Journal de Théorie des Nombres de Bordeaux

Takashi Fukuda

Takashi FUKUDA
Relative class numbers of subfields of the p-th cyclotomic field
Tome 34, no 1 (2022), p. 135-140.
https://doi.org/10.5802/jtnb.1195
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Journal de Théorie des Nombres
de Bordeaux 34 (2022), 135–140
Relative class numbers of subfields of the p-th
cyclotomic field
par Takashi FUKUDA
Résumé. Dans cet article, nous généralisons le résultat récent d’Ichimura sur
les nombres de classes relatifs des sous-corps du p-ième corps cyclotomique.
Abstract. We generalize the recent result of Ichimura concerning relative
class numbers of subfields of the p-th cyclotomic field.
1. Introduction
For a natural number n, we denote by ζn a primitive n-th root of unity
contained in the complex number field C. Let p be a prime number. It is
observed that the relative class number of the p-th cyclotomic field Q(ζp)
grows rapidly as p grows. How about the relative class numbers of subfields?
Let ` be an odd prime divisor of p−1 with the highest power `f dividing p−1
and q a divisor of p−1 prime to `. We fix q and denote byKt (0 ≤ t ≤ f) the
unique subfield of Q(ζp) with degree q`t. We are interested in the relative
class number h−t of Kt.
Several authors observed that the `-part of h−t (0 ≤ t ≤ f) behaves like in
Iwasawa’s class number formula. Recently, Ichimura [3] gave a confirmation
of this phenomenon. Namely, he proved the following which is an analogue
to Corollary 3 in [1].
Let `et be `-part of h−t and let ϕ denote the Euler function as usual.
Theorem 1.1 (Ichimura). Let p be a prime number with p ≡ 3 (mod 4)
and q = 2. Let `f be the highest power of an odd prime number ` dividing
p − 1. If es − es−1 < ϕ(`s) for some s with 1 ≤ s ≤ f , then es − es−1 =
et − et−1 for all t with s ≤ t ≤ f .
Ichimura also gave examples. For example, let (p, q, `) = (7860079, 2, 3).
Then e0 = 3, e1 = 7, e2 = 17, e3 = 25 and et = 8t+ 1 (2 ≤ t ≤ 8).
In this paper, we generalize Ichimura’s result and provide an another
proof for Ichimura’s theorem.
Manuscrit reçu le 20 août 2020, révisé le 30 octobre 2020, accepté le 13 octobre 2020.
Mathematics Subject Classification. 11R18, 11R23.
Mots-clefs. Cyclotomic field, class number.
136 Takashi Fukuda
Theorem 1.2. Let p be a prime number, ` an odd prime divisor of p − 1
with the highest power `f dividing p− 1 and q a divisor of p− 1 prime to `.
(1) If e0 = 0, then et = 0 for all t with 0 ≤ t ≤ f .
(2) If e0 > 0, then et−1 < et for all t with 1 ≤ t ≤ f .
(3) If es − es−1 < ϕ(`s) for some s with 1 ≤ s ≤ f , then es − es−1 =
et − et−1 for all t with s ≤ t ≤ f .
Let v2 denote the additive 2-adic valuation normalized by v2(2) = 1. If
v2(q) < v2(p − 1), then Kt (0 ≤ t ≤ f) is real and h−t = 1 (0 ≤ t ≤ f). So
the theorem has a meaning only when v2(q) = v2(p−1). For the proof of the
theorem, we recall a class number formula of an abelian number field. Let
F be an imaginary abelian number field of finite degree. Then the relative
class number h−(F ) of F is given by
h−(F ) = Q(F )w(F )
∏
K⊂F
NQ(ζd)/Q
(
−12B1,χ
)
,
where Q(F ) is the unit index of F , w(F ) is the number of roots of unity
contained in F and K runs through all imaginary cyclic subfields of F . For
each K, d = [K : Q] and χ is an arbitrary injective character of G(K/Q) to
C× (cf. [4, p. 294]). We also denote by χ the primitive Dirichlet character
induced by χ. Then the Bernoulli number B1,χ is given by
B1,χ =
1
m
m∑
a=1
aχ(a) ,
where m is the conductor of K.
Let pt = (1 − ζ`t) be the prime ideal of Q(ζ`t) lying above `. For an
integer x in Q(ζ`t), let ordpt x denote the maximal integer n such that pnt
divides (x). The following is a key lemma for our work.
