Journal de Théorie des Nombres de Bordeaux

S. Mohammad Hadi Hedayatzadeh

Exterior Powers of Lubin-Tate Groups

Amanda Knecht

Degree of Unirationality for del Pezzo Surfaces over Finite Fields

Reese Scott

Robert Styer

Bennett’s Pillai theorem with fractional bases and negative exponents allowed

Mirela Ciperiani

Jakob Stix

Mirela CIPERIANI et Jakob STIX
Galois sections for abelian varieties over number fields
Tome 27, no 1 (2015), p. 47-52.
<http://jtnb.cedram.org/item?id=JTNB_2015__27_1_47_0>
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Journal de Théorie des Nombres
de Bordeaux 27 (2015), 47–52
Galois sections for abelian varieties
over number fields
par Mirela CIPERIANI et Jakob STIX
Résumé. Pour une variété abélienne A sur un corps de nombres
k, on considère l’espace des sections de l’extension π1(A/k) in-
duite par le groupe fondamental. En étudiant le sous-groupe divi-
sible maximal de H1(k,A), on montre que l’espace des sections de
π1(A/k) contient un sous-groupe isomorphe à Ẑ[k:Q]·dim(A) et n’est
jamais en bijection avec A(k). C’est essentiellement un résultat de
structure pour H1(k,T`(A)).
Abstract. For an abelian variety A over a number field k, we
discuss the space of sections of its fundamental group extension
π1(A/k). By analyzing the maximal divisible subgroup of H1(k,A)
we show that the space of sections of π1(A/k) contains a copy of
Ẑ[k:Q]·dim(A) and is never in bijection with A(k). This is essentially
a result about the structure of H1(k,T`(A)).
1. Galois sections for abelian varieties
1.1. Galois sections. The étale fundamental group π1(X, x̄) of a geomet-
rically connected variety pr : X → Spec(k) forms an extension, see [4] IX
Theorem 6.1,
(1.1) 1→ π1(Xk̄, x̄)→ π1(X, x̄)
pr∗−−→ Galk → 1,
where Galk = π1(Spec(k),Spec(k̄)) is the absolute Galois group of k with
respect to an algebraic closure k̄/k compatible with the geometric base
point x̄ ∈ Xk̄. By functoriality of π1 for pointed spaces, the rational point
a ∈ X(k) gives rise to a continuous homomorphism
sa : Galk → π1(X, ā)
γ(−)γ−1−−−−−→ π1(X, x̄),
where ā is a geometric point of Xk̄ above a, and γ is an étale path on Xk̄
from ā to x̄. We denote the set of π1(Xk̄, x̄)-conjugacy classes of sections of
Manuscrit reçu le 22 août 2013, accepté le 14 janvier 2014.
The authors acknowledge the hospitality and support provided by MATCH.
The first author was partially supported by NSF Grant DMS-07-58362 and by NSA Grant
H98230-12-1-0208.
Mathematics Subject Classification. 11G10, 11S25.
48 Mirela Ciperiani, Jakob Stix
(1.1) by
Sπ1(X/k) = {s : Galk → π1(X, x̄) ; pr∗ ◦ s = idGalk} /π1(Xk̄,x̄)-conjugacy.
The well defined map a 7→ [sa] is called the profinite Kummer map.
By considering only conjugacy classes of maps we may from now on safely
ignore base points.
The section conjecture of anabelian geometry [2] predicts, in the case
of proper smooth curves over number fields of genus at least 2, that the
profinite Kummer map is a bijection. We refer to [6] for details of the above
construction and the section conjecture in general. The main result of this
note states that the profinite Kummer map is never surjective for abelian
varieties.
Theorem 1. For every abelian variety A/k over a number field k there
exist sections of
1→ π1(Ak̄)→ π1(A)→ Galk → 1
that do not come from a rational point of A. More precisely, the space of
