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\(p\)-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)

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
Full Text: DOI EuDML
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