Carlitz exponential explained

In mathematics, the Carlitz exponential is a characteristic p analogue to the usual exponential function studied in real and complex analysis. It is used in the definition of the Carlitz module – an example of a Drinfeld module.

Definition

We work over the polynomial ring F_q[''T''] of one variable over a finite field F_q with q elements. The completion C_∞ of an algebraic closure of the field F_q((T^-1)) of formal Laurent series in T^-1 will be useful. It is a complete and algebraically closed field.

First we need analogues to the factorials, which appear in the definition of the usual exponential function. For i > 0 we define

[i] : = T ^{q ^{i}} - T, D _{i} : = \prod _{1} [j] ^{q ^{i}}

and D₀ := 1. Note that the usual factorial is inappropriate here, since n! vanishes in F_q[''T''] unless n is smaller than the characteristic of F_q[''T''].

Using this we define the Carlitz exponential e_C:C_∞ → C_∞ by the convergent sum

e _{C(x)} : = \sum ^{inf ty} _{i =0} (x ^{q ^{i}})/(D _{i}) .

Relation to the Carlitz module

The Carlitz exponential satisfies the functional equation

e _{C(Tx)} = Te _{C(x)} + \left(e ^{q} _{C(x)\right)} = (T + \tau)e _{C(x),}

where we may view

\tau as the power of q map or as an element of the ring F _{q(T)\{\tau\}} of noncommutative polynomials. By the universal property of polynomial rings in one variable this extends to a ring homomorphism ψ:F_q[''T'']→C_∞, defining a Drinfeld F_q[''T'']-module over C_∞. It is called the Carlitz module.

References

• Book: Goss . D. . David Goss . Basic structures of function field arithmetic . . Berlin, New York . Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] . 978-3-540-61087-8 . 1423131 . 1996 . 35.
• Book: Thakur . Dinesh S. . Function field arithmetic . . New Jersey. 978-981-238-839-1 . 2091265 . 2004.