Lemma 1.1. Let g(X) ∈ Z[X].
(1) If g(1) 6≡ 0 (mod `), then ordp1 g(ζ`) = 0.
(2) If g(1) ≡ 0 (mod `), then ordp1 g(ζ`) > 0.
(3) If ordpt g(ζ`t) > 0 for some t ≥ 1, then
ordpt+1 g(ζ`t+1) > 0 .
(4) If ordpt g(ζ`t) < ϕ(`t) for some t ≥ 1, then
ordpt g(ζ`t) = ordpt+1 g(ζ`t+1) .
Proof. We recall the well known fact ordpt ` = ϕ(`t) and write
g(X) =
∑
i
ai(X − 1)i
with ai ∈ Z. Then (1) is an immediate consequence of a0 = g(1) 6≡ 0
(mod `). Also (2) and (3) are immediate consequences of a0 ≡ 0 (mod `).
relative class numbers 137
In order to prove (4), we assume r = ordpt g(ζ`t) < ϕ(`t). If ai ≡ 0 (mod `)
for all i with 0 ≤ i ≤ r, then we have pr+1t | g(ζ`t) which contradicts the
assumption. Hence there exists some i ≤ r such that ai 6≡ 0 (mod `). We
take i to be minimal. If 0 ≤ i < r, then we have ordpt g(ζ`t) = i which
also contradicts the assumption. Hence we have i = r which immediately
implies the conclusion. 
2. Proof of Theorem 1.2
Let q0 be the highest power of 2 dividing p − 1. First we prove the
theorem when q = q0. Let | · |` denote the multiplicative `-adic valuation
normalized by |`|` = 1/`. We show that es − es−1 < ϕ(`s) with 1 ≤ s < f
implies es − es−1 = es+1 − es. Let χ be an arbitrary injective character of
G(Ks+1/Q). Then, the conductor of χ is p and
B1,χ =
1
p
∑
1≤a≤p
aχ(a) .
Since Ki (0 ≤ i ≤ s+ 1) are all the imaginary cyclic subfields of Ks+1, we
have
`es+1−es =
∣∣∣NQ(ζ`s+1 )/Q ◦NQ(ζq`s+1 )/Q(ζ`s+1 )(B1,χ) ∣∣∣−1`
and
es+1 − es = ordps+1 NQ(ζq`s+1 )/Q(ζ`s+1 )
 ∑
1≤a≤p
aχ(a)
 .
If we fix ζq and ζ`s+1 , then there exist two functions ρ1, ρ2 : N→ N∪{0}
satisfying
χ(a) = ζρ1(a)q ζ
ρ2(a)
`s+1 .
We define g(X) ∈ Q(ζq)[X] by
g(X) =
∏
σ∈G(Q(ζq)/Q)
 ∑
1≤a≤p
aζρ1(a)σq X
ρ2(a)
 .
Then g(X)∈Z[X] because g(X) is invariant under the action ofG(Q(ζq)/Q).
Now we have
NQ(ζq`s+1 )/Q(ζ`s+1 )
 ∑
1≤a≤p
aχ(a)
 = g(ζ`s+1)
and hence
es+1 − es = ordps+1 g(ζ`s+1) .
138 Takashi Fukuda
Let n be an integer satisfying n ≡ 1 (mod q) and n ≡ ` (mod `s+1).
Then ψ = χn is an injective character of G(Ks/Q) and
es − es−1 = ordps NQ(ζq`s )/Q(ζ`s )
 ∑
1≤a≤p
aψ(a)

= ordps g(ζ`s) .
Then Lemma 1.1(4) yields es − es−1 = es+1 − es. Hence we can prove (3)
inductively. Next we consider (1) and (2). Let χ be an arbitrary injective
character of G(K1/Q) and ψ = χn with n satisfying n ≡ 1 (mod q) and
n ≡ 0 (mod `). Then,
e1 − e0 = ordp1 NQ(ζq`)/Q(ζ`)
 ∑
1≤a≤p
aχ(a)

= ordp1 g(ζ`)
with g(X) ∈ Z[X] defined similarly as above and
|g(1)|−1` =
∣∣∣∣∣∣NQ(ζq`)/Q(ζ`)
 ∑
1≤a≤p
aψ(a)
∣∣∣∣∣∣
−1
`
=
∣∣∣∣∣∣NQ(ζq)/Q
 ∑
1≤a≤p
aψ(a)
∣∣∣∣∣∣
−1
`
=
∣∣∣NQ(ζq)/Q(B1,ψ)∣∣∣−1
`
= `e0
because K0 is the only imaginary cyclic subfield of K0. Assume e0 = 0.
