In algebraic number theory, Leopoldt's conjecture, introduced by , states that the p-adic regulator of a

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

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 . Leopoldt proposed a definition of a

p-adic regulator In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of an abelian group, rank of the group of units in the ring (mathematics), ring of algebraic intege ...

''R''_''p'' attached to ''K'' and a prime number ''p''. The definition of ''R''_''p'' uses an appropriate determinant with entries the

p-adic logarithm In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension ...

of a generating set of units of ''K'' (up to torsion), in the manner of the usual regulator. The conjecture, which for general ''K'' is still open , then comes out as the statement that ''R''_''p'' is not zero.

Formulation

Let ''K'' be a

and for each

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

''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_ U_.

Then let ''E''₁ denote the set of global units ''ε'' that map to ''U''₁ via the

diagonal embedding In algebraic geometry, given a morphism of schemes p: X \to S, the diagonal morphism :\delta: X \to X \times_S X is a morphism determined by the universal property of the fiber product X \times_S X of ''p'' and ''p'' applied to the identity 1_X : X ...

of the global units in ''E''. Since

E_1

is a finite-

index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...

subgroup of the global units, it is an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...

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

\mathbb_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 In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...

\mathbb

or an abelian extension of an imaginary

quadratic number field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...

: reduced the abelian case to a p-adic version of

Baker's theorem In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by , subsumed many earlier results in transcendenta ...

, which was proved shortly afterwards by . has announced a proof of Leopoldt's conjecture for all CM-extensions of

\mathbb

. expressed the residue of the ''p''-adic

Dedekind zeta function In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It ...

of a

totally real field In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polyn ...

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

* * * * * *. * * * *{{Citation , first=Lawrence C. , last=Washington , title=Introduction to Cyclotomic Fields , edition=Second , year=1997 , location=New York , publisher=Springer , isbn=0-387-94762-0 , zbl=0966.11047 . Algebraic number theory Conjectures Unsolved problems in number theory