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The \(p\)-adic \(L\)-functions of evil Eisenstein series. (English) Zbl 1339.11059
The authors compute the \(p\)-adic \(L\)-functions of evil Eisenstein series of weight \(k+2\), showing that they factor as product of two Kubota-Leopoldt \(p\)-adic \(L\)-functions times a logarithmic term (Theorem 1.1). The logarithmic term \(\log^{[k]}_p\) is the analytic function defined by \[ \log^{[k]}_p:= w(\sigma)(w(\sigma)- 1)\cdots(w(\sigma)- k+ 1), \] where \(\sigma: \mathbb Z^\times_p\to \mathbb C^\times_p\) is a continuous character, and \(w(\sigma):={\log_p(\sigma(1+p))\over\log_p(1+p)}\).
This proves in particular a conjecture of G. Stevens [“Rigid analytic modular symbols”, Preprint, http://math.bu.edu/people/ghs/research.d.]. The proof of this result (given in Sections 6 and 7) is quite technical, and uses both classical and recent facts concerning overconvergent modular symbols (described in Sections 2–5).

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
Full Text: DOI arXiv
[1] doi:10.1112/jlms/jds057 · Zbl 1317.11051
[3] doi:10.1007/BF01388731 · Zbl 0699.14028
[4] doi:10.1007/978-1-4613-9649-9_5
[5] doi:10.1017/CBO9780511623691
[6] doi:10.1215/S0012-7094-04-12615-6 · Zbl 1070.11016
[7] doi:10.1007/BF01231294 · Zbl 0778.11034
[8] doi:10.1017/CBO9780511721267.004
[9] doi:10.1007/s00222-005-0467-7 · Zbl 1093.11065
[10] doi:10.1017/S1474748006000028 · Zbl 1095.11025
[12] doi:10.1007/s00222-011-0358-z · Zbl 1318.11067
[13] doi:10.4007/annals.2006.163.301 · Zbl 1130.11030
[15] doi:10.1007/s002220050051 · Zbl 0851.11030
[16] doi:10.1007/BF01359701 · Zbl 0177.34901
[17] doi:10.1215/S0012-7094-86-05346-9 · Zbl 0618.10026
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