sections Sπ1(A/k) has a group structure and is isomorphic to H
1(k,T(A))
which contains a copy of Ẑd for d = [k : Q] dim(A).
Remark 2. Some remarks are in order to put Theorem 1 in perspective.
(1) As we recall below in Section §1.2, the profinite Kummer map for
an abelian variety A(k)→ Sπ1(A/k) factors as a map to a profinite
group through the profinite completion A(k)⊗Ẑ. Therefore the first
part of Theorem 1 follows by comparing the cardinality of both sides
in case A(k) is an infinite group. So the key difficulty is to handle
abelian varieties of algebraic rank 0. The second statement on the
Galois cohomology of the Tate module of an abelian variety is of
independent interest.
Let us emphasize that, for example, considering elliptic curves
over the rational numbers Q, the case of algebraic rank 0 occurs
with positive probability by [1] Theorem 4, and conjecturally in
50% of all cases. So the case of finite Mordell–Weil group is not an
exceptional one.
(2) Knowing the prospects of the profinite Kummer map being a bijec-
tion for abelian varieties has some significance for attemps to prove
the section conjecture.
Although the section conjecture avoids a prediction for abelian
varieties, the abelianization of sections by composition with the Al-
banese mapX → AlbX may help to understand the case of arbitrary
smooth projective curves X, see [6] §3.1.
In fact, the known cases of the birational section conjecture for
curves over number fields use in a crucial way (after passing to
Galois sections for abelian varieties over number fields 49
a neighbourhood of the section) a map to an abelian variety of
algebraic rank 0 and argue that the section there corresponds to a
rational point.
(3) The negative outcome concerning the section conjecture for abelian
varieties may turn our attention also in the genuine section conjec-
ture towards spaces of sections with local conditions, see for example
[5]. If we impose a Selmer condition, i.e., we require that the locali-
sation at all places comes from a rational point, and if X(A/k) has
no nontrivial divisible subgroup, then the natural map
A(k)⊗ Ẑ→ S Selmerπ1(A/k)
to the space of conjugacy classes of such Selmer sections is in fact
bijective. In this regard, Theorem 1 may be seen as a statement to
shape our thinking about the section conjecture.
(4) Last but not least, if we consider a finite field Fq and an abelian
variety A/Fq, then results of Tate on Galois cohomology of such
abelian varieties show that in fact A(Fq) is in natural bijection
with the space of sections Sπ1(A/Fq). In some sense, surprisingly,
the section conjecture holds for abelian varieties over finite fields!
This case also shows that unexpected things can happen in the
section conjecture.
1.2. The Kummer map for abelian varieties. Let k be a field of
characteristic 0. For an abelian variety A/k with Tate module
T(A) = Hom(Q/Z, A(k̄)) = lim←−
n
An(k̄),
where An denotes the n-torsion of A, we have a natural identification
Sπ1(A/k) = H
1(k,T(A)) = lim←−
n
H1(k,An)
under which the profinite Kummer map κ : A(k) → Sπ1(A/k) factors over
the limit of the maps
δn : A(k)→ H1(k,An)
of the multiplication by n sequence for A, see [6] Corollary 71. The long
exact sequence of Galois cohomology leads to a diagram with an exact row
(1.2)
A(k)