Then we have e1 − e0 = 0 by Lemma 1.1(1) and et − et−1 = 0 (1 ≤ t ≤ f)
by Theorem 1.2(3). We note that this is also an immediate consequence
of [6, Theorem 10.4(a)]. Assume e0 > 0. Then we have e1 − e0 > 0 by
Lemma 1.1(2) and et − et−1 > 0 (1 ≤ t ≤ f) by Lemma 1.1(3).
Now we return to a general q with q0 | q. In this case, q is a product of
q0 and q1, where q1 is an odd divisor of p − 1 prime to `. If σ(n) denotes
the number of divisors of an integer n, then Kt has σ(q1)(t+ 1) number of
imaginary cyclic subfields. These subfields are determined by the degree.
Namely, if we denote by Q(p,m), for a divisor m of p− 1, a unique subfield
of Q(ζp) with degree m, then all the imaginary cyclic subfields of Kt are
Q(p, q0d`i) (d | q1, 0 ≤ i ≤ t). For each d and i, there exists an injective
character χd,i of G(Q(p, q0d`i)/Q). Then we have
es+1 − es =
∑
d|q1
ordps+1 NQ(ζq0d`s+1 )/Q(ζ`s+1 )
 ∑
1≤a≤p
aχd,s+1(a)

relative class numbers 139
and
es − es−1 =
∑
d|q1
ordps NQ(ζq0d`s )/Q(ζ`s )
 ∑
1≤a≤p
aχd,s(a)

because we may assume χd,s = χnd,s+1 with n ∈ N satisfying n ≡ 1
(mod q0d) and n ≡ ` (mod `s+1). Now, es − es−1 < ϕ(`s) implies
ordps NQ(ζq0d`s )/Q(ζ`s )
 ∑
1≤a≤p
aχd,s(a)
 < ϕ(`s)
for each d. Hence we can follow the argument for q = q0. 
3. Examples
We show some examples having large f . These were calculated by
PARI/GP [5]. All the examples behave like Theorem 1.2.
(p, q, `, f) e0 e1 e2 e3 e4 e5 e6 e7 e8 e9
(1194103, 26, 3, 8) 3 6 9 12 15 18 21 24 27
(1268623, 2, 3, 4) 1 3 9 16 23
(1924561, 80, 3, 7) 4 8 12 16 20 24 28 32
(3123037, 4, 3, 8) 2 6 14 22 30 38 46 54 62
(5826169, 8, 3, 9) 2 4 6 8 10 12 14 16 18 20
(7295833, 8, 3, 8) 4 8 12 16 20 24 28 32 36
(7715737, 8, 3, 9) 4 10 16 22 28 34 40 46 52 58
(9723403, 2, 3, 9) 1 3 5 7 9 11 13 15 17 19
(9723403, 26, 3, 9) 4 9 14 19 24 29 34 39 44 49
(1637501, 4, 5, 5) 2 5 8 11 14 17
(3797501, 4, 5, 4) 5 10 15 20 25
(8365001, 8, 5, 4) 6 10 14 18 22
(1190897, 16, 7, 4) 8 12 16 20 24
(1336337, 16, 17, 4) 1 3 5 7 9
Finally, in connection with the Corollary in [2, p. 237], we would like
to make some comments about further investigations on h−t . The condition
es−es−1 < ϕ(`s) in Theorem 1.2 (3) can not be replaced by es−es−1 ≤ ϕ(`s)
because the case p = 1268623 gives a counterexample. On the other hand,
there are many examples satisfying es−es−1 = ϕ(`s) and es−es−1 = es+1−
es, for example (p, `, s) = (5826169, 3, 1), (7715737, 3, 2), (9723403, 3, 1),
(8365001, 5, 1). It may be interesting to consider under which conditions,
es − es−1 = ϕ(`s) implies es − es−1 = es+1 − es.
Acknowledgments. The author would like to express his gratitude for
the anonymous referee who read the manuscript very carefully and gave
advice how to clarify some ambiguous expressions.
140 Takashi Fukuda
References
[1] R. Gold, “Examples of Iwasawa invariants”, Acta Arith. 26 (1974), p. 21-32.
[2] ———, “Examples of Iwasawa invariants. II”, Acta Arith. 26 (1975), p. 233-240.
