View source for Euler systems for number fields

Towards the end of 1980s, F. Thaine [[#References|[a28]]] discovered a new method for investigating the class groups (cf. also [[Class field theory|Class field theory]]) of real Abelian extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e1202401.png" /> (cf. also [[Extension of a field|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e1202402.png" /> and Abelian extensions of quadratic fields [[#References|[a10]]] (see also [[#References|[a20]]]). In addition, Kolyvagin showed that this method extends to problems concerning Mordell–Weil groups and Tate–Shafarevich groups of modular elliptic curves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e1202403.png" /> [[#References|[a9]]], [[#References|[a10]]] (cf. also [[Elliptic curve|Elliptic curve]]; [[Galois cohomology|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e1202404.png" />. These elements satisfy certain compatibility conditions. Generally, almost all known Euler systems satisfy the condition ES) described below. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e1202405.png" /> be a number field. Fix a [[Prime number|prime number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e1202406.png" /> and consider a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e1202407.png" /> of square-free ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e1202408.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e1202409.png" /> which are relatively prime to some fixed ideal divisible by the primes over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024010.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024011.png" /> be a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024012.png" />-module with action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024013.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024014.png" />, let there be an Abelian extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024016.png" /> with the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024017.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024018.png" />. Then one wants to construct elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024019.png" /> such that:

ES) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024020.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024021.png" /> is the Frobenius homomorphism (cf. [[Frobenius automorphism|Frobenius automorphism]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024022.png" /> is a [[Polynomial|polynomial]] with integral coefficients depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024024.png" /> is the transfer mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024025.png" /> down to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024026.png" />. Next to condition ES), any given Euler system may have additional properties, cf. [[#References|[a4]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a17]]], [[#References|[a20]]], [[#References|[a22]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024027.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024029.png" />-function connected with the corresponding arithmetic object. In this way Euler systems establish (the sought for) relations between arithmetic objects and corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024030.png" />-values.

==Examples.==
Some specific Euler systems and objects they compute are listed below.

===Cyclotomic units.===
This Euler system [[#References|[a10]]], [[#References|[a20]]] computes eigenspaces (for even characters) of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024031.png" />-part of the class group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024032.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024033.png" /> odd. K. Rubin [[#References|[a20]]] extended Kolyvagin's method to give an elementary proof of the main conjecture in Iwasawa theory for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024035.png" /> Abelian (with some restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024036.png" />). In addition, C. Greither [[#References|[a6]]] proved the main conjecture (using Kolyvagin's method) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024037.png" /> Abelian and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024038.png" />, including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024039.png" />.

===Twisted Gauss sums.===
In this case, the eigenspaces (for odd characters) of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024040.png" />-part of the class group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024041.png" /> have been computed [[#References|[a10]]], [[#References|[a25]]].

===Heegner points.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024042.png" /> be a modular elliptic curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024043.png" /> (cf. also [[Modular curve|Modular curve]]). In [[#References|[a9]]], Kolyvagin used Euler systems of Heegner points to show finiteness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024045.png" /> under the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024046.png" /> is non-zero at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024047.png" /> (cf. also [[Dirichlet-L-function|Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024048.png" />-function]]). This result was further generalized to certain higher-dimensional modular Abelian varieties (see [[#References|[a12]]] and [[#References|[a13]]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024049.png" /> be an imaginary quadratic field of discriminant relatively prime to the conductor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024050.png" />. Kolyvagin applied the Euler system of Heegner points [[#References|[a10]]] in case the Heegner point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024051.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024052.png" /> is of infinite order (see also [[#References|[a7]]] and [[#References|[a15]]] for descriptions of this work). He proved that the following statements hold:

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024053.png" /> has rank one;

b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024054.png" /> is finite;

c) under certain assumptions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024055.png" /> (see [[#References|[a15]]], pp. 295–296) the following inequality holds:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024056.png" /></td> </tr></table>

Subsequently, in [[#References|[a11]]] Kolyvagin proved that the inequality above is actually an equality and determined the structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024057.png" />. This Euler system is constructed in cohomology with coefficients in the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024058.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024059.png" /> torsion points on the elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024060.png" />.

M. Bertolini and H. Darmon also constructed cohomology classes based on Heegner points [[#References|[a2]]]. Using these classes they proved finiteness of certain twisted Mordell–Weil groups for an Abelian variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024061.png" /> (see [[#References|[a2]]]) under the assumption that the corresponding twist of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024062.png" /> function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024063.png" /> is non-zero at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024064.png" />.

===Elliptic units.===
K. Rubin considered an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024065.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024066.png" /> which has complex multiplication (cf. [[Elliptic curve|Elliptic curve]]) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024067.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024068.png" />):

A) finiteness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024069.png" />;

B) finiteness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024070.png" />;

C) a Birch–Swinnerton-Dyer formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024071.png" /> up to some very small explicit factors. Rubin proved that the Birch–Swinnerton-Dyer conjecture holds unconditionally for curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024072.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024073.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024074.png" />.

In the above examples (of cyclotomic units, twisted Gauss sums and elliptic units), the module of coefficients equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024075.png" />. A number of problems in arithmetic involve the construction of Euler systems with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024076.png" /> different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024077.png" />, as is the case for Heegner points.

===Soulé's cyclotomic elements.===
M. Kurihara [[#References|[a14]]] found an Euler system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024078.png" /> based on a construction done by C. Soulé [[#References|[a26]]]. The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024079.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024080.png" /> in terms of the index of the Soulé cyclotomic elements inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024081.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024082.png" /> odd.

