Reference Neukirch25, Chapter II(4.6)]).
Lemma 3.2 Let
$a \in {\mathbb {Z}}$
with
$p^2\nmid a$
. Then, we have
$\sqrt {a}\in {\mathbb {Z}}_{p}$
iff
i)
$p=2$
and
$a\equiv 1\ \mathrm {mod}\ 8$
or
ii)
$p\neq 2$
and
$(\frac {\,a\,}{p})=1$
.
Proof (i) Suppose
$p=2$
. If
$\sqrt {a}\in {\mathbb {Z}}_2$
, then
$\sqrt {a}=\sum _i b_i 2^i$
with
$b_i\in \{0,1\}$
, and
$a\equiv (b_0+2b_1+4b_2)^2 \equiv b_0^2+4b_1(b_0+b_1)$
mod 8. Hence, by the assumption
$4\nmid a$
, we have
$b_0=1$
. In both cases
$b_1=0,1$
, we have that
$a\equiv 1$
mod
$8$
. Conversely, suppose that
$a\equiv 1$
mod 8 and write
$a=8b+1$
with
$b\in {\mathbb {Z}}$
. If
$X^2-a=0$
have a solution
$X=\alpha $
in
${\mathbb {Z}}_2$
, then
$\alpha =2\beta +1$
for some
$\beta \in {\mathbb {Z}}_2$
, since otherwise we have
$\alpha =2\beta $
for some
$\beta \in {\mathbb {Z}}_2$
and
$a\hspace{-0.5pt}=\hspace{-0.5pt}\alpha ^2\hspace{-0.5pt}=\hspace{-0.5pt}4\beta ^2\not \equiv 1$
mod 8. Thus, we have
$\alpha ^2-a\hspace{-0.5pt}=\hspace{-0.5pt}(2\beta +1)^2-(8b+1)\hspace{-0.5pt}=\hspace{-0.5pt}4(\beta ^2+\beta -2b)$
. Since the polynomial
$Y^2+Y-2b$
mod 2 has single roots in
${\mathbb {F}}_2$
, Hensel’s lemma yields roots of
$Y^2+Y-2b$
in
${\mathbb {Z}}_2$
, and hence those of
$X^2-a$
in
${\mathbb {Z}}_2$
.
ii) Suppose
$p\neq 2$
. If
$p\mid a$
, then, by the assumption
$p^2\nmid a$
, we have
$a=pu$
with
$u\in {\mathbb {Z}}_p^\times $
, and hence
$\sqrt {a}\not \in {\mathbb {Z}}_{p}$
. If
$p\nmid a$
, then, since
$p\neq 2$
,
$X^2-a$
mod p has single roots in
${\mathbb {F}}_{p^2}$
and they lift to a quadratic extension of
${\mathbb {Z}}_{p}$
by Hensel’s lemma. In addition, the following are equivalent:
$(\frac {\,a\,}{p})=1$
, the roots of
$X^2-a$
mod p belong to
${\mathbb {F}}_p$
, and the roots of
$X^2-a$
belong to
${\mathbb {Z}}_{p}$
. This completes the proof.
Lemma 3.3 There is a bijective correspondence between liminal
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-characters of
$\pi _1(S^3-J(2k,2l))$
and the intersection points of
$f_{k,l}(x,y)=0$
and
$y-2=0$
in
${\mathbb {Z}}_{p}^{\,2}$
.
Proof Recall that the varieties of reducible and irreducible characters are given by
$y-2=0$
and
$f_{k,l}(x,y)=0$
. If
$\rho $
is a liminal representation with
$\mathrm {Im}\,\mathrm {tr}\,\rho \subset {\mathbb {Z}}_{p}$
, then
$\mathrm {tr}\,\rho $
is on
$y-2=0$
in
${\mathbb {Z}}_{p}^{\,2}$
. In addition, for every
$n\in {\mathbb {Z}}_{>0}$
, there is an irreducible representation
$\rho '$
with
$\rho \equiv \rho '$
mod
$p^n$
, so
$\mathrm {tr}\,\rho $
is on
$f_{k,l}(x,y)\equiv 0$
mod
$p^n$
. Thus,
$\mathrm {tr}\,\rho $
is on
$f_{k,l}(x,y)=0$
in
${\mathbb {Z}}_{p}^{\,2}$
as well.
