Splitting of Abelian varieties

44 Pages·2013·0.27 MB·English

Introduction
EndomorphismsandCohomology
ConnectionstoMonodromy
RepresentationTheoryformulation
TateCycles
Splitting of Abelian varieties
V. Kumar Murty
University of Toronto
GEOCRYPT
October 2013
V.KumarMurtyUniversityofToronto SplittingofAbelianvarieties
Introduction
EndomorphismsandCohomology
Aclassicalproblem
ConnectionstoMonodromy
Ageometricanalogue
RepresentationTheoryformulation
TateCycles
A question about polynomials
(cid:73) Question: Given an irreducible polynomial f(T) ∈ Z[T], and
a prime p, does it necessarily remain irreducible modulo a
given prime p?
(cid:73) Answer: Obviously not.
(cid:73) Question: Given an irreducible polynomial f(T) ∈ Z[T], is
there a prime p such that f(T) (mod p) is irreducible?
(cid:73) Answer: Not necessarily!
(cid:73) A simple example is
f(T) = T4+1.
V.KumarMurtyUniversityofToronto SplittingofAbelianvarieties
Introduction
EndomorphismsandCohomology
Aclassicalproblem
ConnectionstoMonodromy
Ageometricanalogue
RepresentationTheoryformulation
TateCycles
Factorization of T4 + 1 (mod p)
(cid:73) If p ≡ 1 (mod 4), there is an a such that a2 ≡ −1 (mod p).
(cid:73) With this a, we have
T4+1 = (T2+a)(T2−a) (mod p).
V.KumarMurtyUniversityofToronto SplittingofAbelianvarieties
Introduction
EndomorphismsandCohomology
Aclassicalproblem
ConnectionstoMonodromy
Ageometricanalogue
RepresentationTheoryformulation
TateCycles
Factorization of T4 + 1 (mod p)
(cid:73) If p ≡ 7 (mod 8), there is a b such that b2 ≡ 2 (mod p).
(cid:73) With this b, we have
T4+1 = (T2+1)2−2T2
= (T2−bT +1)(T2+bT +1) (mod p).
(cid:73) If p ≡ 3 (mod 8), there is a c such that c2 ≡ −2 (mod p)
and
T4+1 = (T2−cT −1)(T2+cT −1) (mod p).
V.KumarMurtyUniversityofToronto SplittingofAbelianvarieties
Introduction
EndomorphismsandCohomology
Aclassicalproblem
ConnectionstoMonodromy
Ageometricanalogue
RepresentationTheoryformulation
TateCycles
Splitting of polynomials
(cid:73) Expressed another way, this means that an irreducible
polynomial f(T) ∈ Z[T] may become reducible (mod p) for
every prime p.
(cid:73) On the other hand, if there is a prime p such that f(T)
(mod p) is irreducible, then there infinitely many such primes
(cid:73) In fact a positive density of primes. We shall see why later.
V.KumarMurtyUniversityofToronto SplittingofAbelianvarieties
Introduction
EndomorphismsandCohomology
Aclassicalproblem
ConnectionstoMonodromy
Ageometricanalogue
RepresentationTheoryformulation
TateCycles
What is behind this?
(cid:73) The answer comes from algebraic number theory.
(cid:73) Let f be a normal polynomial and let E be the splitting field
of f. Let O be the ring of integers.
(cid:73) Dedekind’s theorem: For all but finitely many p, the
factorization of f (mod p) is identical to the splitting of the
ideal pO in the Dedekind domain O.
(cid:73) In other words,
f(T) = f (T)e1···f (T)er (mod p)
1 r
pO = pe1···per.
1 r
V.KumarMurtyUniversityofToronto SplittingofAbelianvarieties
Introduction
EndomorphismsandCohomology
Aclassicalproblem
ConnectionstoMonodromy
Ageometricanalogue
RepresentationTheoryformulation
TateCycles
The Frobenius automorphism
(cid:73) We have
f(T) = f (T)e1···f (T)er (mod p)
1 r
pO = pe1···per
1 r
(cid:73) To each p = p , there is an automorphism Frob in the Galois
i p
group of E/Q.
(cid:73) For most primes p, this is the unique automorphism σ which
satisfies
σ(x) ≡ xp (mod p).
V.KumarMurtyUniversityofToronto SplittingofAbelianvarieties
Introduction
EndomorphismsandCohomology
Aclassicalproblem
ConnectionstoMonodromy
Ageometricanalogue
RepresentationTheoryformulation
TateCycles
The Frobenius automorphism
(cid:73) This automorphism Frob is an element which is of order
p
equal to degf .
i
(cid:73) In particular, if f is irreducible (mod p), then pO stays prime
in E and the order of Frob is n = degf.
p
(cid:73) Thus, Frob generates Gal(E/Q) and so, this group must be
p
cyclic.
(cid:73) In the case of T4+1, the splitting field is Q(ζ ) which has
8
Galois group Z/2Z × Z/2Z.
V.KumarMurtyUniversityofToronto SplittingofAbelianvarieties
Introduction
EndomorphismsandCohomology
Aclassicalproblem
ConnectionstoMonodromy
Ageometricanalogue
RepresentationTheoryformulation
TateCycles
A converse
(cid:73) Conversely, suppose that f is normal and generates a cyclic
extension.
(cid:73) Then there are a positive density of primes p such that f
(mod p) is irreducible.
(cid:73) This follows from the Chebotarev density theorem.
V.KumarMurtyUniversityofToronto SplittingofAbelianvarieties
Introduction
EndomorphismsandCohomology
Aclassicalproblem
ConnectionstoMonodromy
Ageometricanalogue
RepresentationTheoryformulation
TateCycles
The general case
(cid:73) What about non-normal polynomials? Consider the general
case: the Galois group of f is the symmetric group S where
n
n = deg f.
(cid:73) Then f (mod p) factors according to the cycle structure of
the conjugacy class of Frobenius automorphisms over p.
(cid:73) In particular, f (mod p) will be irreducible whenever the
Frob is an n-cycle.
p
(cid:73) To find such a prime, we need only check p (cid:28) (logd )2 (if
f
we believe the Generalized Riemann Hypothesis).
V.KumarMurtyUniversityofToronto SplittingofAbelianvarieties