[3] H. Ichimura, “Relative class numbers inside the pth cyclotomic field”, Osaka J. Math. 57
(2020), no. 4, p. 949-959.
[4] S. Louboutin, “Computation of relative class numbers of imaginary abelian number fields”,
Exp. Math. 7 (1998), no. 4, p. 293-303.
[5] The PARI Group, “PARI/GP version 2.12.1”, 2019, available from http://pari.math.
u-bordeaux.fr/.
[6] L. C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Graduate Texts in Mathe-
matics, vol. 83, Springer, 1997.
Takashi Fukuda
Department of Mathematics
College of Industrial Technology
Nihon University
2-11-1 Shin-ei, Narashino, Chiba, Japan
E-mail: fukuda.takashi@nihon-u.ac.jp


Relative class numbers of subfields of the $p$-th cyclotomic field

Manfred G. Madritsch

Jörg M. Thuswaldner

The level of distribution of the sum-of-digits function of linear recurrence number systems

Matthew Bisatt

Matthew BISATT
Root number of the Jacobian of y2 = xp + a
Tome 34, no 2 (2022), p. 575-582.
https://doi.org/10.5802/jtnb.1217
© Les auteurs, 2022.
Cet article est mis à disposition selon les termes de la licence
CREATIVE COMMONS ATTRIBUTION – PAS DE MODIFICATION 4.0 FRANCE.
http://creativecommons.org/licenses/by-nd/4.0/fr/
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e-ISSN : 2118-8572
Journal de Théorie des Nombres
de Bordeaux 34 (2022), 575–582
Root number of the Jacobian of y2 = xp + a
par Matthew BISATT
Résumé. Soit C/Q une courbe hyperelliptique donnée par un modèle affine
de la forme y2 = xp + a. Nous déterminons le signe de la Jacobienne de C, en
particulier nous nous concentrons sur le signe local en p, où C est sauvagement
ramifiée.
Abstract. Let C/Q be a hyperelliptic curve with an affine model of the
form y2 = xp + a. We explicitly determine the root number of the Jacobian
of C, with particular focus on the local root number at p where C has wild
ramification.
1. Global case
Let A/Q be an abelian variety. One of the consequences of the famed
Birch and Swinnerton-Dyer conjecture is that the parity of the rank of A/Q
should be controlled by the expected sign in its functional equation. The
expected sign is known as the (global) root number of A and denoted by
W (A/Q); the parity conjecture states that W (A/Q) = (−1)rankA(Q).
Root numbers of elliptic curves are completely understood, but the same
cannot be said in higher dimensions. Whilst root numbers only conjec-
turally control the rank, current computer algebra packages are often un-
able to compute either the L-function or rank explicitly in high dimension;
when these computations are coupled with the root number however, one
is more likely to be able to predict the rank exactly. For example, a root
number calculation may predict that the Jacobian of a curve has rank 0,
and thus make the curve more amenable to Chabauty-type arguments since
one generally needs to know explicit generators of a finite index subgroup
to apply them.
The author has previously described the root number when A has ev-
erywhere tame reduction [2], with an explicit description when A arises as
the Jacobian of a hyperelliptic curve. The purpose of this note is to com-
pute the root number of the Jacobian of a hyperelliptic curve of the form
y2 = xp + a when p is an odd prime. We concentrate on the case when
a is not a p-th power in Qp although this restriction is not necessary (see
Manuscrit reçu le 19 février 2021, révisé le 13 août 2021, accepté le 29 octobre 2021.
2020 Mathematics Subject Classification. 11G20, 11G40.
Mots-clefs. Root numbers, hyperelliptic curves.
This work was funded by the Heilbronn Institute for Mathematical Research.
576 Matthew Bisatt
Remark 1.3). The main theoretical contributions of this paper are that we
extend a result of Kobayashi’s to deal with the wild ramification at p and
also give an approach to treat the reduction at 2 where one cannot use the
theory of cluster pictures of hyperelliptic curves to determine the Galois
representation.
We will not list the various properties of root numbers that we use here,
but instead refer the reader to [2, §2] for an overview or [11, §3] for more
details.
For a prime p, we write vp for the p-adic valuation,
(∗
p
)
for the Legendre
symbol and (−,−)Qp for the Hilbert symbol at p.