Let us prove the converse. Note that Riley’s representation
$\rho ^{\mathrm {R}}$
in Section 2 may be seen as a representation over a quadratic extension of
${\mathbb {Z}}_{p}[x,y]$
, via
$s^2-xs+1=0$
.
By Proposition 2.1, if
$f_{k,l}(x,y)=0$
and
$y-2=0$
has an intersection point in
${\mathbb {Z}}_p^{\,2}$
, then
$f_{k,l}(x,2)=(f_{k,l}(x,y)\ \mathrm {mod}\ (y-2))$
has a single root
$\alpha $
in
${\mathbb {Z}}_{p}$
, and Hensel’s lemma yields the implicit function
$x=x_f(y) \in {\mathbb {Z}}_{p}[\![y-2]\!]$
around
$(x,y)=(\alpha ,2)$
. By substituting
$x=x_f(y)$
in Riley’s representation, we obtain an irreducible representation
${\boldsymbol {\rho }}^R$
over a quadratic extension of
${\mathbb {Z}}_{p}[\![y-2]\!]$
with
$\mathrm {Im}\,\mathrm {tr}\,{\boldsymbol {\rho }}^R \subset {\mathbb {Z}}_{p}[\![y-2]\!]$
.
Note that every element of
${\mathbb {Z}}_{p}[\![y-2]\!]$
converges in
${\mathbb {Z}}_p$
at any
$y\in {\mathbb {Z}}_p$
in the p-adic unit disc
$|y-2|_p<1$
. If we substitute
$y=2$
, then this
${\boldsymbol {\rho }}^R$
yields a reducible representation
$\rho $
with
$\mathrm {tr}\,\rho \subset {\mathbb {Z}}_{p}$
at
$(\alpha ,2)$
. If we instead substitute
$y=\eta \in {\mathbb {Z}}_{p}$
with
$0\neq |2-\eta |_p<1$
, namely,
$2\equiv \eta $
mod
$p^n$
for some
$n\in {\mathbb {Z}}_{>0}$
, then
${\boldsymbol {\rho }}^R$
yields an absolutely irreducible representation
$\rho _\eta $
with
$\mathrm { tr}\,\rho _\eta \subset {\mathbb {Z}}_{p}$
. Thus,
$\mathrm {tr}\,\rho $
is a liminal
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-character.
Proof of Theorem 3.1
Since
$f_{k,l}(x,y)=0$
and
$y-2=0$
have an intersection point in
${\mathbb {Z}}_{p}^{\,2}$
if and only if
$p\nmid kl$
and
$\sqrt {4k^2l^2-kl}\in {\mathbb {Z}}_{p}$
, Lemmas 3.2 and 3.3 yield the assertion.
Example 3.4 The condition for
$\pi _1(S^3-J(2,2l))$
admitting a liminal
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-character becomes i)
$p=2$
and
$4l^2-l\equiv 1$
mod 8 or ii)
$p\neq 2$
and
$(\frac {4l^2-l}{p})=1$
. By elementary calculation, we obtain the following:
-
(i)
$J(2,2)=3_1$
(trefoil):
$(\frac {\,3\,}{p})=1$
, i.e.,
$p\equiv \pm 1$
mod 12. -
(ii)
$J(2,-2)=4_1$
(figure-eight):
$(\frac {\,5\,}{p})=1$
, i.e.,
$p\equiv \pm 1$
mod 5. -
(iii)
$J(2,4)$
:
$(\frac {14}{p})=1$
, i.e.,
$p\equiv \pm 1,\pm 9,\pm 25, \pm 5, \pm 11, \pm 13$
mod 56. -
(iv)
$J(2,-4)$
:
$(\frac {18}{p})=1$
, i.e.,
$p\equiv \pm 1$
mod 8. -
(v)
$J(2,6)$
:
$p=2$
or
$(\frac {33}{p})=1$
, i.e.,
$p\equiv \pm 1,\pm 2,\pm 4,\pm 8,\pm 16$
mod 33. -
(vi)
$J(2,-6)$
:
$(\frac {39}{p})=1$
, i.e.,
$\pm p \equiv 1,5,7,19,23,25,26, 35,41,49,61,67$
mod 156.
