Mennicke symbol

Mennicke symbol

In mathematics, a Mennicke symbol is a map from pairs of elements of a number field to an abelian group satifying some identities found by Menncike (1965). They were named by Bass, Milnor & Serre (1967), who used them in their solution of the congruence subgroup problem.

Definition

Suppose that A is a Dedekind domain and q is a non-zero ideal of A. The set Wq is defined to be the set of pairs (ab) with a = 1 mod q, b = 0 mod q, such that a and b generate the unit ideal.

A Mennicke symbol on Wq with values in a group C is a function (ab) → [b
a
] from Wq to C such that

  • [0
    1
    ] = 1, [bc
    a
    ] = [b
    a
    ][c
    a
    ]
  • [b
    a
    ] = [b + ta
    a
    ] if t is in q, [b
    a
    ] = [b
    a + tb
    ] if t is in A.

There is a universal Mennicke symbol with values in a group Cq such that any Mennicke symbol with values in C can be obtained by composing the universal Mennicke symbol with a unique homomorphism from Cq to C.

References


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