κ // Sπ1(A/k)
0 // lim←−nA(k)/nA(k)
lim←− δn// H1(k,T(A)) // Hom
(
Q/Z,H1(k,A)
)
// 0.
We immediately deduce the following lemma.
Lemma 3. The profinite Kummer map for an abelian variety A/k is
50 Mirela Ciperiani, Jakob Stix
(1) injective if and only if
⋂
n≥1 nA(k) = 0, and
(2) surjective if and only if both Div(H1(k,A)) is trivial (the maximal
divisible subgroup of H1(k,A)), and the map
(1.3) A(k)→ lim←−
n
A(k)/nA(k)
is surjective.
Remark 4. If k/Q is a finitely generated field, then A(k) is a finitely
generated Z-module by the theorems of Mordell–Weil and of Lang–Néron.
The profinite Kummer map for A/k is in fact injective over such fields by
Lemma 3 (1). Moreover, if A(k) is not finite, then (1.3) is not surjective
and the profinite Kummer map fails to be surjective by Lemma 3 (2).
2. The maximal divisible subgroup of the Weil–Châtelet group
2.1. Rephrasing Theorem 1. From now on let k be a number field.
By Remark 4 above, the proof of the first part of Theorem 1 consists of
showing that Div(H1(k,A)) 6= 0 if A(k) is finite and thus follows from the
next theorem.
Theorem 5. Let k be a number field, and let A/k be an abelian variety.
(1) If A/k has algebraic rank r = rkZA(k) < d = dim(A) · [k : Q], then
Div(H1(k,A)) contains a copy of (Q/Z)d−r.
(2) The space of sections Sπ1(A/k) contains a copy of Ẑd.
2.2. Generalized Selmer groups. For a place v of k we denote the com-
pletion of k in v by kv. We shall be working with the following generalized
Selmer groups for n ∈ N or n =∞. The Selmer group free at p is defined
as
H1Selp(k,Apn) = ker
(
H1(k,Apn)→
∏
v-p
H1(kv, A)pn
)
and the Selmer group trivial at p is defined as
H1Selp(k,Apn) = ker
(
H1(k,Apn)→
∏
v-p
H1(kv, A)pn ×
∏
v|p
H1(kv, Apn)
)
.
Let At denote the dual abelian variety. The local conditions Selp for Apn
and Selp for Atpn are dual local conditions, see [3] Definition (8.7.8), and
the Euler characteristic formula of Wiles, see [3] Theorem (8.7.9), reads
(n ∈ N)
(2.1)
|H1Selp(k,Apn)|
|H1Selp(k,Atpn)|
= |Ap
n(k)|
|Atpn(k)|
·
∏
v
λv =
|Apn(k)|
|Atpn(k)|
· pnd ·
∏
v|p
|Atpn(kv)|
where the local factor λv is 1 for v - p∞, for v | ∞ it is
λv = p−n·[kv :R] dim(A),
Galois sections for abelian varieties over number fields 51
and for v | p by local Tate duality and Mattuck–Tate
λv =
|H1(kv, Apn)|
|Apn(kv)|
= |A(kv)/p
nA(kv)| · |H1(kv, A)pn |
|Apn(kv)|
= p2n·[kv :Qp] dim(A) · |Atpn(kv)|.
2.3. Proof of Theorem 5. We can focus on the p-primary part
Div(H1(k,A)p∞)
for a prime number p. Since H1(kv, A) has finite p-part if v - p, the group
Div(H1(k,A)p∞) is locally trivial outside v | p, in other words
(2.2) Div
(
H1(k,A)p∞
)
⊆ im
(
H1Selp(k,Ap∞)→ H
1(k,A)
)
.
