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A $p$-adic interpolation of the Cogdell lift

Francesco Maria Iudica

TL;DR

The paper develops a comprehensive $p$-adic interpolation framework for the Cogdell lift on Picard modular surfaces by integrating the adjoint Kudla lift, higher-weight cycle constructions on Kuga–Sato varieties, and Loeffler’s $\Lambda$-adic machinery. It establishes a $p$-adic Rallis inner product formula linking the adjoint and Kudla lifts, constructs higher-weight cycles whose generating series are modular, and assembles a $\Lambda$-adic family of special cycles with a big pairing that yields a $\Lambda$-adic Cogdell theorem. The results collectively interpolate Cogdell lifts across weight and level, connecting geometric cycle theory, unitary theta correspondences, and $p$-adic families. This advances the arithmetic understanding of how special cycles on Picard modular surfaces encode automorphic forms in a $p$-adic analytic family with potential applications to $p$-adic $L$-functions and unitary Shimura varieties.

Abstract

In this paper we obtain several results related to the $p$-adic interpolation of the classical Cogdell lift, mapping special cycles on Picard modular surfaces to elliptic modular forms. The results have a three-fold nature: in the first part of the paper, we $p$-adically interpolate the adjoint Kudla lift, exploiting the previously constructed $Λ$-adic Kudla lift. In the second part, we construct higher weight cycles in Kuga-Sato varieties attached to Picard modular surfaces, and show modularity of the generating series of these cycles, thus obtaining a higher weight analogue of the Cogdell lift. Finally, we apply the formalism introduced by Loeffler to construct $p$-adic analytic cohomology classes of special cycles, whose generating series is proved to be a Hida family interpolating the Cogdell lifts in the weight and level variables.

A $p$-adic interpolation of the Cogdell lift

TL;DR

The paper develops a comprehensive -adic interpolation framework for the Cogdell lift on Picard modular surfaces by integrating the adjoint Kudla lift, higher-weight cycle constructions on Kuga–Sato varieties, and Loeffler’s -adic machinery. It establishes a -adic Rallis inner product formula linking the adjoint and Kudla lifts, constructs higher-weight cycles whose generating series are modular, and assembles a -adic family of special cycles with a big pairing that yields a -adic Cogdell theorem. The results collectively interpolate Cogdell lifts across weight and level, connecting geometric cycle theory, unitary theta correspondences, and -adic families. This advances the arithmetic understanding of how special cycles on Picard modular surfaces encode automorphic forms in a -adic analytic family with potential applications to -adic -functions and unitary Shimura varieties.

Abstract

In this paper we obtain several results related to the -adic interpolation of the classical Cogdell lift, mapping special cycles on Picard modular surfaces to elliptic modular forms. The results have a three-fold nature: in the first part of the paper, we -adically interpolate the adjoint Kudla lift, exploiting the previously constructed -adic Kudla lift. In the second part, we construct higher weight cycles in Kuga-Sato varieties attached to Picard modular surfaces, and show modularity of the generating series of these cycles, thus obtaining a higher weight analogue of the Cogdell lift. Finally, we apply the formalism introduced by Loeffler to construct -adic analytic cohomology classes of special cycles, whose generating series is proved to be a Hida family interpolating the Cogdell lifts in the weight and level variables.
Paper Structure (20 sections, 33 theorems, 175 equations)

This paper contains 20 sections, 33 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. The unitary Shimura variety of signature $(2,1)$
  3. PEL type moduli spaces
  4. Cogdell's Theorem
  5. The adjoint Kudla lift
  6. The analogue of a result of Zagier and Oda
  7. The $p$-adic interpolation of the adjoint Kudla lift
  8. Analytic weight spaces
  9. A $p$-adic Rallis inner product formula
  10. Fiber varieties
  11. Compactifications
  12. Cohomology
  13. Higher weight cycles
  14. A $\Lambda$-adic Family of Special Cycles
  15. Loeffler's machinery
  16. ...and 5 more sections

Key Result

Theorem 1

If the level $\mathcal{U}^p$ is neat, the above moduli problem is represented by a smooth, quasi-projective scheme $S_{\mathcal{U}_f}$ over $\mathcal{O}_{K,\mathfrak{p}}$.

Theorems & Definitions (65)

  • Definition 1
  • Theorem 1
  • Lemma 2
  • proof
  • Remark 1
  • Theorem 3: Cogdell
  • Theorem 4
  • proof
  • Remark 2
  • Proposition 1
  • ...and 55 more