## $$L$$-functions of $$p$$-adic characters.(English)Zbl 1292.11074

The authors define a $$p$$-adic character to be a continuous homomorphism from $$1+t\mathbb{F}_q[[t]]$$ to $$\mathbb{Z}_p^*$$. They give a correspondence between $$p$$-adic characters and sequences $$(c_i)_{(i,p)=1}$$ with $$c_i \in \mathbb{Z}_q$$, such that $$c_i$$ tends to zero.
The authors associate an $$L$$-function to $$p$$-adic characters and discuss the convergence of these $$L$$-series in terms of the corresponding sequence $$(c_i)$$.

### MSC:

 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 13F35 Witt vectors and related rings

### Keywords:

$$p$$-adic character; $$L$$-function
Full Text:

### References:

 [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. · Zbl 0918.11026 [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. · Zbl 0615.14013 [5] B. Dwork, Normalized period matrices, I: Plane curves , Ann. of Math. (2) 94 (1971), 337-388. · Zbl 0241.14011 [6] B. Dwork, Normalized period matrices, II , Ann. of Math. (2) 98 (1973), 1-57. · Zbl 0265.14008 [7] M. Emerton and M. Kisin, Unit $$L$$-functions and a conjecture of Katz , Ann. of Math. (2) 153 (2001), 329-354. · Zbl 1076.14027 [8] M. Hazewinkel, Formal Groups and Applications , Pure Appl. Math. 78 , Academic Press, New York, 1978. · Zbl 0454.14020 [9] L. Hesselholt, The big de Rham-Witt complex , preprint, (accessed 27 August 2013). · Zbl 1316.13028 [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, [math.NT]. · Zbl 1314.11028 [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. · Zbl 1270.11123 [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. · Zbl 0868.14011 [18] D. Wan, Dwork’s conjecture on unit root zeta functions , Ann. of Math. (2) 150 (1999), 867-927. · Zbl 1013.11031 [19] D. Wan, Higher rank case of Dwork’s conjecture , J. Amer. Math. Soc. 13 (2000), 807-852. · Zbl 1086.11030 [20] D. Wan, Rank one case of Dwork’s conjecture , J. Amer. Math. Soc. 13 (2000), 853-908. · Zbl 1086.11031 [21] D. Wan, Variation of $$p$$-adic Newton polygons for $$L$$-functions of exponential sums , Asian J. Math. 8 (2004), 427-471. · Zbl 1084.11067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.