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Transactions of the American Mathematical Society

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Derivative of the standard $ p$-adic $ L$-function associated with a Siegel form


Author: Giovanni Rosso
Journal: Trans. Amer. Math. Soc. 370 (2018), 6469-6491
MSC (2010): Primary 11F33, 11F67
DOI: https://doi.org/10.1090/tran/7138
Published electronically: April 4, 2018
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Abstract: In this paper we first construct a two-variable $ p$-adic $ L$-function for the standard representation associated with a Hida family of parallel weight genus $ g$ Siegel forms, using a method developed by Böcherer-Schmidt in one variable. When a form $ f$ has weight $ g+1$ a non-crystalline trivial zero could appear. In this case, using the two-variable $ p$-adic $ L$-function we have constructed, we can apply the method of Greenberg-Stevens to calculate the first derivative of the $ p$-adic $ L$-function for $ f$ and show that it has the form predicted by a conjecture of Greenberg on trivial zeros.


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Additional Information

Giovanni Rosso
Affiliation: DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
Address at time of publication: Department of Mathematics and Statistics, Concordia University, Montreal H3G 1M8, Canada
Email: gr385@cam.ac.uk

DOI: https://doi.org/10.1090/tran/7138
Received by editor(s): September 2, 2016
Received by editor(s) in revised form: November 11, 2016, November 23, 2016, and November 28, 2016
Published electronically: April 4, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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