4 Liminal
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-representations
Let
$\rho $
be a representation over a finite extension of
${\mathbb {Z}}_{p}$
such that
$\overline {\rho }:=\rho $
mod p is absolutely irreducible and suppose that
$\mathrm {Im}\,\mathrm {tr}\,\overline {\rho }\subset {\mathbb {F}}_p$
. Since the Brauer group of a finite field is trivial, the Skolem–Noether theorem assures that such
$\overline {\rho }$
is conjugate to some representation
$\overline {\rho }'$
over
${\mathbb {F}}_p$
(cf.[Reference Marché15, Subsection 3.5]). In addition, Nyssen’s theorem [Reference Nyssen26, Théorèm 1] asserts that
$\overline {\rho }'$
lifts to some
$\rho '$
over
${\mathbb {Z}}_{p}$
such that
$\mathrm {tr}\,\rho =\mathrm {tr}\,\rho '$
, and such
$\rho '$
is unique up to conjugate.
One might wonder whether this argument extends to representations on the Zariski closure of the variety
$X_{\mathrm {irr}}(S^3-J(2k,2l))=\{f_{f,l}(x,y)=0\}\setminus \{y-2=0\}$
of irreducible characters. Nyssen’s proof uses the fact that being absolutely irreducible is equivalent to the surjectivity of the corresponding algebra homomorphism, so the result does not necessarily extend to residually reducible representations. In addition, as pointed out in [Reference Kitayama, Morishita, Tange and Terashima11, Subsection 2.3], in the setting of knot group representations, the deformation problem is not unobstructed in the sense that the 2nd cohomology of the adjoint representation does not vanish. The following question seems subtle.
Question 4.1 Given a liminal character over
${\mathbb {Z}}_{p}$
, can we find a liminal
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-representation?
We may say at least the following.
Proposition 4.2 Under the conditions of Theorem 3.1, if
$p\neq 2$
and
$(\frac {-kl}{p})=1$
, then the group of
$J(2k,2l)$
admits a liminal
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-representaion.
Proof Let
${\boldsymbol {\rho }}^R$
be the one in Lemma 3.3 and note that
$s=\frac {x-\sqrt {x^2-4}}{2}$
. By Proposition 2.1, we have
$\sqrt {x^2-4}\,|_{y=2}=\sqrt {\frac {-1}{kl}}$
. If
$(\frac {-kl}{p})=1$
, then by Lemma 3.2 (ii),
$\sqrt {x^2-4}=\sqrt {\frac {-1}{kl}}\in {\mathbb {Z}}_{p}$
, and
${\boldsymbol {\rho }}^R$
mod
$y-2$
becomes a liminal
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-representation.
Remark 4.3 We remark that most argument above persists for any two-bridge knot K with the
$\mathrm {SL}_2$
-character variety
$f(x,y)(y-2)=0$
, yielding that, if
$f(x,2)$
has a single root in
${\mathbb {F}}_p$
, then K admits a liminal
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-character. The existence of a liminal
$\mathrm {SL}_2{\mathbb {Z}}_{p}$
-representation remains in mystery.
5 Lucas- and Fibonacci-type sequences
Let
$m\in {\mathbb {Z}}$
and write
$t^2-t+m=(t-a)(t-b)$
, so that we have
$a+b=1$
and
$ab=m$
. We define the Lucas-type sequence and the Fibonacci-type sequence by
$L_n=a^n+b^n$
and
$F_n=\frac {a^n-b^n}{a-b}$
, respectively. Then, we have
$L_0=2$
,
$L_1=1$
,
$L_2=1-2m$
,
$L_{n+2}=L_{n+1}-mL_n$
,
$F_0=0$
,
$F_1=1$
,
$F_2=1$
, and
$F_{n+2}=F_{n+1}-mF_n$
, and hence
$L_n,F_n\in {\mathbb {Z}}$
. We have
$$\begin{align*}L_n^{\,2}+(4m-1)F_n^{\,2}=4m^n.\ \ \cdots (\star).\end{align*}$$
Indeed, by
$(a-b)^2=(a+b)^2-4ab=1-4m$
, we have
$(a^n+b^n)^2+(4m-1)(\frac {a^n-b^n}{a-b})^2 =(a^n+b^n)^2-(a^n-b^n)^2 =4a^nb^n=4m^n$
.