Theorem 1.1. Fix 0 6= a ∈ Z and an odd prime p. Let Ca/Q be the
hyperelliptic curve y2 = xp + a and let Ja be its Jacobian. Assume that a is
not a p-th power in Qp and let a′ = 2−v2(a)a.
(1) Let W2 =

(2
p
)
if v2(a) is even, p - v2(4a), and a′ ≡ 1 mod 4,(−1
p
)
if a′ ≡ 3 mod 4,
1 else.
(2) Assume there exists a model of Ca of the form y2 = g(x), where
p - vp(g(0)). Let a0 = g(0) and let ∆g be the discriminant of g.
Define
Wp = −
(−2
p
)
(a0(p− 1)vp(a0),∆g)Qp
(−1
p
) 1+vp(∆g)
2
.
Write
S1 =
{
primes ` | a : ` 6∈ {2, p}, v`(a) odd
}
,
S2 =
{
primes ` | a : ` 6∈ {2, p}, p - v`(a), v`(a) even
}
.
Then the global root number of Ja is
W (Ja/Q) = W2Wp
(−1
p
) ∏
`∈S1
(−1
`
) p−1
2 ∏
`∈S2
(
`
p
)
.
Remark 1.2. As in the elliptic curve setting, a suitable model for Ca
always exists; see [7, Proposition 5.1b] for the genus two case or [8, Propo-
sition III.1.2] for the case under consideration here. In practice, it appears
that taking g(x) = f(x + r) for some r ≡ −a mod p suffices to obtain a
suitable model.
Remark 1.3. The assumption that a is not a p-th power in Qp is only used
to compute the local root number Wp = W (Ja/Qp). If a is a p-th power in
Qp then one can still compute that the inertia representation at p is tamely
ramified and isomorphic to γ ⊗ C[Cp−1], where γ is a faithful character of
C2(p−1), and then deduce the root number by [1, §11, §13]. We choose to
omit the result here in the interest of brevity.
Root number of y2 = xp + a 577
Proof. Note that the discriminant of Ca is (−1)
p−1
2 22(p−1)ppap−1 and hence
W (Ja/Q) = W (Ja/R)W (Ja/Q2)W (Ja/Qp)
∏
`|a, ` 6=2,p
W (Ja/Q`).
Moreover, one has that W (Ja/R) =
(−1
p
)
(see for example [9, Lemma 2.1]).
Now suppose that ` 6= p is an odd prime. We will use the theory of clusters
to compute the corresponding inertia representation and refer the reader
to the user’s guide [1] for the relevant definitions and theorem statements.
The set of roots is R = {ζjp p
√
−a | 0 ≤ j ≤ p − 1}, for ζp a primitive
p-th root of unity; this is the only proper cluster of Ca and has depth v`(a)p .
Moreover, inertia acts on R through a Cp-quotient and the action is trivial
if and only if p | v`(a).
If p - v`(a), then H1ét(Ja/Q`,Q2) decomposes as an inertia representation
into p − 1 characters of order either 2p if v`(a) is odd and order p other-
wise. The corresponding local root number is then either
(−1
`
) p−1
2 or
(
`
p
)
respectively by [1, Theorem 13.4]. On the other hand, if p | v`(a), then
H1ét(Ja/Q`,Q2) decomposes into p − 1 characters of order either 2 if v`(a)
is odd and order 1 otherwise; in this case the local root number is then(−1
`
) p−1
2 or 1 respectively.
Next we compute the local root number at 2, for which we can no longer
apply the cluster machinery. First suppose that v2(a) = 2k is even and
a′ = a2−2k ≡ 1 mod 4. Using the substitution (x, y) 7→ (x, y+2k), we change
our model for Ca/Q2 to y2 +2k+1y = xp+4k(a′−1). Let π be a fixed choice
of p
√
2 in some algebraic closure of Q2 and let L = Q2(π2(k+1)). Consider the
base change CL of Ca to L with the substitution x = π2(k+1)X, y = 2k+1Y ;
this gives us a model for CL as Y 2+Y = Xp+ a
′−1
4 which has good reduction
over L. Now L = Q2 if and only if p | (k + 1); in this case the root number
is 1. If p - (k + 1), then L = Q2(π) and using the lift-act-reduce procedure
of [5, Thereom 1.5] (this example is analogous to Example 1.9 of loc. cit.)
one can moreover see that the inertia representation is C[Cp]	1 and hence
the local root number is W (Ja/Q2) =
(2
p
)
.