The corank of a p-primary torsion group M = (Qp/Zp)m×M0 with finite
M0, is the well-defined number m = corankZp(M). We set
sp = corankZp(H1Selp(k,Ap∞)),
sp = corankZp(H1Selp(k,Atp∞)).
The snake lemma applied to the defining sequences together with the finite-
ness of Ap∞(k) and Ap∞(kv) shows that both natural maps
H1Selp(k,Apn)→ H
1
Selp(k,Ap∞)pn
H1Selp(k,Atpn)→ H1Selp(k,Atp∞)pn
have kernel and cokernel bounded independently of n. Therefore there is a
constant c > 0 independent of n such that
pnsp/c ≤ |H1Selp(k,Apn)| ≤ p
nsp · c, and
pns
p
/c ≤ |H1Selp(k,Atpn)| ≤ pns
p · c.
Since |Apn(k)|/|Atpn(k)| and
∏
v|p |Atpn(kv)| are bounded independently of
n, analyzing the asymptotic for n 0 in (2.1) we obtain
sp − sp = d.
The exact sequence from Kummer theory of A/k
0→ A(k)⊗Qp/Zp → H1Selp(k,Ap∞)→ im
(
H1Selp(k,Ap∞)→ H
1(k,A)
)
→ 0
splits, because A(k)⊗Qp/Zp is a divisible group, and it therefore remains
exact upon applying the functor Div(−). By (2.2) we find a split exact
sequence
0→ A(k)⊗Qp/Zp → Div
(
H1Selp(k,Ap∞)
)
→ Div
(
H1(k,A)p∞
)
→ 0
which allows to compute coranks as
corankZp
(
Div
(
H1(k,A)p∞)
))
= sp − r = d+ sp − r ≥ d− r
52 Mirela Ciperiani, Jakob Stix
This shows assertion (1) of Theorem 5. For assertion (2) we compute the
rank of Sπ1(A/k) via (1.2) as
rkẐ Sπ1(A/k) = rkẐ H
1(k,T(A))
= r + rkẐ Hom
(
Q/Z,Div(H1(k,A))
)
= d+ sp ≥ d.
References
[1] M. Bhargava, A. Shankar, Ternary cubic forms having bounded invariants, and the ex-
istence of a positive proportion of elliptic curves having rank 0, Annals of Math., 181, 2,
(2015), 587–621.
[2] A. Grothendieck, Brief an Faltings (27/06/1983), In Geometric Galois Action 1, ed.
L. Schneps, P. Lochak, LMS Lecture Notes, 242, Cambridge, (1997), 49–58.
[3] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number fields, Second edition,
Grundlehren der Mathematischen Wissenschaften, 323, Springer, 2008.
[4] A. Grothendieck, Séminaire de Géométrie Algébrique du Bois Marie (SGA 1) 1960-61:
Revêtements étales et groupe fondamental, Documents Mathématiques, 3, Société Mathé-
matique de France, (2003).
[5] J. Stix, On the birational section conjecture with local conditions, Inventiones mathemati-
cae, 199, 1, (2015), 239–265.
[6] J. Stix, Rational points and arithmetic of fundamental groups: Evidence for the section
conjecture, Springer Lecture Notes in Mathematics, 2054, Springer, Heidelberg, (2013).
Mirela Ciperiani
Department of Mathematics
The University of Texas at Austin
1 University Station
C1200, Austin, Texas 78712, USA
E-mail: mirela@math.utexas.edu
Jakob Stix
Institut für Mathematik
Goethe–Universität Frankfurt
Robert-Mayer-Straße 6–8, 60325 Frankfurt am Main, Germany
E-mail: stix@math.uni-frankfurt.de