Now suppose that
$p\neq 2$
divides
$L_{2n+1}$
for some
$n\in {\mathbb {Z}}_{\geq 0}$
. Then, we have
$(4m-1)F_{2n+1}^{\,2}\equiv (2m^{n})^2m$
mod p. This implies that
$m(4m-1)$
must be a square mod p. In addition, we have
$p\nmid m(4m-1)$
. Indeed, if
$p\mid m$
, then the recurrence formula yields that
$L_n\equiv 1$
mod p for all
$n\in {\mathbb {Z}}_{>0}$
, contradicting
$p\mid L_{2n+1}$
. If instead
$p\mid 4m-1$
, then
$(\star )$
yields
$p=2$
or
$p\mid m$
, hence again contradiction.
Together with a direct calculation, we obtain the following.
Theorem 5.1
-
(i) If
$2^3$
divides
$L_{2n+1}$
for some
$n\in {\mathbb {Z}}_{\geq 0}$
, then
$4m^2-m\equiv 1$
mod 8 holds. -
(ii) If an odd prime number p divides
$L_{2n+1}$
for some
$n\in {\mathbb {Z}}_{\geq 0}$
, then the Legendre symbol satisfies
$\left (\dfrac {4m^2-m}{p}\right )=1$
.
Proof The assertion (ii) is done. To verify (i), note that
$(L_n\, \mathrm {mod}\, 8)_n$
with
$n\geq 0$
is periodic for
$n\gg 0$
. In mod 8, we have
Thus, we have
$2^3\mid L_{2n+1}$
for some
$n\in {\mathbb {Z}}_{\geq 0}$
if and only if
$m\equiv 3$
mod 8 holds. Since the condition
$4m^2-m\equiv 1$
mod 8 is equivalent to that
$m\equiv 3$
mod 8, we obtain the assertion.
Example 5.2 The following may be compared with Example 3.4.
Let
$m=1$
. Then, the condition
$(\frac {12}{p})=1$
becomes
$p\equiv \pm 1$
mod 12. By
$L_{n+1}=L_n-L_{n-1}$
, we see that
$L_n$
is periodic as
so only
$p=2$
actually appears.
Let
$m=-1$
. Then,
$L_n$
and
$F_n$
become the classical ones, and the condition
$(\frac {\,5\,}{p})=1$
becomes
$p\equiv \pm 1$
mod 5. We actually have
We list the value of
$L_{2n+1}$
with
$0\leq n\leq 10$
for
$m=-1,2,-2,3,-3,4$
:
We may often find a large prime number there.
Remark 5.3 At least when
$m=\pm 1$
, the converse of theorem 5.1 does not hold. When
$m=-1$
(the classical case), we can say more: Lagarias [Reference Lagarias13] proved that the density of
$$\begin{align*}S_A:=\{p\mid p\equiv \pm1\ \text{mod}\ 5\ \text{and}\ p\ \text{divides}\ L_n\ \text{for\ some}\ n\in {\mathbb{Z}}_{>0}\}\end{align*}$$
in the set of prime numbers is 5/12. So, the density of
$$\begin{align*}\{p\mid p\equiv \pm1\ \text{mod}\ 5\ \text{and}\ p\ \text{divides}\ L_{2n+1}\ \text{for\ some}\ n\in {\mathbb{Z}}_{\geq0}\}\end{align*}$$
is smaller than the density 1/2 of the set
$\{p\mid p\equiv \pm 1\ \text {mod}\ 5\}$
.