The remaining cases at 2 are quadratic twists of the one above so we will
make use of the well-known formula
W (A/Q`)W (AD/Q`) = W (A/Q`(
√
D))χD(−1)dimA,
where A/Q` is an abelian variety, AD its quadratic twist by D, and χD is
the quadratic character of Gal(Q`(
√
D)/Q`).
Note that if ã ≡ 1 mod 4, then Ca is a quadratic twist of a curve of the
form C22kã for some D ∈ {−1, 2,−2}; if a′ = 2−v2(a)a, then the required
quadratic twist is given by the squarefree part of (−1)
a′−1
2 2v2(a).
578 Matthew Bisatt
Since C22kã obtains good reduction over L which has ramification degree
dividing p and Q2(
√
D)/Q2 is ramified of degree coprime to p, one has
that W (J22kã/Q2) = W (J22kã/Q2(
√
D)) (note it does not matter whether
L = Q2 here). Hence W (Jac(CD22kã)/Q2) = χD(−1)
p−1
2 . One can compute
via class field theory that χ−1(−1) = χ−2(−1) = −1 and χ2(−1) = 1, and
hence the local root number W2 = W (Ja/Q2) follows.
Now let ` = p and consider the model change to y2 = g(x). Note that the
two torsion field of Ja is independent of the choice of model and equal to the
splitting fields of f and g. Since a is not a p-th power in Qp, g satisfies the
hypotheses of Theorem 2.1 and the computation of the local root number
Wp = W (Ja/Qp) follows. 
Remark 1.4. When ` = 2, one could also obtain the inertia representation
in the tame case via [4, §1.3].
2. Local case at p
We now compute the local root number at p. Our method closely fol-
lows that of [6], adapting for higher genus. We first collate the bulk of the
notation that we will use throughout this section for ease of reading.
Notation. For a finite extension F/Qp, we denote
OF the valuation ring of F ,
vF the normalised valuation of F so that vF (F×) = Z,
FF the residue field of F ,
qF = |FF | the cardinality of the residue field of F ,
F un the maximal unramified extension of F ,
W(F ′/F ) the Weil group of an extension F ′/F ,
TrF ′/F the trace map for a finite extension F ′/F .
For an irreducible squarefree polynomial f ∈ K[x] of degree p, we also
define:
Cf the hyperelliptic curve y2 = f(x),
ρJ the `-adic representation of Jac(Cf ) for ` 6= p,
α a root of f ,
∆f the discriminant of f ,
M the splitting field of f ,
σ a fixed element of order p in the Galois group Gal(M/K),
N = M(
√
α− σα).
For nonzero complex numbers a, b, we write a ≈ b if ab−1 ∈ C is in
the multiplicative group generated by the positive real numbers and the
p-power roots of unity. We are now in a position to state our theorem.
Root number of y2 = xp + a 579
Theorem 2.1. Let p be an odd prime, K/Qp a finite extension and let
f ∈ K[x] be a squarefree polynomial of degree p. Suppose that:
(1) f is irreducible;
(2) the valuation of the discriminant of f , vK(∆f ), is coprime to p−1;
(3) the splitting field M of f has degree p(p− 1);
(4) p - vK(a0) where a0 = f(0).
Then, for the hyperelliptic curve Cf : y2 = f(x), the local root number is
W (Jac(Cf )/K) = −
(−2
qK
)
(a0(p− 1)vK(a0),∆f )K
(−1
qK
) 1+vK (∆f )
2
.
We will implicitly assume these hypotheses from now on in order to prove
this theorem through a series of lemmas.
Proposition 2.2.
(1) Let σ be an element of order p of Gal(M/K) ∼= Cp o Cp−1 and let
N = M(
√
α− σα). Then ρJ is a faithful representation of the Weil
group W(Nun/K).
(2) Let H ⊂ M be such that [H : K] = p − 1. Then ρJ = IndH/K χ is
the induction of a character χ from W(Nun/H);
(3) The conductor exponent of χ, a(χ), is even.
Proof. (1)–(2) follow from Proposition 2.2 and §5.1 of [3].
Let’s now prove (3). Note that a(ρJ) = vK(∆K(α)/K), where ∆K(α)/K
is the relative discriminant of the valuation ring of K(α), by [3, §6]; we
claim this is odd. Since the discriminant of the lattice OK [α] is equal to
∆f and has square index inside the valuation ring, vK(∆K(α)/K) ≡ vK(∆f )
mod 2 from which the claim follows. By the conductor exponent of induced
representations formula [10, p. 101], a(χ) is even. 