Galois sections for abelian varieties over number fields

Michael A. Bennett

Amir Ghadermarzi

Extremal families of cubic Thue equations

Hiromichi Yanai

Hiromichi YANAI
On the index of degeneracy of a CM abelian variety
Tome 27, no 3 (2015), p. 815-820.
<http://jtnb.cedram.org/item?id=JTNB_2015__27_3_815_0>
© Société Arithmétique de Bordeaux, 2015, tous droits réservés.
L’accès aux articles de la revue « Journal de Théorie des Nom-
bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord
avec les conditions générales d’utilisation (http://jtnb.cedram.
org/legal/). Toute reproduction en tout ou partie de cet article sous
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tion pénale. Toute copie ou impression de ce fichier doit contenir la
présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Journal de Théorie des Nombres
de Bordeaux 27 (2015), 815–820
On the index of degeneracy of a CM abelian
variety
par Hiromichi YANAI
Résumé. Nous considérons une variété abélienne dégénérée A
de type CM. Alors il existe m > 0 tel que l’anneau des cycles
de Hodge sur Am n’est pas engendré par les classes de diviseurs.
Nous appelons le plus petit m vérifiant cette propriété l’indice de
dégénérescence de A.
Dans cet article, nous déterminons l’indice de dégénérescence
d’un certain type de variétés abéliennes de type CM. Cela com-
plète un résultat antérieur de H. W. Lenstra, Jr.
Abstract. We consider a degenerate abelian variety A of CM
type. Then there exists m > 0 such that the ring of Hodge cy-
cles on Am is not generated by the divisor classes. We call the
minimum of such m the index of degeneracy of A.
In this paper, we determine the index of degeneracy for a certain
type of CM abelian varieties. This supplements a former result of
H. W. Lenstra, Jr.
1. Introduction
For an abelian variety A defined over the complex number field C, we call
the elements ofH2p(A,Q)∩Hp,p the Hodge cycles (of codimension p), where
Hp,q = Hq(A,Ωp). It is conjectured that the Hodge cycles are algebraic (this
is the Hodge conjecture; for a survey see [1]). Since the divisor classes (the
Hodge cycles of codimension 1) are algebraic, we are interested in Hodge
cycles that are not generated by divisor classes.
We call such cycles exceptional (in [9] they are called sporadic). Ex-
ceptional Hodge cycles cause certain degeneration on various arithmetic
objects (cf. [10]). Moreover, constructing abelian varieties with exceptional
Hodge cycles is related to various combinatorial or group theoretic concepts
(cf. [4, 5]).
Let A be a CM abelian variety of dimension d. By definition, EndA⊗Q
contains a CM field K of degree 2d. Let S ⊂ Hom(K,C) be the CM type of
A; the representation of K on H0(A,Ω1) is isomorphic to
⊕
σ∈S σ. We say
Manuscrit reçu le 28 février 2014, révisé le 28 juin 2014, accepté le 28 juin 2014.
Mathematics Subject Classification. 11G15.
Mots-clefs. Abelian variety, CM-type, Hodge cycle.
816 Hiromichi Yanai
that A is of CM type (K,S). Let MT(A) be the Mumford-Tate group of
A (cf. [1]). MT(A) is an algebraic subtorus of ResK/QGm. The dimension
of MT(A) is called the rank of A (or the rank of the CM type (K,S) ) in
certain contexts (cf. [7]).
When A is simple, the next two conditions (i) and (ii) are equivalent
(cf. [1]).
(i) For each positive integer m, the ring of the Hodge cycles on Am
is generated by the divisor classes.
(ii) dim MT(A) = d+ 1.
When A satisfies (one of) these conditions, A is called stably nondegenerate
and the Hodge conjecture holds for every power of A (cf. [2]). If A is not
stably nondegenerate (we simply say A is degenerate) then there exists an
integer m > 0 such that Am holds an exceptional Hodge cycle. Following F.
Hazama [3], we call the minimum of such m the index of degeneracy of A.
From this point on, we assume that K is an abelian extension over Q
with the Galois group G = Gal(K/Q). Then dim MT(A) is equal to the
number of characters χ of G satisfying χ(S) =
∑
σ∈S χ(σ) 6= 0 (see [1] for
a reference). Note that χ(S) = 0 for every nontrivial even character χ and
χ0(S) = d 6= 0 for the trivial character χ0. Hence, in this case, the above
conditions (i) and (ii) are equivalent to the next (iii).