6 Cyclic covers
Let
$\Delta _K(t)$
denote the Alexander polynomial of a knot K in
$S^3$
. Then, Fox–Weber’s formula [Reference Weber36] asserts that the
${\mathbb {Z}}/n{\mathbb {Z}}$
-cover
$M_n\to S^3$
branched over K satisfies
$$\begin{align*}r_n:=|H_1(M_n;{\mathbb{Z}})|=|\mathrm{Res}(t^n-1, \Delta_K(t))|\end{align*}$$
for each
$n\in {\mathbb {Z}}_{>0}$
. Here, if G is a finite group, then
$|G|$
denotes the order
$\#G$
of G, and if G is an infinite group, then we put
$|G|=0$
. For polynomials
$f(t), g(t)\in {\mathbb {Z}}[t]$
,
$\mathrm {Res}(f(t),g(t))\in {\mathbb {Z}}$
denotes their resultant. We remark that if K is a two-bridge knot of genus one, then the precise group structure of
$H_1(M_n)$
is known by Fox and others ([Reference Fox4], see also [Reference Mednykh18]). Now, we assert the following.
Theorem 6.1 Suppose
$\Delta _K(t)=mt^2+(1-2m)t+m$
with
$m\in {\mathbb {Z}}$
:
(i) If
$2^3\mid r_n$
for some odd n, then
$4m^2-m\equiv 1$
mod 8 holds.
(ii) If
$p\neq 2$
and
$p\mid r_n$
for some odd n, then
$\left (\dfrac {4m^2-m}{p}\right )=1$
holds.
Proof If
$m=0$
, then
$r_n=1$
and
$p\nmid r_n$
. Suppose
$m\neq 0$
. Let us write
$\Delta _K(t)=mt^2+(1-2m)t+m=m(t-\alpha )(t-\beta )$
. Then, we have
$\alpha \beta =1$
,
$\alpha +\beta =\frac {2m-1}{m}$
, and
$\mathrm {Res}(\Delta _K(t),t^n-1) =m^n(\alpha ^n-1)(\beta ^n-1) =m^n(2-\alpha ^n-\beta ^n)$
. In addition, let us write
$t^2-t+m=(t-a)(t-b)$
, so that we have
$a+b=1$
,
$ab=m$
,
$a^2+b^2=1-2m=-m(\alpha +\beta )$
,
$a^2b^2=m^2=(-m\alpha )(-m\beta )$
, and hence
$\{a^2,b^2\}=\{-m\alpha ,-m\beta \}$
. We may assume
$a^2=-m\alpha $
,
$b^2=-m\beta $
. If n is odd, then we have
$L_n^{\,2}=(a^n+b^n)^2 =((-m\alpha )^n+(-m\beta )^n+2m^n) \kern1.5pt{=}\kern1.5pt m^n(2-\alpha ^n-\beta ^n) \kern1.5pt{=}\kern1.5pt\mathrm {Res}(\Delta _K(t),t^n-1)\kern1.5pt{=}\kern1.5pt-\mathrm {Res}(t^n-1, \Delta _K(t)) = r_n$
. Hence, the assertion follows from Theorem 5.1.
Lemma 6.2 Let
$\Delta _K(t)=mt^2+(1-2m)t+m$
with
$m \in {\mathbb {Z}}$
. Then,
$H_1(M_n)$
is an infinite group if and only if
$(m,n)=(1,6)$
.
Proof For each
$n\in {\mathbb {Z}}_{>0}$
, let
$\Phi _n(t)\in {\mathbb {Z}}[t]$
denote the n-th cyclotomic polynomial over
${\mathbb {Q}}$
, so we have
$[{\mathbb {Q}}(\zeta _n):{\mathbb {Q}}]=\mathrm {deg}\,\Phi _n(t)=\varphi (n)$
, where
$\varphi (n)$
denotes Euler’s totient function. Since
$[{\mathbb {Q}}(\zeta _n):{\mathbb {Q}}]=2[{\mathbb {Q}}(\cos \frac {2\pi }{n}):{\mathbb {Q}}]$
, we have
$\varphi (n)\leq 2$
if and only if
$n\in \{1,2,3,4,6\}$
.
If
$H_1(M_n)$
is an infinite group, then
$r_n=|\mathrm {Res}(t^n-1,\Delta _K(t))|=0$
, so we have
$\Phi _n(t)\mid \Delta _K(t)$
and
$m\neq 0$
. By
$\Delta _K(t)/m=t^2-(2-\frac {1}{m})t+1=(t-\zeta _n)(t-\zeta _n^{-1})$
with
$\zeta _n=e^{2\pi \sqrt {-1}/n}$
, we must have
$\Delta _K(t)/m\in \{(t+1)^2,(t-1)^2,t^2+t+1,t^2+1,t^2- t+1\}$
. By
$2-\frac {1}{m}=2\cos \frac {2\pi }{n} \in {\mathbb {Z}}$
, we have
$m=1$
and
$n=6$
, namely,
$\Delta _K(t)=t^2-t+1=\Phi _6(t)$
.
