Leopoldt's conjecture explained

In algebraic number theory, Leopoldt's conjecture, introduced by, states that the p-adic regulator of a number field does not vanish. The p-adic regulator is an analogue of the usual regulator defined using p-adic logarithms instead of the usual logarithms, introduced by .

Formulation

Let K be a number field and for each prime P of K above some fixed rational prime p, let U_P denote the local units at P and let U_1,P denote the subgroup of principal units in U_P. Set

U _{1} = \prod _{P|p} U _{1,P} .

Then let E₁ denote the set of global units ε that map to U₁ via the diagonal embedding of the global units in E.

Since

E _{1} is a finite-index subgroup of the global units, it is an abelian group of rank r _{1} + r _{2} - 1 , where r _{1} is the number of real embeddings of K and r _{2} the number of pairs of complex embeddings. Leopoldt's conjecture states that the Z _{p} -module rank of the closure of E _{1} embedded diagonally in U _{1} is also r _{1} + r _{2} - 1 .

Leopoldt's conjecture is known in the special case where

K is an abelian extension of Q or an abelian extension of an imaginary quadratic number field: reduced the abelian case to a p-adic version of Baker's theorem, which was proved shortly afterwards by . has announced a proof of Leopoldt's conjecture for all CM-extensions of Q .

expressed the residue of the p-adic Dedekind zeta function of a totally real field at s = 1 in terms of the p-adic regulator. As a consequence, Leopoldt's conjecture for those fields is equivalent to their p-adic Dedekind zeta functions having a simple pole at s = 1.

References

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