(iii) For each odd character χ of G, one has χ(S) 6= 0.
In this paper, we prove that if there exists an odd character χ satisfying
χ(S) = 0 and the order of χ is a power of 2, then the index of degeneracy
of A is equal to 1 (see Theorem 4.1).
Remark. In the CM case, one can provide an example of variety A having
index of degeneracy strictly bigger than 1, which was obtained by S. P.
White [9].
2. Abelian varieties of Weil type
Let A be a CM abelian variety of type (K,S). We assume that K is an
abelian extension overQ andK contains a proper sub CM field k.H denotes
the subgroup of G = Gal(K/Q) corresponding to k. Put r = #H = [K : k].
For τ ∈ G/H = Gal(k/Q), put
H1,0τ = {ω ∈ H1,0
∣∣ ∀a ∈ k, a(ω) = aτω},
nτ = dimH1,0τ = #{σ ∈ S
∣∣ σ|k = τ}.
If nτ does not depend on τ ∈ Gal(k/Q), we say that A is of Weil type. It
is easy to see that the condition is equivalent to nτ = nτρ = r2 for each τ ,
where ρ denotes the complex conjugation. For such A, χ(S) = 0 for the odd
characters χ of G which are trivial on H. If this is the case, A is degenerate
Index of degeneracy 817
and there exists an exceptional Hodge cycle of codimension r2 (cf. [6]). In
particular, the index of degeneracy of A is equal to 1.
3. Lenstra’s result
When a CM abelian variety A is degenerate, we want to determine the
index of degeneracy of A. H. W. Lenstra, Jr. gives an answer for some
cases. Here we recall his argument briefly. For more information, see [9].
(The expressions in [9] are slightly different from ours.)
Let A be a simple CM abelian variety of type (K,S). Let us assume that
K is an abelian extension over Q and that there exists an odd character χ
of G = Gal(K/Q) with χ(S) = 0; hence A is degenerate.
When χ is faithful, G is cyclic. We denote a generator of G by γ and the
order of γ by 2t. Put
h =
∏
p|t
(ε+ ργ
2t
p ),
where p runs over the odd primes dividing t and ε is the unit element of G.
The element h lies in the group ring Z[G]. We can see that the coefficients
in h are 1 or 0, hence h is naturally regarded as a subset of G. For each
σ ∈ G, we take a nonzero ωσ ∈ H1(A,C) such that a(ωσ) = aσωσ for each
a ∈ K. Such ωσ is unique up to a constant multiple; we have H1(A,C) =⊕
σ∈GCωσ.
Then the element ∧
σ∈h
ωσ
(one implicitly fixes an order for taking this wedge product) is an excep-
tional Hodge cycle on A. More precisely, it is an element of (H2q(A,Q) ∩
Hq,q) ⊗ C for some q and is not generated by divisor classes. This implies
that the index of degeneracy of A is equal to 1.
When χ is not faithful, we take the above h for G/Kerχ, then its pullback
to G works.
In the definition of h, we have used the odd prime factors of t. When t is
a power of 2, such a prime number does not exist and the above argument
doesn’t work.
4. The 2-power case
In this section, we consider the case where t is a power of 2. In this case,
the index of degeneracy of A is also 1, but the reason is different from that
of Lenstra’s case.
818 Hiromichi Yanai
Theorem 4.1. Let A be a CM abelian variety of dimension d as in
Section 3. Assume that there exists an odd character χ of G = Gal(K/Q)
with χ(S) = 0 and the order of χ (= 2t) is a power of 2. Then A is of Weil
type; the index of degeneracy of A is equal to 1.
Proof. Put t = 2s−1 with s > 0. Let H be the kernel of χ and k be the
subfield ofK corresponding toH. The quotient G/H = Gal(k/Q) is a cyclic
group of order 2t. Let us denote by γ (a representative of) a generator of
G/H. Then ζ = χ(γ) is a primitive 2s-th root of unity. Let G =
⋃2t−1
i=0 Hγ
i
be the coset decomposition of G by H, where Hγt is the coset of the
complex conjugation ρ. For each i, put ni = #(S ∩Hγi) = dimH1,0τ , where
τ ∈ Gal(k/Q) is corresponding to Hγi. Then ni +nt+i = #H = dt (0 ≤ i ≤
t− 1).
We have
0 =
∑
σ∈S
χ(σ) =
2t−1∑
i=0
niχ(γi) =
t−1∑
i=0
(ni − nt+i)ζi.
Since ζ is a primitive 2s-th root of unity, its degree over Q is 2s−1 = t.
This implies ni = nt+i (0 ≤ i ≤ t − 1). If H is trivial (i.e. k = K), then
ni = 0 or 1 for each i, but this is impossible because ni + nt+i = 1. So k
is a proper CM subfield of K and A is of Weil type with respect to k; the
index of degeneracy of A is equal to 1. 
5. Example: CM Jacobians
