## $$p$$-adic $$L$$-functions and $$p$$-adic periods of modular forms.(English)Zbl 0778.11034

Let $$E$$ be a modular elliptic curve over $$\mathbb{Q}$$, $$p$$ be a prime for which $$E$$ has split multiplicative reduction, and $$L_ p(E,s)$$ be the $$p$$-adic $$L$$-function of $$E$$. From the interpolation property of $$L_ p(E,s)$$, it is automatically the case that $$L_ p(E,1)=0$$.
This paper proves the following formula conjectured by Mazur, Tate, and Teitelbaum: $L_ p'(E,1)= {{\log_ p(q_ E)} \over {\text{ord}_ p(q_ E)}} {{L_ \infty(E,1)} \over {\Omega_ E}}.$ Here, $$q_ E$$ is the Tate period of $$E$$ at $$p$$, $$\log_ p$$ is Iwasawa’s $$p$$-adic logarithm, $$\text{ord}_ p$$ is the normalized valuation at $$p$$, $$L_ \infty(E,s)$$ is the Hasse-Weil $$L$$-function of $$E$$, and $$\Omega_ E$$ is the real period of $$E$$. The paper actually works in the more general setting of a “split multiplicative” weight 2 newform, but the main motivation is the situation described above.
The proof studies a two variable $$p$$-adic $$L$$-function which specializes to $$L_ p(E,s)$$. The authors are actually able to determine the constant term of the two-variable $$p$$-adic $$L$$-function, from which they derive their result.
Reviewer: J.Jones (Tempe)

### MSC:

 11G05 Elliptic curves over global fields 11R23 Iwasawa theory 14G20 Local ground fields in algebraic geometry 11F85 $$p$$-adic theory, local fields 11S40 Zeta functions and $$L$$-functions
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