Function Field Arithmetic

Hypergeometric functions for function fields and transcendence

“…These polynomials [j ], D j and L j are fundamental for the arithmetic on F q [T ] and also occur in the formulas for Carlitz module structure, and in the expansions of its exponential, logarithm, etc. For more details on this subject, see [8] and [19]. Now we recall the definition of r F s , the hypergeometric function for function fields introduced in [14].…”

“…In [14] and [19], there is another analog r F s (just as there are two kinds of Zeta, Gamma functions and cyclotomic theories in the function field context [19]) of hypergeometric functions, where the parameters are in C ∞ . For these analogs, (a) n := e n (a) := (a − f ), with f running over all polynomials of degree less than n. Since this is zero for large n, when a itself is a polynomial, the bad or trivial cases now are for a ∈ F q [T ].…”

“…Theorem 4 is proved using some results of the first author and Anderson on the 'solitons' of Anderson (see [3,19,16], and especially [18] for the background on the name and references).…”

Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic

“…Note that when we regard C r as a Drinfeld ‫ކ‬ p [t]-module, it is of rank r [Goss 1996;Thakur 2004]. One has the Carlitz exponential associated to C r :…”

Quasi-holonomic modules in positive characteristic

“…Analysis over K and K c , which was initiated in the great paper by Carlitz [5] and developed subsequently by Wagner, Goss, Thakur, the author, and many others (see the bibliography in [9,25]) is very different from the classical calculus. An important feature is the availability of many non-trivial additive (actually, F q -linear) polynomials and power series of the form u(t) = a k t q k .…”