Theta functions can be viewed as the canonical section of the determinant of cohomology. The corresponding notion in the etale topology would be L-series. “All known constructions of p-adic L-functions seem to be related to some situation where some etale cohomology has vanishing Euler char.
p-adic regulator and p-adic L-functions intro in Washington, in cyclotomic folder
p-adic L-functions: lectures of Kato, and of Iwasawa, in Iwasawa theory folder
[arXiv:1207.2289] On special zeros of -adic -functions of Hilbert modular forms fra arXiv Front: math.NT av Michael Spiess Let be a modular elliptic curve over a totally real number field . We prove the weak exceptional zero conjecture which links a (higher) derivative of the -adic -function attached to to certain -adic periods attached to the corresponding Hilbert modular form at the places above where has split multiplicative reduction. Under some mild restrictions on and the conductor of we deduce the exceptional zero conjecture in the strong form (i.e.\ where the automorphic -adic periods are replaced by the -invariants of defined in terms of Tate periods) from a special case proved earlier by Mok. Crucial for our method is a new construction of the -adic -function of in terms of local data.
MR591682 (82c:12014) 12B40 Koblitz, Neal Fp-adic analysis: a short course on recentwork. LondonMathematical Society Lecture Note Series, 46. Cambridge University Press, Cambridge-New York, 1980. 163 pp. $14.95. ISBN 0-521-28060-5 This book gives a nice survey of some recent work on p-adic analysis and p-adic L-functions. The first chapter sketches, usually without proofs, the basics of p-adic analysis, including Newton polygons and p-adic functions. In this and later chapters it is noted how “correcting by Frobenius” extends the domain of many p-adic functions. In the second chapter, the author introduces p-adic measures, including a useful replacement for the nonexistent translation-invariant measure. He constructs the Kubota-Leopoldt p-adic L-functions and evaluates them at 1. He then defines the p-adic gamma function and expresses the derivative of the p-adic L-function at 0 in terms of the p-adic log gamma function (a result due to B. Ferrero and R. Greenberg). The proof, using twisted log gamma functions, is the same as that in a paper of the author [Duke Math. J. 46 (1979), no. 2, 455–468; MR0534062 (80f:12011)]. However, the relation also follows trivially from the easily proved formula of the reviewer http://www.ams.org/mathscinet-getitem?mr=0406982 expressing the p-adic L-functions in terms of the p-adic log gamma function [see S. Lang, Cyclotomic fields, II, Springer, New York, 1980; MR0566952 (81i:12004)]. In the third chapter the author first treats Gauss and Jacobi sums, L-series for algebraic varieties over finite fields, and Fermat and Artin-Schreier curves and their cohomology. He then proves the Gross-Koblitz formula [B. H. Gross and N. Koblitz, Ann. of Math. (2) 109 (1979), no. 3, 569–581; MR0534763 (80g:12015)], relating Gauss sums and the p-adic gamma function. Stickelberger’s theorem on the factorization of Gauss sums follows as a corollary (Stickelberger’s name is uniformly misspelled). The proof of the Gross-Koblitz formula is similar to the proof given by Lang [op. cit.]; however, the present exposition asks the reader to believe certain cohomological facts, while Lang’s book uses the Dwork trace formula to obtain the Gauss sums, hence is self-contained. The fourth chapter discusses the p-adic regulators of Leopoldt and of Gross. It is proved that Leopoldt’s regulator does not vanish for abelian extensions of Q, and Gross’s conjectures on the behavior of p-adic L-functions at 0 are discussed. The book concludes with an appendix which treats the ˇ Snirel0man integral and the p-adic Stieltjes transform, the Viˇsik transform (the inverse of the Stieltjes transform), and the p-adic spectral theorem.
Good article: MR1327803 (96e:11062) Perrin-Riou, Bernadette(F-PARIS11) Fonctions -adiques des représentations -adiques. (French. English, French summary) [-adic -functions of -adic representations] Astérisque No. 229 (1995), 198 pp.
MR2096804 (2005h:11135) Bars, Francesc(E-BARA) A relation between -adic -functions and the Tamagawa number conjecture for Hecke characters.
MR2111647 (2005i:11080) Colmez, Pierre La conjecture de Birch et Swinnerton-Dyer -adique (Looks like an excellent survey)
http://mathoverflow.net/questions/10603/does-p-adic-l-function-determine-the-l-function
http://mathoverflow.net/questions/13287/special-values-of-p-adic-l-functions
Soule on cyclotomic units in Asterisque 147-148 http://www.ams.org/mathscinet-getitem?mr=891430, seems to be a really nice paper, on groups of cyclotomic units in odd K-groups of rings of integers, and relation to p-adic regulators and p-adic L-functions.
nLab page on p-adic L-function