p-adic Numbers

Hensel introduced p-adic numbers in 1897. Ostrowski's theorem (1916) completed the classification. The number 6 is 'small' 2-adically (1/2) and 'large' ordinarily (6).

• Number theory: Q_p is the main tool for the Hasse-Minkowski local-global principle
• Representation theory: GL(n,Q_p) is the local component of automorphic forms
• Physics: p-adic string theory

Предварительные знания

• Previous lesson

The p-adic Norm and Ultrametric

Kurt Hensel introduced p-adic numbers in 1897 by analogy with Laurent series. Ostrowski's theorem (1916): every nontrivial norm on Q is equivalent to |.|_inf or |.|_p. Product formula: |r|_inf * prod_p |r|_p = 1 for any r in Q^times. The number 6 is 'small' 2-adically: ||6||_2 = 1/2.

Hensel's Lemma and Ostrowski's Theorem

Ostrowski's theorem (1916): all norms on Q are |.|_inf and |.|_p. Product formula: |r|_inf * prod_p |r|_p = 1 for r in Q^times. Hensel's lemma: a simple root of a polynomial mod p lifts uniquely to a root in Z_p -- the p-adic implicit function theorem.

Key Ideas

• v_p(n): exponent of p; ||n||_p = p^{-v_p(n)}
• Ultrametric: ||x+y||_p <= max(||x||_p, ||y||_p)
• Z_p: completion of Z; Q_p: completion of Q
• Ostrowski: norms on Q are |.|_inf and |.|_p
• Hensel: simple root mod p lifts to Z_p

Further Directions

These ideas open paths to deeper number theory.

• arith-29-adeles — extends

Вопросы для размышления

• Give a concrete example.
• How does this connect to other areas of number theory?