Proof of Theorem 1.1
Since
$\Delta _{J(2k,2l)}(t)=klt^2+(1-2kl)t+kl$
, by putting
$m=kl$
, Theorems 3.1 and 6.1, together with Lemma 6.2, yield the assertion.
Remark 6.3 If n is even, then we have
$\mathrm { Res}(t^n-1,\Delta _k(t))=m^n(\alpha ^n+\beta ^n-2)=((-m\alpha )^n\ {+}\ (-m\beta )^n\ {-}\ 2m^n)\ {=}\ (a^n\ {-}\ b^n)^2\ {=}\ (a\ {-}\ b)^2F_n^{\,2}\ {=}\ (1\ {-}\ 4m)F_n^{\,2}\ {=}\ L_n^{\,2}\ {-}(2m^{n/2})^2$
. Thus,
$p\mid r_n$
does not lead to a similar condition of the Legendre symbol.
Example 6.4 If
$4m^2-m$
has a square factor, then the converse of Theorem 1.1 does not hold. For instance, if
$m=16$
and
$p=3$
, then
$4m^2-m=16\!\cdot \! 63=7\cdot 12^2$
, and
$(7/3)=(1/3)=1$
. So
$J(2,32)$
admits a liminal character over
${\mathbb {Z}}_3$
. In this case,
$3\nmid r_{2n-1}$
for all
$n\in {\mathbb {Z}}_{>0}$
.
Remark 6.5 The following are still a mystery.
(1) Does Theorem 1.1 extend to some wider class of knots? We may further verify by using Mathematica [37] and SageMath [Reference Stein31] that a similar assertion holds for genus two two-bridge knots:
-
•
$K=6_2$
with
$n< 500$
and
$p<10,000$
except for
$p=2,7,53;$
-
•
$K=6_3$
with
$n<500$
and
$p<2,000$
except for
$p=2,53,101.$
For instance, we find that
$K=6_{2}$
with
$\Delta _K(t)=-t^4+3t^3-3t^2+3t-1$
satisfies
$r_5=|\mathrm {Res}\,(t^5-1,\Delta _K(t))|=2^4$
, while the intersection of two varieties is empty in
${\mathbb {F}}_2^{\,2}$
. We expect that there is a correct formulation or interpretation in a more general setting.
(2) One may believe that the Burde–de Rham theory is in the background of Theorem 1.1. However, there is no obvious implication, and we rather expect that our result will shed new light around there.
(3) The Burde–de Rham theory has been extended to higher-dimensional representations by Heusener–Porti [Reference Heusener and Porti7]. We wonder if our study also extends to higher-dimensional cases. The study of
$\mathrm {SL}_3{\mathbb {C}}$
-character variety of the figure-eight knot [Reference Heusener, Muñoz and Porti6] might be a clue.
(4) As we asked in Question 4.1, it seems to be a subtle question to ask if there always exists a liminal
$\mathrm { SL}_2{\mathbb {Z}}_{p}$
-representation under the condition of Theorem 3.1.
(5) Prof. Ohshita recently pointed out that from a viewpoint of number theory, our work seems like a partial analog of the converse of the Herbrand–Ribet theorem, or an analog of Sharifi’s conjecture (cf.[Reference Fukaya and Kato5]) that suggests a relation between the divisibility of the class numbers by p and a certain congruence of certain cusp forms.
Acknowledgements
We are grateful to Léo Bénard, Tomoki Mihara, Tatsuya Ohshita, Shin-ichiro Seki, Adam Sikora, Motoo Tange, Yuji Terashima, Anh T. Tran, Yoshikazu Yamaguchi, and Hyuga Yoshizaki for their useful comments. Numerical calculations of examples in this article were partially enhanced by ChatGPT and Grok.
