188 research outputs found
Zeta-like Multizeta Values for higher genus curves
We prove or conjecture several relations between the multizeta values for
positive genus function fields of class number one, focusing on the zeta-like
values, namely those whose ratio with the zeta value of the same weight is
rational (or conjecturally equivalently algebraic). These are the first known
relations between multizetas, which are not with prime field coefficients. We
seem to have one universal family. We also find that interestingly the
mechanism with which the relations work is quite different from the rational
function field case, raising interesting questions about the expected motivic
interpretation in higher genus. We provide some data in support of the guesses.Comment: Expository revisions plus appendices containing proofs of more cases
of conjecture
Zeta measure associated to Fq[T]
AbstractThe object of this paper is to identify the divided power series corresponding to the zeta measure associated to Fq[T]. The first section introduces the zeta function for Fq[T] and describes some of its interesting properties. In the second section, we describe results on interpolations and measures and state our main result (Theorem VII). The third section summarizes various results about power sums and zeta functions. The last section contains two proofs of the main result. In the appendix, we elaborate on the existence of zeta and beta measures and give alternate descriptions for them
Fermat-Wilson Supercongruences, arithmetic derivatives and strange factorizations
In [Tha15], we looked at two (`multiplicative' and `Carlitz-Drinfeld
additive') analogs each, for the well-known basic congruences of Fermat and
Wilson, in the case of polynomials over finite fields. When we look at them
modulo higher powers of primes, i.e. at `supercongruences', we find interesting
relations linking them together, as well as linking them with arithmetic
derivatives and zeta values. In the current work, we expand on the first analog
and connections with arithmetic derivatives more systematically, giving many
more equivalent conditions linking the two, now using `mixed derivatives' also.
We also observe and prove remarkable prime factorizations involving derivative
conditions for some fundamental quantities of the function field arithmetic.Comment: 8 page
Diophantine approximation and deformation
We associate certain curves over function fields to given algebraic power
series and show that bounds on the rank of Kodaira-Spencer map of this curves
imply bounds on the exponents of the power series, with more generic curves
giving lower exponents. If we transport Vojta's conjecture on height inequality
to finite characteristic by modifying it by adding suitable deformation
theoretic condition, then we see that the numbers giving rise to general curves
approach Roth's bound. We also prove a hierarchy of exponent bounds for
approximation by algebraic quantities of bounded degree
Algebraic independence of arithmetic gamma values and Carlitz zeta values
We consider the values at proper fractions of the arithmetic gamma function
and the values at positive integers of the zeta function for F_q[theta] and
provide complete algebraic independence results for them.Comment: 15 page
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