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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).

MSC:
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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