2016 Theses Doctoral

# Nearly Overconvergent Forms and p-adic L-Functions for Symplectic Groups

We reformulate Shimura's theory of nearly holomorphic forms for Siegel modular forms using automorphic sheaves over Siegel varieties. This sheaf-theoretic reformulation allows us to define and study basic properties of nearly overconvergent Siegel modular forms as well as their 𝑝-adic families. Besides, it finds applications in the construction, via the doubling method, of 𝑝-adic partial standard 𝐿-functions associated to Siegel cuspidal Hecke eigensystems.

We illustrate how the sheaf-theoretic definition of nearly holomorphic forms and Maass--Shimura differential operators helps with the choice of the archimedean sections for the Siegel Eisenstein series on the doubling group Sp(4𝑛) and the study of the p-adic properties of their restrictions to Sp(2𝑛)/𝕢 × Sp(2𝑛)/𝕢. The selection of archimedean sections, together with p-adic interpolation considerations, then naturally gives the sections at the place p. We compute p-adic zeta integrals corresponding to those sections.

Finally, we construct the 𝑝-adic standard 𝐿-functions associated to ordinary families of Siegel Hecke eigensystems and obtain their interpolation properties.

## Subjects

## Files

- Liu_columbia_0054D_13342.pdf application/pdf 1.19 MB Download File

## More About This Work

- Academic Units
- Mathematics
- Thesis Advisors
- Urban, Eric Jean-Paul
- Degree
- Ph.D., Columbia University
- Published Here
- May 5, 2016