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

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Romain Branchereau at ETH Zurich
Romain Branchereau
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Abstract

Let V be quadratic space of even dimension and of signature (p, q) with p≥q>0p \ge 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.
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Mathematische Zeitschrift (2025) 309:57
https://doi.org/10.1007/s00209-025-03683-0
Mathematische Zeitschrift
Kudla Millson lift of toric cycles and restriction of Hilbert
modular forms
Romain Branchereau1
Received: 3 February 2024 / Accepted: 10 January 2025 / Published online: 12 February 2025
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2025
Abstract
Let Vbe quadratic space of even dimension and of signature (p,q)with p≥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.
Contents
1 Introduction ............................................... 2
1.1 Intersection numbers of geodesics on modular curves ....................... 2
1.2 Main result ............................................. 3
1.3 Spans of diagonal restrictions and toric cycles ........................... 5
1.4 Examples .............................................. 6
2 Étale algebras with involutions and algebraic tori ............................ 7
2.1 Étale algebras as quadratic spaces ................................. 7
2.2 Étale algebras of maximal tori ................................... 10
3 Kudla-Millson theta correspondence .................................. 12
3.1 Weil representation ......................................... 12
3.2 (Co)homology of adelic spaces ................................... 14
3.3 Kudla-Millson theta lift ....................................... 16
3.4 Restriction of scalars and seesaw .................................. 17
3.5 Computations of the Hilbert modular form ............................. 21
3.6 Generating series of intersection numbers ............................. 23
4 Example of biquadratic fields ...................................... 25
4.1 Locally symmetric space ...................................... 26
4.2 Maximal torus in GSpinV...................................... 27
5 Spans of diagonal restrictions and toric cycles ............................. 30
References .................................................. 32
BRomain Branchereau
branchereauromain.math@gmail.com
1McGill, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec H3A 0B9, Canada
123
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