Arc (projective geometry)

In mathematics, the Artin–Hasse exponential, named after Emil Artin and Helmut Hasse, is the power series given by

$E_p(x)=\exp (x+\frac{x^p}{p}+\frac{x^{p^2}}{p^2}+\frac{x^{p^3}}{p^3}+\cdots ).$

Motivation

Unlike the traditional exponential series, the coefficients are p-integral; in other words, their denominators are not divisible by p. This follows from Dwork's lemma, which says that a power series f(x) = 1 + ... with rational coefficients has p-integral coefficients if and only if f(x^p)/f(x)^p ≡ 1 mod p.

Following this definition, by Möbius inversion it can be written as the infinite product

$E_p(x)=\prod _{(p,n)=1}(1-x^n{)}^{-\mu (n)/n}.$

(The function μ is the Möbius function.) This follows from the analogous relation for the regular exponential series, in the sense that taking this product over all n rather than only n prime to p is an infinite product which converges (in the ring of formal power series) to the exponential series.

Combinatorial interpretation

The Artin–Hasse exponential is the generating function for the probability a uniformly randomly selected element of S_n (the symmetric group with n elements) has p-power order (the number of which is denoted by t_p,n):

$E_p(x)=\sum _{n\ge 0}\frac{t_{p,n}}{n!}{x}^n.$

This gives another proof that the coefficients are p-integral, using the fact that in a finite group of order divisible by d the number of elements of order dividing d is also divisible by d.

More generally, for any topologically finitely generated profinite group G there is an identity

$\exp (\sum _{H\subset G}{x}^{[G:H]}/[G:H])=\sum _{n\ge 0}\frac{a_{G,n}}{n!}{x}^n,$

where H runs over open subgroups of G with finite index (there are finitely many of each index since G is topologically finitely generated) and a_G,n is the number of continuous homomorphisms from G to S_n. Two special cases are worth noting. (1) If G is the p-adic integers, it has exactly one open subgroup of each p-power index and a continuous homomorphism from G to S_n is essentially the same thing as choosing an element of p-power order in S_n, so we have recovered the above combinatorial interpretation of the Taylor coefficients in the Artin–Hasse exponential series. (2) If G is a finite group then the sum in the exponential is a finite sum running over all subgroups of G, and continuous homomorphisms from G to S_n are simply homomorphisms from G to S_n. The result in this case is due to Wohlfahrt (1977). The special case when G is a finite cyclic group is due to Chowla, Herstein, and Scott (1952), and takes the form

$\exp (\sum _{d|m}{x}^d/d)=\sum _{n\ge 0}\frac{a_{m,n}}{n!}{x}^n,$

where a_m,n is the number of solutions to g^m = 1 in S_n.

David RobertsTemplate:Dn provided a natural combinatorial link between the Artin–Hasse exponential and the regular exponential in the spirit of the ergodic perspective (linking the p-adic and regular norms over the rationals) by showing that the Artin–Hasse exponential is also the generating function for the probability that an element of the symmetric group is unipotent in characteristic p, whereas the regular exponential is the probability that an element of the same group is unipotent in characteristic zero.

Conjectures

At the 2002 PROMYS program, Keith Conrad conjectured that the coefficients of $E_p(x)$ are uniformly distributed in the p-adic integers with respect to the normalized Haar measure, with supporting computational evidence. The problem is still open.

Dinesh Thakur has also posed the problem of whether the Artin–Hasse exponential reduced mod p is transcendental over ${\mathbb{𝔽}}_p(x)$.

Various other relatively simple properties of the functions are also open, including whether it satisfies the traditional exponential functional equation $E_p(a{)}^b=E_p(ab)$ and analogous relations defined for a general exponential by utilizing its inverse, the Artin–Hasse logarithm.

See also

• Witt vector
• Formal group

References

• A course in p-adic analysis, by Alain M. Robert
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