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Kudla-Millson lift of toric cycles and restriction of Hilbert modular forms

Romain Branchereau

TL;DR

The paper develops a Kudla-Millson theta lift framework for cycles attached to anisotropic maximal $\\mathbb{Q}$-tori in an orthogonal group of signature $(p,q)$ with $p\\ge q>0$, showing that the lift of toric cycles is always a cusp form and arises as a diagonal restriction of a parallel-weight Hilbert modular form. Using a seesaw argument and restriction-of-scalars, the authors express these diagonal restrictions as products of diagonally restricted Hilbert modular forms, leading to a dimension formula relating the space of cusp forms to the spans of toric and special cycles. They provide explicit computations of the Hilbert modular forms in the torus setting, prove cusp-ness via positivity conditions, and illustrate the constructions with biquadratic and more general étale-algebra examples. The results connect geometric intersection theory on locally symmetric spaces with arithmetic information encoded in Hilbert modular forms, enabling a precise comparison of the spans of diagonal restrictions and toric cycles. Overall, the work clarifies how toric cycles contribute to the cusp-spectrum via KM lifts and offers concrete mechanisms to compute and compare their contributions across S-arithmetic settings.

Abstract

Let $V$ be quadratic space of even dimension and of signature $(p, q)$ with $p \geq q > 0$. We show that the Kudla-Millson lift of toric cycles - attached to algebraic tori - is a cusp form that is the diagonal restriction of a Hilbert modular form of parallel weight one. We deduce a formula relating the dimension of the span of such diagonal restrictions and the dimension of the span of toric and special cycles.

Kudla-Millson lift of toric cycles and restriction of Hilbert modular forms

TL;DR

The paper develops a Kudla-Millson theta lift framework for cycles attached to anisotropic maximal -tori in an orthogonal group of signature with , showing that the lift of toric cycles is always a cusp form and arises as a diagonal restriction of a parallel-weight Hilbert modular form. Using a seesaw argument and restriction-of-scalars, the authors express these diagonal restrictions as products of diagonally restricted Hilbert modular forms, leading to a dimension formula relating the space of cusp forms to the spans of toric and special cycles. They provide explicit computations of the Hilbert modular forms in the torus setting, prove cusp-ness via positivity conditions, and illustrate the constructions with biquadratic and more general étale-algebra examples. The results connect geometric intersection theory on locally symmetric spaces with arithmetic information encoded in Hilbert modular forms, enabling a precise comparison of the spans of diagonal restrictions and toric cycles. Overall, the work clarifies how toric cycles contribute to the cusp-spectrum via KM lifts and offers concrete mechanisms to compute and compare their contributions across S-arithmetic settings.

Abstract

Let be quadratic space of even dimension and of signature with . We show that the Kudla-Millson lift of toric cycles - attached to algebraic tori - is a cusp form that is the diagonal restriction of a Hilbert modular form of parallel weight one. We deduce a formula relating the dimension of the span of such diagonal restrictions and the dimension of the span of toric and special cycles.
Paper Structure (20 sections, 19 theorems, 193 equations)

This paper contains 20 sections, 19 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Intersection numbers of geodesics on modular curves
  3. Main result
  4. Spans of diagonal restrictions and toric cycles
  5. Examples
  6. Acknowledgments.
  7. Étale algebras with involutions and algebraic tori
  8. Étale algebras as quadratic spaces
  9. Étale algebras of maximal tori
  10. Kudla-Millson theta correspondence
  11. Weil representation
  12. (Co)homology of adelic spaces
  13. Kudla-Millson theta lift
  14. Restriction of scalars and seesaw
  15. Computations of the Hilbert modular form
  16. ...and 5 more sections

Key Result

Theorem 1

If $p$ splits in $\mathbb{Q}(\sqrt{D})$, the modular form of weight $2$ and level $\Gamma_0(p)$ is the diagonal restriction of a $p$-stabilized Hilbert-Eisenstein series $E^{(p)}(\psi,\tau,\tau)$ for a subgroup $\operatorname{SL}_2(\mathbb{Q}(\sqrt{D}))$.

Theorems & Definitions (42)

  • Theorem : Darmon-Pozzi-Vonk
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.2.1
  • Remark 1.4
  • Proposition 2.1
  • proof
  • ...and 32 more