Euler systems for number fields
Towards the end of 1980s, F. Thaine [a28] discovered a new method for investigating the class groups (cf. also Class field theory) of real Abelian extensions of (cf. also Extension of a field). His method turned out to be the first step of a descent procedure introduced by V.A. Kolyvagin, shortly after Thaine's result. Kolyvagin used this procedure to investigate class groups of Abelian extensions of
and Abelian extensions of quadratic fields [a10] (see also [a20]). In addition, Kolyvagin showed that this method extends to problems concerning Mordell–Weil groups and Tate–Shafarevich groups of modular elliptic curves over
[a9], [a10] (cf. also Elliptic curve; Galois cohomology). The key idea of Kolyvagin's method is to construct a family of cohomology classes indexed by an infinite set of square-free integral ideals of the base field
. These elements satisfy certain compatibility conditions. Generally, almost all known Euler systems satisfy the condition ES) described below. Let
be a number field. Fix a prime number
and consider a set
of square-free ideals
in
which are relatively prime to some fixed ideal divisible by the primes over
. Let
be a finite
-module with action of
. For each
, let there be an Abelian extension
of
with the property that
if
. Then one wants to construct elements
such that:
ES) . Here
is the Frobenius homomorphism (cf. Frobenius automorphism),
is a polynomial with integral coefficients depending on
and
is the transfer mapping from
down to
. Next to condition ES), any given Euler system may have additional properties, cf. [a4], [a9], [a10], [a17], [a20], [a22], [a23].
To discover an Euler system is usually a difficult task. Once an Euler system has been identified, one figures out local conditions that the global cohomology classes satisfy. Then Kolyvagin's descent procedure gives good control over corresponding arithmetic objects such as the class group of a number field or the Selmer group of an elliptic curve. On the other hand, an Euler system encodes values of the
-function connected with the corresponding arithmetic object. In this way Euler systems establish (the sought for) relations between arithmetic objects and corresponding
-values.
Examples.
Some specific Euler systems and objects they compute are listed below.
Cyclotomic units.
This Euler system [a10], [a20] computes eigenspaces (for even characters) of the -part of the class group of
for
odd. K. Rubin [a20] extended Kolyvagin's method to give an elementary proof of the main conjecture in Iwasawa theory for
and
Abelian (with some restrictions on
). In addition, C. Greither [a6] proved the main conjecture (using Kolyvagin's method) for all
Abelian and all
, including
.
Twisted Gauss sums.
In this case, the eigenspaces (for odd characters) of the -part of the class group of
have been computed [a10], [a25].
Heegner points.
Let be a modular elliptic curve over
(cf. also Modular curve). In [a9], Kolyvagin used Euler systems of Heegner points to show finiteness of
and
under the assumption that
is non-zero at
(cf. also Dirichlet
-function). This result was further generalized to certain higher-dimensional modular Abelian varieties (see [a12] and [a13]).
Let be an imaginary quadratic field of discriminant relatively prime to the conductor of
. Kolyvagin applied the Euler system of Heegner points [a10] in case the Heegner point
in
is of infinite order (see also [a7] and [a15] for descriptions of this work). He proved that the following statements hold:
a) has rank one;
b) is finite;
c) under certain assumptions on (see [a15], pp. 295–296) the following inequality holds:
![]() |
Subsequently, in [a11] Kolyvagin proved that the inequality above is actually an equality and determined the structure of . This Euler system is constructed in cohomology with coefficients in the module
, the
torsion points on the elliptic curve
.
M. Bertolini and H. Darmon also constructed cohomology classes based on Heegner points [a2]. Using these classes they proved finiteness of certain twisted Mordell–Weil groups for an Abelian variety (see [a2]) under the assumption that the corresponding twist of the
function of
is non-zero at
.
Elliptic units.
K. Rubin considered an elliptic curve over
which has complex multiplication (cf. Elliptic curve) by
. He applied the Euler system of elliptic units to prove one- and two-variable main conjectures in Iwasawa theory. Using this he obtained (under the assumption that
):
A) finiteness of ;
B) finiteness of ;
C) a Birch–Swinnerton-Dyer formula for up to some very small explicit factors. Rubin proved that the Birch–Swinnerton-Dyer conjecture holds unconditionally for curves
for
modulo
.
