Period(d)ness of L-Values
In our recent work with Rogers on resolving some of Boyd’s conjectures on two-variate Mahler measures, a new analytical machinery was introduced to write the values L(E, 2) of L-series of elliptic curves as periods in the sense of Kontsevich and Zagier. Here we outline, in slightly more general settings, the novelty of our method with Rogers and provide two illustrative period evaluations of L(E, 2) and L(E, 3) for a conductor 32 elliptic curve E.
Key wordsModular form L-series Period Arithmetic differential equation Dedekind’s eta function
I am thankful to Anton Mellit, Mat Rogers, Evgeny Shinder, Masha Vlasenko and James Wan for fruitful conversations on the subject and to Don Zagier for his encouragement to isolate the L-series transformation part from [8, 9]. I thank the anonymous referee for his careful reading of an earlier version and for his suggesting some useful corrections.
The principal part of this work was done during my visit in the Max Planck Institute for Mathematics (Bonn). I would like to thank the staff of the institute for hospitality and enjoyable working conditions.
This work is supported by the Max Planck Institute for Mathematics (Bonn, Germany) and the Australian Research Council (grant DP110104419).
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