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 $\mathbf{Q}$ (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 $\mathbf{Q}$ 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 $\mathbf{Q}$ [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 $K$. These elements satisfy certain compatibility conditions. Generally, almost all known Euler systems satisfy the condition ES) described below. Let $K$ be a number field. Fix a prime number $p$ and consider a set $\mathcal{S}$ of square-free ideals $L$ in $\mathcal{O} _ { K }$ which are relatively prime to some fixed ideal divisible by the primes over $p$. Let $A$ be a finite ${\bf Z} / p ^ { m }$-module with action of $G ( \overline { K } / K )$. For each $L$, let there be an Abelian extension $K ( L )$ of $K$ with the property that $K ( L ) \subset K ( L ^ { \prime } )$ if $L | L ^ { \prime }$. Then one wants to construct elements $c _ { L } \in H ^ { 1 } ( G ( \overline { K } / K ( L ) ) ; A )$ such that:

ES) . Here $\operatorname { Fr}_l$ is the Frobenius homomorphism (cf. Frobenius automorphism), $P _ { l } ( x ) \in \mathbf{Z} [ x ]$ is a polynomial with integral coefficients depending on $l$ and $\operatorname { Tr } _ { L l / L }$ is the transfer mapping from $K ( L l )$ down to $K ( L )$. 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 $c_L$ 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 $L$-function connected with the corresponding arithmetic object. In this way Euler systems establish (the sought for) relations between arithmetic objects and corresponding $L$-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 $p$-part of the class group of ${\bf Q} ( \mu _ { p } )$ for $p$ odd. K. Rubin [a20] extended Kolyvagin's method to give an elementary proof of the main conjecture in Iwasawa theory for $p > 2$ and $F / \mathbf Q $ Abelian (with some restrictions on $F$). In addition, C. Greither [a6] proved the main conjecture (using Kolyvagin's method) for all $F / \mathbf Q $ Abelian and all $p$, including $p = 2$.

Twisted Gauss sums.

In this case, the eigenspaces (for odd characters) of the $p$-part of the class group of ${\bf Q} ( \mu _ { p } )$ have been computed [a10], [a25].

Heegner points.

Let $E / \mathbf{Q}$ be a modular elliptic curve over $\mathbf{Q}$ (cf. also Modular curve). In [a9], Kolyvagin used Euler systems of Heegner points to show finiteness of $E ( {\bf Q} )$ and $\square ( E / \mathbf{Q} )$ under the assumption that $L ( E / {\bf Q }; s )$ is non-zero at $s = 1$ (cf. also Dirichlet $L$-function). This result was further generalized to certain higher-dimensional modular Abelian varieties (see [a12] and [a13]).

Let $K$ be an imaginary quadratic field of discriminant relatively prime to the conductor of $E$. Kolyvagin applied the Euler system of Heegner points [a10] in case the Heegner point $y_{ K }$ in $E ( K )$ is of infinite order (see also [a7] and [a15] for descriptions of this work). He proved that the following statements hold:

a) $E ( K )$ has rank one;

b) $\square ( E / K )$ is finite;

c) under certain assumptions on $p$ (see [a15], pp. 295–296) the following inequality holds:

\begin{equation*} \operatorname{ord} _ { p } \square ( E / K ) \leq 2 \text { ord } _ { p } [ E ( K ) : {\bf Z} y _ { K } ]. \end{equation*}

Subsequently, in [a11] Kolyvagin proved that the inequality above is actually an equality and determined the structure of $\square ( E / K )$. This Euler system is constructed in cohomology with coefficients in the module $A = E [ p ^ { m } ]$, the $p ^ { m }$ torsion points on the elliptic curve $E$.

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 $A_f$ (see [a2]) under the assumption that the corresponding twist of the $L$ function of $A_f$ is non-zero at $s = 1$.

Elliptic units.

K. Rubin considered an elliptic curve $E$ over $\mathbf{Q}$ which has complex multiplication (cf. Elliptic curve) by $K$. 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 $L ( E / K , 1 ) \neq 0$):

A) finiteness of $E ( K )$;

B) finiteness of $\square ( E / K )$;

C) a Birch–Swinnerton-Dyer formula for $E$ up to some very small explicit factors. Rubin proved that the Birch–Swinnerton-Dyer conjecture holds unconditionally for curves $y ^ { 2 } = x ^ { 3 } - p ^ { 2 } x$ for $p \equiv 3$ modulo $8$.

In the above examples (of cyclotomic units, twisted Gauss sums and elliptic units), the module of coefficients equals $A = \mathbf{Z} / p ^ { m } ( 1 )$. A number of problems in arithmetic involve the construction of Euler systems with $A$ different from ${\bf Z} / p ^ { m } ( 1 )$, as is the case for Heegner points.

Soulé's cyclotomic elements.