In the above examples (of cyclotomic units, twisted Gauss sums and elliptic units), the module of coefficients equals . A number of problems in arithmetic involve the construction of Euler systems with
different from
, as is the case for Heegner points.
Soulé's cyclotomic elements.
M. Kurihara [a14] found an Euler system based on a construction done by C. Soulé [a26]. The elements
are made of cyclotomic units twisted by the Tate module and sent down to an appropriate field level by the co-restriction mapping. Kurihara used this Euler system to estimate
in terms of the index of the Soulé cyclotomic elements inside
for
odd.
Analogues of Gauss sums for higher
-groups.
G. Banaszak and W. Gajda [a1] found an Euler system for higher -groups of number fields. It is given in terms of transfer (to an appropriate field level) applied to Gauss sums (as above) multiplied by Bott elements. This system of elements is used to estimate from above the order of the
part of the group of divisible elements in
for
even. One can map this Euler system via the Dwyer–Fiedlander homomorphism and obtain an Euler system in cohomology. Actually, one obtains elements
which form an Euler system.
Heegner cycles.
J. Nekovaŕ [a18] discovered an Euler system for a submodule of the
-module
, where
is a Kuga–Sato variety attached to a modular form of weight
. He used Heegner cycles in
. The elements thus constructed live in
, where
. Similarly to Kolyvagin, he could prove that the Tate–Shafarevich group for the module
is finite and that its order divides the square of the index
![]() |
which is also proven to be finite. Recently (1997), A. Besser [a3] refined the results of Nekovaŕ. He defined the Tate–Shafarevich group considering also the "bad primes" . For each away from the "bad primes" , he found annihilators (determined by the Heegner cycles) of the
part of the Tate–Shafarevich group.
Euler systems for
-adic representations.
Assuming the existence of an Euler system for a -adic representation
of
, K. Kato [a8], B. Perin-Riou [a19] and K. Rubin [a24] derived bounds for the Selmer group of the dual representation
. K. Kato constructed such an Euler system, the Kato Euler system, in the case when
, the Tate module of a modular elliptic curve without complex multiplication (cf. [a24], [a27]). Let
be a quotient of an open modular curve
(see [a27]). To start with, Kato constructed an element in
which is a symbol of two carefully chosen modular units. Then, by a series of natural mappings and a clever twisting trick, he mapped these elements to the group
![]() |
where is a
equivariant
-lattice in a
-vector space
and
is the
th power root of unity. The vector space
is a quotient of
![]() |
where is the natural mapping from the universal elliptic curve down to
and
. Under the assumption that
, Kato proved the finiteness of the Tate–Shafarevich and Mordel–Weil groups. In this way, he also reproved Kolyvagin's result on Heegner points (see above). Nevertheless, the work of Kato avoided reference to many analytic results (see [a24], Chap. 7; 8).
Work of M. Flach.
Interesting and useful cohomology classes were constructed by M. Flach [a5]. These elements were independently found by S. Bloch and were used by S.J.M. Mildenhall in [a16]. Flach considered a modular elliptic curve with a modular parametrization
. Let
be the set of prime numbers containing
and the primes where
has bad reduction. For each prime number
, Flach constructed an element
which is the image (via a series of natural mappings) of an element in
. The elements
seem to be a first step of some (still unknown, 1998) Euler system. Nevertheless, Flach was able to prove the finiteness of the Selmer and Tate–Shafarevich groups associated with the module
. Actually, he proved that these groups are annihilated by
.
Constructing interesting elements in cohomology, especially Euler system elements, is a major task of contemporary arithmetic. The interplay between arithmetic and algebraic geometry, analysis (both -adic and complex), number theory, etc. has brought about many interesting examples.
