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Constructions of p-adic L-functions and admissible measures for Hermitian modular forms. Pantchichkine, Alexei

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For a prime p and a positive integer n, the standard zeta function LF (s) is consid- ered, attached to an Hermitian modular form F = â A(H)qH on the Hermitian upper half H plane Hm of degree n, where H runs through semi-integral positive definite Hermitian matrices of degree n, i.e. H â Î m(O) over the integers O of an imaginary quadratic field K, where qH = exp(2Ï iTr(HZ)). Analytic p-adic continuation of their zeta functions constructed by A.Bouganis in the ordinary case, is extended to the admissible case via growing p-adic measures. Previously this problem was solved for the Siegel modular forms. Main result is stated in terms of the Hodge polygon PH(t) : [0,d] â R and the Newton polygon PN(t) = PN,p(t) : [0,d] â R of the zeta function LF (s) of degree d = 4n. Main theorem gives a p-adic analytic interpolation of the L values in the form of certain integrals with respect to Mazur-type measures.

Related references:
[BS00] B Ì ocherer, S., and Schmidt, C.-G., p-adic measures attached to Siegel modular forms, Ann. Inst. Fourier 50, N. 5, 1375â 1443 (2000).

[Bou16] Bouganis T., p-adic Measures for Hermitian Modular Forms and the Rankinâ Selberg Method. in Elliptic Curves, Modular Forms and Iwasawa Theory â Conference in honour of the 70th birthday of John Coates, pp 33â 86

[CourPa] Courtieu M, Panchishkin A. A, Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms, Lecture Notes in Mathematics 1471, Springer-Verlag, 2004 (2nd augmented ed.)

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