We now fix an additive character ψ of K with conductor −1 such that
ψ(x) = χ(σ)−TrFK /Fp (x) whenever x ∈ OK , where TrFK/Fp is the trace map
from FK to Fp and x is the image of x in FK . We further let ψH be the
composition of ψ with the trace map TrH/K , also of conductor −1. We also
choose an arithmetic Frobenius element Frob ofW(Nun/N) ⊂ W(Nun/K)
and extend σ to W(Nun/K) by imposing that it acts trivially on Kun and√
α− σα.
Lemma 2.3. Let signH/K be the nontrivial quadratic character of
Gal(H/K). Then
W (ρJ , ψ) =
(−2
qK
)
W (signH/K , ψH)W (χ, ψH).
Proof. For F ⊂ N , let 1F denote the trivial character of W(Nun/F ).
Since root numbers are inductive in degree 0, one has W (IndH/K χ, ψ) =
W (IndH/K 1H , ψ)W (χ, ψH).
580 Matthew Bisatt
We now compute W (IndH/K 1H , ψ). Let µ be a faithful character of
W(Nun/K)/W(Nun/H) ∼= Gal(H/K) ∼= Cp−1. Then
IndH/K 1H = 1K ⊕ signH/K ⊕
⊕
16j< p−12
(µj ⊕ µ−j),
and hence
W (IndH/K 1, ψ) = W (1K , ψ)W (signH/K , ψ)
∏
16j< p−12
µj(−1).
Now W (1K , ψ) = 1 and µj(−1) = 1 ⇔ qK ≡ 1 mod 2 p−1gcd(j,p−1) by [2,
Lemma 3.8]. If qK is an even power of p, then this is always satisfied since
2(p − 1) divides (qK − 1). On the other hand, if qK is an odd power of p,
then this congruence is satisfied if and only if j is even so it suffices to count
the number of odd integers in the range 1 6 j < p−12 ; the parity of this is
precisely given by the Legendre symbol
(−2
p
)
. 
Remark 2.4. One of our running assumptions is that the splitting field
of f has degree p(p − 1) over K; the preceding lemma is the only place
we made use of this fact since this implies that H/K is a cyclic Galois
extension.
We briefly generalise Propositions 5.7 and 5.13 of [6] now.
Lemma 2.5. Let C/Fq : y2 = xp − x and let Frob be the Frobenius endo-
morphism of C given by (x, y) 7→ (xq, yq) and let ρ be the endomorphism
(x, y) 7→ (x + 1, y). By Albanese functoriality, these descend to endomor-
phisms of the Jacobian J of C which we again denote by Frob and ρ. Then
in the endomorphism ring of J , one has
Frob = −
∑
u∈F×q
(
u
q
)
ρTrFq/Fp (u).
Proof. By the Hasse–Davenport Theorem, it suffices to prove this when
q = p. Let ∞ denote the point at infinity of C and embed ι : C ↪→ J via
P 7→ [(P )− (∞)]. Let P = (x, y); it suffices to prove this equality for ι(P )
since J is generated by the image of ι. Using the hyperelliptic involution,
we haveFrob + ∑
u∈F×p
(
u
p
)
ρu
 (ι(P )) = (xp, yp) + ∑
u∈F×p
(x+ u,
(
u
p
)
y)− p(∞).
One can check that this is the divisor of Y − y(X − x)
p−1
2 and hence is
trivial on J , which proves the equality. 
Root number of y2 = xp + a 581
Proposition 2.6. As operators of H1ét(Cf/K,Q`), we have an equality
Frob = −
∑
u∈F×K
(
u
qK
)
σ−TrFK /Fp (u).
Proof. Note that over N , the model of Cf determined by the substitution
(x, y) 7→ ((α − σα)x + α,
√
α− σαpy) reduces to y2 = xp − x (see [3,
Lemma 4.1]). Moreover, one can show via the lift-act-reduce method of [5,
Theorem 1.5], that Frob and σ−1 act on H1ét(Cf/FK ,Q`) as (x, y) 7→
(xqK , yqK ) and (x, y) 7→ (x+1, y) respectively (equivalently use the function
field approach of [6, Proposition 5.12]). The Galois module isomorphism
of [5, Corollary 1.6] then transfers the operator equality of Lemma 2.5 to
H1ét(Cf/K,Q`). 
