Abstract
The p-adic L-function for modular forms of integral weight is well-known. For certain weights the p-adic L-function for modular forms of half-integral weight is also known to exist, via a correspondence, established by Shimura, between them and forms of integral weight. However, we construct it here without any recourse to the Shimura correspondence, allowing us to establish its existence for all weights, including those exempt from the Shimura correspondence. We do this by employing the Rankin–Selberg method, and proving explicit p-adic congruences in the resultant Rankin–Selberg expression.
Highlights
The centrality of general p-adic L-functions to the Iwasawa main conjectures almost goes without saying, as they form the backbone of the analytic side of the conjecture
Due to insufficiently developed theory the conjectures for when the modular forms are of half-integral weight are not possible to even state
The main issue in the case where f is a modular form of half-integral weight is the difficulty of developing a ‘Galois side’, which forms the second and last backbone of the Iwasawa main conjectures
Summary
The centrality of general p-adic L-functions to the Iwasawa main conjectures almost goes without saying, as they form the backbone of the analytic side of the conjecture. The main issue in the case where f is a modular form of half-integral weight is the difficulty of developing a ‘Galois side’, which forms the second and last backbone of the Iwasawa main conjectures. This paper gives some impetus to extend this to half-integral weight Siegel modular forms of higher degree n > 1 for which, crucially, there is no longer a Shimura correspondence The paper is concluded in the final section by a comparison of the integer and half-integer weight p-adic measures in accordance with the Shimura correspondence, and a discussion on the integrality of this p-adic measure
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