Nagoya Mathematical Journal

L-functions of p-adic characters

Christopher Davis and Daqing Wan

Full-text: Open access


We define a p-adic character to be a continuous homomorphism from 1+tFq[[t]] to Zp. For p>2, we use the ring of big Witt vectors over Fq to exhibit a bijection between p-adic characters and sequences (ci)(i,p)=1 of elements in Zq, indexed by natural numbers relatively prime to p, and for which lim ici=0. To such a p-adic character we associate an L-function, and we prove that this L-function is p-adic meromorphic if the corresponding sequence (ci) is overconvergent. If more generally the sequence is Clog-convergent, we show that the associated L-function is meromorphic in the open disk of radius qC. Finally, we exhibit examples of Clog-convergent sequences with associated L-functions which are not meromorphic in the disk of radius qC+ϵ for any ϵ>0.

Article information

Nagoya Math. J., Volume 213 (2014), 77-104.

First available in Project Euclid: 31 October 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
Secondary: 13F35: Witt vectors and related rings


Davis, Christopher; Wan, Daqing. $L$ -functions of $p$ -adic characters. Nagoya Math. J. 213 (2014), 77--104. doi:10.1215/00277630-2379114.

Export citation


  • [1] N. Bourbaki, Éléments de mathématique: Algèbre commutative, chapitres 8 et 9, reprint of the 1983 original, Springer, Berlin, 2006.
  • [2] R. F. Coleman, $p$-adic Banach spaces and families of modular forms, Invent. Math. 127 (1997), 417–479.
  • [3] R. Coleman and B. Mazur, “The eigencurve” in Galois Representations in Arithmetic Algebraic Geometry (Durham, England, 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge University Press, Cambridge, 1998, 1–113.
  • [4] R. Crew, $L$-functions of $p$-adic characters and geometric Iwasawa theory, Invent. Math. 88 (1987), 395–403.
  • [5] B. Dwork, Normalized period matrices, I: Plane curves, Ann. of Math. (2) 94 (1971), 337–388.
  • [6] B. Dwork, Normalized period matrices, II, Ann. of Math. (2) 98 (1973), 1–57.
  • [7] M. Emerton and M. Kisin, Unit $L$-functions and a conjecture of Katz, Ann. of Math. (2) 153 (2001), 329–354.
  • [8] M. Hazewinkel, Formal Groups and Applications, Pure Appl. Math. 78, Academic Press, New York, 1978.
  • [9] L. Hesselholt, The big de Rham–Witt complex, preprint, (accessed 27 August 2013).
  • [10] N. Katz, “Travaux de Dwork” in Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 409, Lecture Notes in Math. 317, Springer, Berlin, 1973, 167–200.
  • [11] K. S. Kedlaya, J. Pottharst, and L. Xiao, Cohomology of arithmetic families of $(\varphi,\Gamma)$-modules, preprint, arXiv:1203.5718 [math.NT].
  • [12] N. Koblitz, $p$-Adic Numbers, $p$-Adic Analysis, and zeta-Functions, 2nd ed., Grad. Texts in Math. 58, Springer, New York, 1984.
  • [13] H. Lenstra, Construction of the ring of Witt vectors, preprint, (accessed 27 August 2013).
  • [14] C. Liu and D. Wan, $T$-adic exponential sums over finite fields, Algebra Number Theory 3 (2009), 489–509.
  • [15] J. Rabinoff, The theory of Witt vectors, preprint, (accessed 27 August 2013).
  • [16] J.-P. Serre, Local Fields, Grad. Texts in Math. 67, Springer, New York, 1979.
  • [17] D. Wan, Meromorphic continuation of $L$-functions of $p$-adic representations, Ann. of Math. (2) 143 (1996), 469–498.
  • [18] D. Wan, Dwork’s conjecture on unit root zeta functions, Ann. of Math. (2) 150 (1999), 867–927.
  • [19] D. Wan, Higher rank case of Dwork’s conjecture, J. Amer. Math. Soc. 13 (2000), 807–852.
  • [20] D. Wan, Rank one case of Dwork’s conjecture, J. Amer. Math. Soc. 13 (2000), 853–908.
  • [21] D. Wan, Variation of $p$-adic Newton polygons for $L$-functions of exponential sums, Asian J. Math. 8 (2004), 427–471.