Lemma 2.7. Let G be the Gauss sum
∑
u∈F×K
(
u
qK
)
χ(σ)−TrFK /Fp (u). Then
W (χ, ψH) ≈ (a0vM (α),∆f )K(−G)vK(∆f ).
Proof. First note that W (χ, ψH) ≈ χ
(−NormM/H(α−σα)
a0vM (α)p
)
by the proofs of
Propositions 4.2 and 5.6 of [6]. The remainder is largely [6, Proposition 5.8].
Note χ|K×= signH/K det ρJ . Since det ρJ is an integer power of the cy-
clotomic character and signH/K is the Hilbert symbol (−,∆f )K , we have
χ(a0vM (α)p) ≈ (a0vM (α)p,∆f )K = (a0vM (α),∆f )K .
Let β = −NormM/H(α−σα). We now claim that vH(β) = vK(∆f ). Since
Cf has potentially good reduction, vM (σiα−σjα) = vM (α−σα) whenever
σi 6= σj by [1, Theorem 5.3]. Hence vM (∆f ) = p(p − 1)vM (α − σα) =
p(p − 1)vH(NormM/H(α − σα)). The claim follows since M/K is totally
ramified of degree p(p− 1).
Observe that β = NormN/H(
√
α− σα) is a norm from N and so by
Proposition 2.6 with the proof of [6, Proposition 5.8], χ(β) = χ(FrobvH(β)) =
(−G)vK(∆f ). 
Finally, we can amalgamate these results to prove Theorem 2.1.
Proof of Theorem 2.1. By direct computation, W (signH/K , ψ) ≈ G. More-
over, W (ρJ , ψ) ≈ −
(−2
qK
)
(a0vM (α),∆f )K(−G)1+vK(∆f ) by Lemmas 2.3
and 2.7, and vM (α) = (p − 1)vK(a0). Since vK(∆f ) is odd and G2 is real
with sign
(−1
qK
)
we get the result up to ≈ a priori; the equality follows since
both sides are ±1. 
Acknowledgments. The author thanks Tim Dokchitser for his suggestion
when ` = 2, Nirvana Coppola for answering questions on her paper and the
referees for their comments.
582 Matthew Bisatt
References
[1] A. J. Best, L. A. Betts, M. Bisatt, R. van Bommel, V. Dokchitser, O. Faraggi,
S. Kunzweiler, C. Maistret, A. Morgan, S. Muselli & S. Nowell, “A user’s guide
to the local arithmetic of hyperelliptic curves”, Bull. Lond. Math. Soc. 54 (2022), no. 3,
p. 825-867.
[2] M. Bisatt, “Explicit root numbers of abelian varieties”, Trans. Am. Math. Soc. 372 (2019),
no. 11, p. 7889-7920.
[3] N. Coppola, “Wild Galois representations: a family of hyperelliptic curves with large inertia
image”, https://arxiv.org/abs/2001.08287, to appear in Math. Proc. Camb. Philos. Soc.,
2022.
[4] T. Dokchitser, “Models of curves over discrete valuation rings”, Duke Math. J. 170 (2021),
no. 11, p. 2519-2574.
[5] T. Dokchitser, V. Dokchitser & A. Morgan, “Tate module and bad reduction”, Proc.
Am. Math. Soc. 149 (2021), no. 4, p. 1361-1372.
[6] S.-i. Kobayashi, “The local root number of elliptic curves with wild ramification”, Math.
Ann. 323 (2002), no. 3, p. 609-623.
[7] Q. Liu, “Modèles minimaux des courbes de genre deux”, J. Reine Angew. Math. 453 (1994),
p. 137-164.
[8] L. Melninkas, “On local root numbers of abelian varieties”, PhD Thesis, Université de
Strasbourg (France), 2021, https://hal.archives-ouvertes.fr/tel-03258699v2.
[9] M. Sabitova, “Root numbers of abelian varieties”, Trans. Am. Math. Soc. 359 (2007),
no. 9, p. 4259-4284.
[10] J.-P. Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer, 1979.
[11] J. Tate, “Number theoretic background”, in Automorphic Forms, Representations and L-
Functions, Proceedings of Symposia in Pure Mathematics, vol. 33-2, American Mathematical
Society, 1979, p. 3-26.
Matthew Bisatt
Fry Building
University of Bristol
Bristol, BS8 1UG, UK
E-mail: matthew.bisatt@bristol.ac.uk
URL: https://sites.google.com/view/mbisatt/


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