A formal Mellin transform in the arithmetic of function fields
Author:
David Goss
Journal:
Trans. Amer. Math. Soc. 327 (1991), 567582
MSC:
Primary 11R58; Secondary 11S80, 11T55
MathSciNet review:
1041048
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The Mellin transform is a fundamental tool of classical arithmetic. We would also like such a tool in the arithmetic of function fields based on Drinfeld modules, although a construction has not yet been found. One formal approach to finding Mellin transforms in classical theory is through adic measures. It turns out that this approach also works for function fields. Thus this paper is devoted to exploring what can be learned this way. We will establish some very enticing connections with gamma functions and the KummerVandiver conjecture for function fields.
 [1]
ErnstUlrich
Gekeler, On power sums of polynomials over finite fields, J.
Number Theory 30 (1988), no. 1, 11–26. MR 960231
(89k:11122), http://dx.doi.org/10.1016/0022314X(88)900236
 [2]
David
Goss, The arithmetic of function fields. II. The
“cyclotomic” theory, J. Algebra 81
(1983), no. 1, 107–149. MR 696130
(86k:11061), http://dx.doi.org/10.1016/00218693(83)902120
 [3]
David
Goss, The theory of totally real function fields, theory,
Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer.
Math. Soc., Providence, RI, 1986, pp. 449–477. MR 862648
(87m:11116), http://dx.doi.org/10.1090/conm/055.2/1862648
 [4]
David
Goss and Warren
Sinnott, Special values of Artin 𝐿series, Math. Ann.
275 (1986), no. 4, 529–537. MR 859327
(87k:11127), http://dx.doi.org/10.1007/BF01459134
 [5]
David
Goss, The Γfunction in the arithmetic of function
fields, Duke Math. J. 56 (1988), no. 1,
163–191. MR
932861 (89d:11106), http://dx.doi.org/10.1215/S0012709488056086
 [6]
David
Goss, Fourier series, measures and divided power series in the
theory of function fields, 𝐾Theory 2 (1989),
no. 4, 533–555. MR 990575
(90i:11138), http://dx.doi.org/10.1007/BF00533281
 [7]
David
Goss, Harmonic analysis and the flow of a Drinfel′d
module, J. Algebra 146 (1992), no. 1,
219–241. MR 1152442
(93a:11052), http://dx.doi.org/10.1016/00218693(92)90065T
 [8]
Nicholas
M. Katz, 𝑝adic 𝐿functions via moduli of elliptic
curves, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29,
Humboldt State Univ., Arcata, Calif., 1974), Amer. Math. Soc., Providence,
R. I., 1975, pp. 479–506. MR 0432649
(55 #5635)
 [9]
Serge
Lang, Cyclotomic fields, SpringerVerlag, New York, 1978.
Graduate Texts in Mathematics, Vol. 59. MR 0485768
(58 #5578)
 [10]
Dinesh
S. Thakur, Zeta measure associated to
𝐹_{𝑞}[𝑇], J. Number Theory 35
(1990), no. 1, 1–17. MR 1054555
(91e:11139), http://dx.doi.org/10.1016/0022314X(90)901006
 [1]
 E.U. Gekeler, On power sums of polynomials over finite fields, J. Number Theory 30 (1988), 1127. MR 960231 (89k:11122)
 [2]
 D. Goss, The arithmetic of function fields : The 'cyclotomic' theory, J. Algebra 81 (1983), 107149. MR 696130 (86k:11061)
 [3]
 , The theory of totallyreal function fields, Applications of Algebraic Theory to Algebraic Geometry, Vol. 55, part II, Amer. Math. Soc., Providence, R. I., 1986, pp. 449477. MR 862648 (87m:11116)
 [4]
 D. Goss and W. Sinnott, Special values of Artin series, Math. Ann. 275 (1986), 529537. MR 859327 (87k:11127)
 [5]
 D. Goss, The function in the arithmetic of function fields, Duke Math. J. 56 (1988), 163191. MR 932861 (89d:11106)
 [6]
 , Fourier series, measures and divided power series in the theory of function fields, Theory 1 (1989), 533555. MR 990575 (90i:11138)
 [7]
 , Harmonic analysis and the flow of a Drinfeld module, J. Algebra (to appear). MR 1152442 (93a:11052)
 [8]
 N. Katz, adic functions via moduli of elliptic curves, Proc. Sympos. Pure Math., vol. 29, Amer. Math. Soc., Providence, R. I., 1975, pp. 479506. MR 0432649 (55:5635)
 [9]
 S. Lang, Cyclotomic fields, Springer, 1978. MR 0485768 (58:5578)
 [10]
 D. Thakur, Zeta measure associated to , J. Number Theory 35 (1990) 117. MR 1054555 (91e:11139)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
11R58,
11S80,
11T55
Retrieve articles in all journals
with MSC:
11R58,
11S80,
11T55
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199110410485
PII:
S 00029947(1991)10410485
Keywords:
Mellin transforms,
nonarchimedean measures,
divided power series,
hyperderivatives,
zeta functions,
gamma functions,
magic numbers
Article copyright:
© Copyright 1991 American Mathematical Society
