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# Factorisation of Two-variable $p$-adic $L$-functions

Published:2013-11-26
Printed: Dec 2014
• Antonio Lei,
Department of Mathematics and Statistics, Burnside Hall, McGill University, Montreal QC, H3A 0B9
 Format: LaTeX MathJax PDF

## Abstract

Let $f$ be a modular form which is non-ordinary at $p$. Loeffler has recently constructed four two-variable $p$-adic $L$-functions associated to $f$. In the case where $a_p=0$, he showed that, as in the one-variable case, Pollack's plus and minus splitting applies to these new objects. In this article, we show that such a splitting can be generalised to the case where $a_p\ne0$ using Sprung's logarithmic matrix.
 Keywords: modular forms, p-adic L-functions, supersingular primes
 MSC Classifications: 11S40 - Zeta functions and $L$-functions [See also 11M41, 19F27] 11S80 - Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.)

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