On two arithmetic theta lifts

Abstract

Our aim is to clarify the relationship between Kudla's and Bruinier's Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type, which play a key role in the arithmetic geometry of these cycles in the context of Kudla's program. In particular, we show that the generating series obtained by taking the differences of the two families of Green functions is a non-holomorphic modular form and has trivial (cuspidal) holomorphic projection.Along the way, we construct a section of the Maaß lowering operator for moderate growth forms valued in the Weil representation using a regularized theta lift, which may be of independent interest, as it in particular has applications to mock modular forms.We also consider arithmetic-geometric applications to integral models of U (n, 1) Shimura varieties. Each family of Green functions gives rise to a formal arithmetic theta function, valued in an arithmetic Chow group, that is conjectured to be modular; our main result is the modularity of the difference of the two arithmetic theta functions.Finally, we relate the arithmetic heights of the special cycles to special derivatives of Eisenstein series, as predicted by Kudla's conjecture, and describe a refinement of a theorem of Bruinier-Howard-Yang on arithmetic intersections against CM points.

The construction of Green currents and singular theta lifts for unitary groups

“…The difference between the two Green functions and its consequences in the context of the Kudla program are fairly subtle. This was studied and clarified by Ehlen and Sankaran [12]. They show in the cases of O(p, 2) respectively U(p, 1) that the difference of the generating series can be viewed as a smooth modular form of weight p 2 +1 respectively p + 1.…”

“…Following Ehlen and Sankaran [12], we generalize this setup by introducing two further spaces of modular forms, A mod k,L − and A ! k,L − .…”

“…In this section, we compare the Green forms of Kudla type G K (m, w, h), for m ∈ Q, h ∈ L /L and w ∈ R >0 , and those of Bruinier type G B (m, h) (see below). The aim is to transfer some of the results of Ehlen and Sankaran from [12] to the present setting.…”

“…Following ideas of Ehlen and Sankaran [12] we then compare the two Green forms in a different way. We show Theorem 1.4.…”

CM values of higher automorphic Green functions for orthogonal groups

“…Write T = GSpin(U ) and let K Proof Arguing as in Proposition 2.12 of [18], there is aG ∈ H ! k with L k (G(τ )) = θ N (τ, h) satisfying (1).…”