2. The Kolyvagin System for the real quadratic field F modulo p^{N} (coming from cyclotomic units). Second discussion, which still leaves out the construction!

The Kolyvagin apparatus will come in three parts: Notation, Data, and Relations satisfied by the Kolyvagin elements. Notation. Recall the notation of last time. Let L denote the set of odd prime numbers ℓ ≡ 1 mod p^{N} and which split in F. In our first discussion we “ignored” the twists occurring in the construction of Kolyvagin elements. To deal with them, we must introduce, for each ℓ ∈ L, a specific cyclic group of order p^{N}. The easiest way of defining C_{ℓ} is as follows. Let λ be a prime of F lying above ℓ (either one will do) and put C_{ℓ} := O_{F_{λ}} ⊗ Z/p^{N}Z. Then C_{ℓ} is well-defined up to canonical isomorphism (i.e., conjugation of F over Q induces a canoncial isomoprhism between O_{F_{λ}} ⊗ Z/p^{N}Z and O_{Fλ̄} ⊗ Z/p^{N}Z). Let R denote the set of squarefree positive integer r whose prime divisors are in L. If r ∈ R then C_{r} is defined to be the tensor product of the C_{ℓ}’s for the prime divisors ℓ of r. Data. Now we can correct our first discussion. For each r ∈ R Kolyvagin will give you an element, κ_{r}, in the minus eigenspace of O_{F} [1/r]^{∗} ⊂ F^{∗} (modulo p^{N}-th powers) “twisted by tensoring with C_{r}; i.e.,

κ_{r} ∈ (O_{F} [1/r]^{∗})^{−} ⊗_{Z} C_{r}.

Local relations between Kolyvagin elements If ℓ is a prime number in L not dividing r and λ is a prime of F dividing ℓ, denote by

κ_{r,λ,f} ∈ O_{F_{λ}}^{∗} ⊗_{Z} C_{r} = C_{ℓ} ⊗_{Z} C_{r} = C_{rℓ}

the image (under the natural map) of κ_{r}.

Denote by

κ_{rℓ,λ,s} ∈ (F_{λ}^{∗}/O_{F_{λ}}^{∗}) ⊗_{Z} C_{rℓ} = Z/p^{N}Z ⊗_{Z} C_{rℓ} = C_{rℓ}

the image (under the natural map) of κ_{rℓ}.

The local relations that the Kolyvagin elements satisfy is that κ_{r,λ,f} is brought to κ_{rℓ,λ,s} under the (above) natural identification of the groups that contain them.

Exercise. Rewrite, in corrected form (using these twisting groups C_{r}), the definition of the “Kolyvagin ideal class group of F modulo p^{N}” introduced in the first discussion.