References
[a1] | G. Banaszak, W. Gajda, "Euler systems for higher K-theory of number fields" J. Number Th. , 58 : 2 (1996) pp. 213–252 MR1393614 Zbl 0851.19003 |
[a2] | M. Bertolini, H. Darmon, "A rigid analytic Gross–Zagier formula and arithmetic applications" preprint MR1469318 Zbl 1029.11027 |
[a3] | A. Besser, "On the finiteness of ![]() |
[a4] | H. Darmon, "Euler systems and refined conjectures of Birch Swinnerton–Dyer type" Contemp. Math. , 165 (1994) pp. 265–276 MR1279613 Zbl 0823.11036 |
[a5] | M. Flach, "A finiteness theorem for the symmetric square of an elliptic curve" Invent. Math. , 109 (1992) pp. 307–327 MR1172693 Zbl 0781.14022 |
[a6] | C. Greither, "Class groups of abelian fields and the main conjecture" Ann. Inst. Fourier (Grenoble) , 42 No 3 (1992) pp. 449–499 MR1182638 Zbl 0729.11053 |
[a7] | B.H. Gross, "Kolyvagin's work on modular elliptic curves" J. Coates (ed.) M.J. Taylor (ed.) , ![]() |
[a8] | K. Kato, "Euler systems, Iwasawa theory and Selmer groups" to appear MR1727298 Zbl 0993.11033 |
[a9] | V.A. Kolyvagin, "Finitness of ![]() ![]() |
[a10] | V.A. Kolyvagin, "Euler Systems" , Grothendieck Festschrift II , Progr. Math. , 87 , Birkhäuser (1990) pp. 435–483 MR1106906 Zbl 0742.14017 |
[a11] | V.A. Kolyvagin, "On the structure of Shafarevich–Tate groups" S. Bloch (ed.) I. Dolgachev (ed.) W. Fulton (ed.) , Algebraic Geometry , Lecture Notes Math. , 1479 (1991) pp. 333–400 MR1181210 Zbl 0753.14025 |
[a12] | V.A. Kolyvagin, D.Y. Logacev, "Finiteness of Shafarevich–Tate group and the group of rational points for some modular Abelian varieties" Algebra i Anal. , 1 (1989) pp. 171–196 Zbl 0728.14026 |
[a13] | V.A. Kolyvagin, D.Y. Logacev, "Finiteness of ![]() |
[a14] | M. Kurihara, "Some remarks on conjectures about cyclotomic fields and K-groups of ![]() |
[a15] | W.G. Mccallum, "Kolyvagin's work on Shafarevich–Tate groups" J. Coates (ed.) M.J. Taylor (ed.) , ![]() |
[a16] | S.J.M. Mildenhall, "Cycles in products of elliptic curves and a group analogous to the class group" Duke Math. J. , 67, No.2 (1992) pp. 387–406 MR1177312 |
[a17] | J. Nekovaŕ, "Values of ![]() ![]() |
[a18] | J. Nekovaŕ, "Kolyvagin's method for Chow groups of Kuga–Sato varieties" Invent. Math. , 107 (1992) pp. 99–125 MR1135466 |
[a19] | B. Perrin-Riou, "Systèmes d'Euler ![]() |
[a20] | K. Rubin, "A proof of some `main conjectures' via methods of Kolyvagin" preprint (1988) |
[a21] | K. Rubin, "On the main conjecture of Iwasawa theory for imaginary quadratic fields" Invent. Math. , 93 (1988) pp. 701–713 MR0952288 Zbl 0673.12004 |
[a22] | K. Rubin, "The `main conjectures' of Iwasawa theory for imaginary quadratic fields" Invent. Math. , 103 (1991) pp. 25–68 |
[a23] | K. Rubin, "Stark units and Kolyvagin's `Euler systems'" J. Reine Angew. Math. , 425 (1992) pp. 141–154 |
[a24] | K. Rubin, "Euler systems and modular elliptic curves" preprint (1997) MR1696493 Zbl 0952.11016 |
[a25] | K. Rubin, "Kolyvagin's systems of Gauss sums" G. van der Geer (ed.) F. Oort (ed.) J. Steenbrink (ed.) , Arithmetic Algebraic Geometry , Progr. Math. , Birkhäuser (1991) pp. 309–324 |
[a26] | C. Soulé, "On higher ![]() |
[a27] | A. Scholl, "Symbols and Euler systems for modular varieties" preprint |
[a28] | F. Thaine, "On the ideal class groups of real abelian number fields" Ann. of Math. , 128 (1988) pp. 1–18 MR0951505 Zbl 0665.12003 |
Euler systems for number fields. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_systems_for_number_fields&oldid=50306