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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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##### References:
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