In this section, we give examples of abelian varieties satisfying the con-
dition in the previous section. These are CM Jacobian varieties treated
in [8]. For these abelian varieties, we can determine the dimensions of the
Mumford-Tate groups.
Let p > 5 be a prime number and ζp be a primitive p-th root of unity.
We denote the minimal polynomial of −(ζp + ζ−1p ) by g(x) ∈ Z[x]. Then
the Jacobian variety A of the algebraic curve y2 = x·g(x2 − 2) is a simple
CM abelian variety of dimension p−12 . Put K = Q(
√
−1, ζp + ζ−1p ) then
G = Gal(K/Q) ∼= Z/2Z×
(
(Z/pZ)× /{±1}
)
and EndA⊗Q ∼= K.
According to [8], the CM type S of A is:
when p ≡ 1 (mod 4), S = {(0, 1), (1, 2), (0, 3), . . . , (1, p− 12 )},
when p ≡ 3 (mod 4), S = {(0, 1), (1, 2), (0, 3), . . . , (0, p− 12 )}.
Let ψ be the nontrivial character of (Z/4Z)× ∼= Z/2Z and ξ0 be the
trivial character of (Z/pZ)×. Then the product ψ× ξ0 can be viewed as an
odd character of G; its order is 2.
Index of degeneracy 819
When p ≡ 1 (mod 4), one has (ψ×ξ0)(S) = 1−1+1−· · ·−1 = 0. Hence
by Theorem 4.1, the abelian variety A is of Weil type with k = Q(
√
−1)
and the index of degeneracy of A is equal to 1.
Moreover, representing an odd character χ of G by χ = ψ× ξ (where ξ is
an even character mod p), we can describe the value χ(S) by the generalized
Bernoulli numbers B1,χ = 14p
∑4p
a=1 χ(a)a. Here we are regarding χ as a
character mod 4p.
In fact, for ξ 6= ξ0, we can deduce:
when p ≡ 1 (mod 4),
χ(S) = ξ(1)− ξ(2) + ξ(3)− · · · − ξ(p− 12 ) = ξ̄(2)B1,χ,
when p ≡ 3 (mod 4),
χ(S) = ξ(1)− ξ(2) + ξ(3)− · · ·+ ξ(p− 12 ) = −ξ̄(2)B1,χ.
We know B1,χ 6= 0. When p ≡ 3 (mod 4), one has (ψ×ξ0)(S) 6= 0. Hence
the dimension of the Mumford-Tate group MT(A) of A is:
when p ≡ 1 (mod 4), dim MT(A) = dimA,
when p ≡ 3 (mod 4), dim MT(A) = dimA+ 1.
In particular, when p ≡ 3 (mod 4), A is stably nondegenerate; the Hodge
conjecture holds for every power of A.
Remark. When p ≡ 1 (mod 4) and when ξ(∗) =
(∗
p
)
(the quadratic
residue symbol), the above calculations imply the following (probably well
known) equalities concerning the class number hp of the imaginary qua-
dratic field Q(
√
−p):(1
p
)
−
(2
p
)
+
(3
p
)
− · · · −
(
p−1
2
p
)
= −
(2
p
)
hp,
(1
p
)
−
(2
p
)
−
(3
p
)
+
(4
p
)
+
(5
p
)
− · · ·+ (−1)
p−1
4
(
p−1
2
p
)
= hp.
References
[1] B. B. Gordon, “A Survey of the Hodge Conjecture for abelian Varieties”, in A survey of
the Hodge conjecture, CRM Monograph Series, vol. 10, American Mathematical Society,
Providence, RI, 1999, p. 297-356.
[2] F. Hazama, “Algebraic cycles on certain abelian varieties and powers of special surfaces”,
J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1985), no. 3, p. 487-520.
[3] ———, “Hodge cycles on abelian varieties of Sn-type”, J. Algebraic Geom. 9 (2000), no. 4,
p. 711-753.
[4] ———, “Discrete tomography and the Hodge conjecture for certain abelian varieties of
CM-type”, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 3, p. 25-29.
820 Hiromichi Yanai
[5] ———, “The Hodge rings of abelian varieties associated to certain subsets of finite fields”,
J. Math. Sci. Univ. Tokyo 14 (2007), no. 1, p. 99-111.
[6] V. K. Murty, “Hodge andWeil classes on abelian varieties”, in The arithmetic and geometry
of algebraic cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer
Acad. Publ., Dordrecht, 2000, p. 83-115.
[7] K. A. Ribet, “Division fields of abelian varieties with complex multiplication”, Mém. Soc.
Math. France (N.S.) (1980/81), no. 2, p. 75-94, Abelian functions and transcendental num-
bers (Colloq., Étole Polytech., Palaiseau, 1979).
[8] W. Tautz, J. Top & A. Verberkmoes, “Explicit hyperelliptic curves with real multiplica-
tion and permutation polynomials”, Canad. J. Math. 43 (1991), no. 5, p. 1055-1064.
[9] S. P. White, “Sporadic cycles on CM abelian varieties”, Compositio Math. 88 (1993), no. 2,
p. 123-142.
[10] H. Yanai, “Hodge cycles and unramified class fields”, in Class field theory—its centenary
and prospect (Tokyo, 1998), Adv. Stud. Pure Math., vol. 30, Math. Soc. Japan, Tokyo, 2001,
p. 307-312.
Hiromichi Yanai
Aichi Institute of Technology
Yakusa-cho
Toyota 470-0392
JAPAN
E-mail: yanai@aitech.ac.jp


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