===Analogues of Gauss sums for higher <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024083.png" />-groups.===
G. Banaszak and W. Gajda [[#References|[a1]]] found an Euler system for higher <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024084.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024085.png" /> part of the group of divisible elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024086.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024087.png" /> even. One can map this Euler system via the Dwyer–Fiedlander homomorphism and obtain an Euler system in cohomology. Actually, one obtains elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024088.png" /> which form an Euler system.

===Heegner cycles.===
J. Nekovaŕ [[#References|[a18]]] discovered an Euler system for a submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024089.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024090.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024091.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024092.png" /> is a Kuga–Sato variety attached to a [[Modular form|modular form]] of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024093.png" />. He used Heegner cycles in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024094.png" />. The elements thus constructed live in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024095.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024096.png" />. Similarly to Kolyvagin, he could prove that the Tate–Shafarevich group for the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024097.png" /> is finite and that its order divides the square of the index

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024098.png" /></td> </tr></table>

which is also proven to be finite. Recently (1997), A. Besser [[#References|[a3]]] refined the results of Nekovaŕ. He defined the Tate–Shafarevich group considering also the  "bad primes" . For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e12024099.png" /> away from the  "bad primes" , he found annihilators (determined by the Heegner cycles) of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240100.png" /> part of the Tate–Shafarevich group.

===Euler systems for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240101.png" />-adic representations.===
Assuming the existence of an Euler system for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240102.png" />-adic representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240103.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240104.png" />, K. Kato [[#References|[a8]]], B. Perin-Riou [[#References|[a19]]] and K. Rubin [[#References|[a24]]] derived bounds for the Selmer group of the dual representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240105.png" />. K. Kato constructed such an Euler system, the Kato Euler system, in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240106.png" />, the Tate module of a modular elliptic curve without complex multiplication (cf. [[#References|[a24]]], [[#References|[a27]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240107.png" /> be a quotient of an open modular curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240108.png" /> (see [[#References|[a27]]]). To start with, Kato constructed an element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240109.png" /> 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240110.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240111.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240112.png" /> equivariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240113.png" />-lattice in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240114.png" />-vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240115.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240116.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240117.png" />th power root of unity. The vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240118.png" /> is a quotient of

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240119.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240120.png" /> is the natural mapping from the universal elliptic curve down to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240121.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240122.png" />. Under the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240123.png" />, 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 [[#References|[a24]]], Chap. 7; 8).

===Work of M. Flach.===
Interesting and useful cohomology classes were constructed by M. Flach [[#References|[a5]]]. These elements were independently found by S. Bloch and were used by S.J.M. Mildenhall in [[#References|[a16]]]. Flach considered a modular elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240124.png" /> with a modular parametrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240125.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240126.png" /> be the set of prime numbers containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240127.png" /> and the primes where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240128.png" /> has bad reduction. For each prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240129.png" />, Flach constructed an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240130.png" /> which is the image (via a series of natural mappings) of an element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240131.png" />. The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240132.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240133.png" />. Actually, he proved that these groups are annihilated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240134.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240135.png" />-adic and complex), number theory, etc. has brought about many interesting examples.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Banaszak,   W. Gajda,   "Euler systems for higher K-theory of number fields"  ''J. Number Th.'' , '''58''' :  2  (1996)  pp. 213–252</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Bertolini,   H. Darmon,   "A rigid analytic Gross–Zagier formula and arithmetic applications"  ''preprint''</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Besser,   "On the finiteness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240136.png" /> for motives associated to modular forms"  ''Doc. Math. J. Deutsch. Math. Ver.'' , '''2'''  (1997)  pp. 31–46</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Darmon,   "Euler systems and refined conjectures of Birch Swinnerton–Dyer type"  ''Contemp. Math.'' , '''165'''  (1994)  pp. 265–276</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Flach,   "A finiteness theorem for the symmetric square of an elliptic curve"  ''Invent. Math.'' , '''109'''  (1992)  pp. 307–327</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C. Greither,   "Class groups of abelian fields and the main conjecture"  ''Ann. Inst. Fourier (Grenoble)'' , '''42 No 3'''  (1992)  pp. 449–499</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  B.H. Gross,   "Kolyvagin's work on modular elliptic curves"  J. Coates (ed.)  M.J. Taylor (ed.) , ''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240137.png" />-Functions and Arithmetic. Proc. Symp. Durham 1989'' , ''London Math. Soc. Lecture Notes'' , '''153'''  (1991)  pp. 235–256</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  K. Kato,   "Euler systems, Iwasawa theory and Selmer groups"  ''to appear''</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  V.A. Kolyvagin,   "Finitness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240138.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240139.png" /> for a class of Weil curves"  ''Izv. Akad. Nauk SSSR'' , '''52'''  (1988)  pp. 522–540</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  V.A. Kolyvagin,   "Euler Systems" , ''Grothendieck Festschrift II'' , ''Progr. Math.'' , '''87''' , Birkhäuser  (1990)  pp. 435–483</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  V.A. Kolyvagin,   D.Y. Logacev,   "Finiteness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240140.png" /> over totally real fields"  ''Izv. Akad. Nauk SSSR Ser. Math.'' , '''55'''  (1991)  pp. 851–876</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  M. Kurihara,   "Some remarks on conjectures about cyclotomic fields and K-groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120240/e120240141.png" />"  ''Compositio Math.'' , '''81'''  (1992)  pp. 223–236</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  W.G. Mccallum,   "Kolyvagin's work on Shafarevich–Tate groups"  J. Coates (ed.)  M.J. 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