M. Kurihara [a14] found an Euler system $c _ { L } \in H ^ { 1 } ( \mathbf{Q} ( \mu _ { L } ) ; \mathbf{Z} / M ( n ) )$ based on a construction done by C. Soulé [a26]. The elements $c_L$ 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 $H ^ { 2 } ( {\bf Z} [ 1 / p ] ; {\bf Z} _ { p } ( n ) )$ in terms of the index of the Soulé cyclotomic elements inside $H ^ { 1 } ( \mathbf{Z} [ 1 / p ] ; \mathbf{Z} _ { p } ( n ) )$ for $n$ odd.

Analogues of Gauss sums for higher $K$-groups.

G. Banaszak and W. Gajda [a1] found an Euler system for higher $K$-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 $p$ part of the group of divisible elements in $K _ { 2 n - 2 } ( \mathbf Q )$ for $n$ even. One can map this Euler system via the Dwyer–Fiedlander homomorphism and obtain an Euler system in cohomology. Actually, one obtains elements $\Lambda_L \in H ^ { 1 } ( \mathbf{Z} [ 1 / p L ] ; \mathbf{Z} / M ( n ) )$ which form an Euler system.

Heegner cycles.

J. Nekovaŕ [a18] discovered an Euler system for a submodule $T$ of the $\mathbf{Z}_l$-module $H ^ { 2 r - 1 } ( \overline{X} ; \mathbf{Z} _{l} ( r ) )$, where $X$ is a Kuga–Sato variety attached to a modular form of weight $2 r > 2$. He used Heegner cycles in $C H ^ { r } ( X \otimes _ { K } K _ { n } )$. The elements thus constructed live in $H ^ { 1 } ( K _ { n } ; A )$, where $A = T / M$. Similarly to Kolyvagin, he could prove that the Tate–Shafarevich group for the module $T$ is finite and that its order divides the square of the index

\begin{equation*} [ H _ { f } ^ { 1 } ( K ; T ) : \mathbf{Z} _ { p } y ], \end{equation*}

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 $p$ away from the "bad primes" , he found annihilators (determined by the Heegner cycles) of the $p$ part of the Tate–Shafarevich group.

Euler systems for $p$-adic representations.

Assuming the existence of an Euler system for a $p$-adic representation $T$ of $G ( \overline { \mathbf{Q} } / \mathbf{Q} )$, K. Kato [a8], B. Perin-Riou [a19] and K. Rubin [a24] derived bounds for the Selmer group of the dual representation $\operatorname { Hom } ( T , \mathbf{Q} _ { p } / \mathbf{Z} _ { p } ( 1 ) )$. K. Kato constructed such an Euler system, the Kato Euler system, in the case when $T = T _ { p } ( E )$, the Tate module of a modular elliptic curve without complex multiplication (cf. [a24], [a27]). Let $Y _ { 1 } ( N )$ be a quotient of an open modular curve $Y ( N )$ (see [a27]). To start with, Kato constructed an element in $K _ { 2 } ^ { M } ( Y ( N ) )$ 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

\begin{equation*} H ^ { 1 } ( G ( \overline { \mathbf{Q} } / \mathbf{Q} ( \xi _ { L } ) ) ; T ( k - r ) ), \end{equation*}

where $T$ is a $G ( \overline { \mathbf{Q} } / \mathbf{Q} )$ equivariant $\mathbf{Z} _ { p }$-lattice in a $\mathbf{Q} _ { p }$-vector space $V$ and $\xi _ { L }$ is the $L$th power root of unity. The vector space $V$ is a quotient of

\begin{equation*} H ^ { 1 } \left( \overline { Y _ { 1 } ( N ) } ; \operatorname { Sym } ^ { k - 2 } R ^ { 1 } \overline { f } *\mathbf{Z} _ { p } \right) \bigotimes \mathbf{Q} _ { p }, \end{equation*}

where $f :{ \cal{E}} \rightarrow Y _ { 1 } ( N )$ is the natural mapping from the universal elliptic curve down to $Y _ { 1 } ( N )$ and $\overline { f } = f \otimes \overline { \mathbf{Q} }$. Under the assumption that $L ( E , 1 ) \neq 0$, 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 $E / \mathbf{Q}$ with a modular parametrization $\phi : X _ { 0 } ( N ) \rightarrow E$. Let $S _ { 0 }$ be the set of prime numbers containing $p$ and the primes where $E$ has bad reduction. For each prime number $l \notin S_0$, Flach constructed an element $c _ { l } \in H ^ { 1 } ( G ( \overline { \mathbf Q } / \mathbf Q ) ; \operatorname { Sym } ^ { 2 } T _ { p } ( E ) )$ which is the image (via a series of natural mappings) of an element in $\epsilon _ { l } \in H ^ { 1 } ( X _ { 0 } ( N ) \times X _ { 0 } ( N ) ; \mathcal{K} _ { 2 } )$. The elements $c_l$ 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 $T = \operatorname { Sym } ^ { 2 } T _ { p } ( E )$. Actually, he proved that these groups are annihilated by $\operatorname { deg } \phi$.

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 $p$-adic and complex), number theory, etc. has brought about many interesting examples.

References

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