Abstract
ith respect to the analytic‐algebraic dichotomy, the theory of Siegel modular forms of half‐integral weight is lopsided; the analytic theory is strong, whereas the algebraic lags behind. In this paper, we capitalise on this to establish the fundamental object needed for the analytic side of the Iwasawa main conjecture — the p ‐adic L ‐function obtained by interpolating the complex L ‐function at special values. This is achieved through the Rankin–Selberg method and the explicit Fourier expansion of non‐holomorphic Siegel Eisenstein series. The construction of the p ‐stabilisation in this setting is also of independent interest.
Highlights
With respect to the analytic-algebraic dichotomy, the theory of Siegel modular forms of halfintegral weight is lopsided; the analytic theory is strong, whereas the algebraic lags behind
This section runs through the very basics of the modular forms that we study and their Fourier expansions are detailed
Elements of Cκ∞(Γ) and Mκ(Γ) have Fourier expansions summing over positive semi-definite symmetric matrices, the precise forms for which are given later
Summary
This section runs through the very basics of the modular forms that we study and their Fourier expansions are detailed. Elements of Cκ∞(Γ) and Mκ(Γ) have Fourier expansions summing over positive semi-definite symmetric matrices, the precise forms for which are given later . If f ∈ Mκ(Γ, ψ), its adelisation fA : pr−1(GA) → C is fA(x) := ψc(|dw|)(f ||κw)(i), where x = αw for α ∈ G and w ∈ pr−1(D[b−1, bc]), and i = iIn. To give the precise Fourier expansions of these forms, define the following spaces of symmetric matrices:. Consider b fixed in the definitions of Γ = Γ[b−1, bc], so that this group depends only on c, and let ψ be a normalised Hecke character satisfying (2.6) and (2.7). This integral is convergent whenever one of f, g belongs to Sκ(Γ, ψ)
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