[Paper Review] Spectacle cycles with coefficients and modular forms of half-integral weight

Jens Funke, John J. Millson|arXiv (Cornell University)|Aug 26, 2011

Advanced Algebra and Geometry16 references3 citations

TL;DR

This paper introduces 'spectacle cycles'—capped modular symbols with nontrivial coefficients—to extend the Shintani lift from cusp forms to arbitrary modular forms, including Eisenstein series. It proves that the generating series of cohomological periods over these cycles is a modular form of half-integral weight, generalizing Kudla-Millson duality to non-compact settings via a new simplicial homology theory with non-locally constant local systems.

ABSTRACT

In this paper we present a geometric way to extend the Shintani lift from even weight cusp forms for congruence subgroups to arbitrary modular forms, in particular Eisenstein series. This is part of our efforts to extend in the noncompact situation the results of Kudla-Millson and Funke-Millson relating Fourier coefficients of (Siegel) modular forms with intersection numbers of cycles (with coefficients) on orthogonal locally symmetric spaces. In the present paper, the cycles in question are the classical modular symbols with nontrivial coefficients. We introduce "capped" modular symbols with coefficients which we call "spectacle cycles" and show that the generating series of cohomological periods of any modular form over the spectacle cycles is a modular form of half-integral weight. In the last section of the paper we develop a new simplicial homology theory with local coefficients (that are not locally constant) that allows us to extend the above results to orbifold quotients of the upper half plane.

Motivation & Objective

• To extend the Shintani lift from cusp forms to arbitrary modular forms, including Eisenstein series, in the non-compact setting.
• To generalize the Kudla-Millson theta lift to cohomology groups capturing boundary behavior via relative cycles.
• To develop a simplicial homology theory with non-locally constant local coefficients to handle orbifold quotients of the upper half-plane.
• To provide a geometric interpretation of the Hirzebruch-Zagier result using capped cycles and correction terms in the theta series.
• To establish that the generating series of cohomological periods over spectacle cycles yields a modular form of half-integral weight.

Proposed method

• Introduce 'spectacle cycles' as capped modular symbols with coefficients in symmetric power local systems $\widetilde{E_{2k}}$.
• Construct a mapping cone de Rham complex on $\overline{X}$ with boundary $\partial\overline{X}$ to model relative cohomology with nontrivial coefficients.
• Define a cohomology class $[a,b]$ in compactly supported cohomology $H^i_c(X,E)$ via a form $\alpha$ satisfying $a - d(f\tilde{b}) = \alpha + d\mu$.
• Use a retraction map $f$ on a collar neighborhood of $\partial\overline{X}$ to extend forms and define the correction term via $f\tilde{b}$.
• Establish an isomorphism $\overline{F}: H^\bullet(C) \to H^\bullet_c(X,E)$ via the map $F(a,b) = [a,b]$, showing compatibility with cohomology.
• Derive integral formulas for Kronecker pairings using $\int_{\overline{X}} \eta \wedge \alpha = \int_{\overline{X}} \eta \wedge a - \int_{\partial\overline{X}} i^*\eta \wedge b$.

Experimental results

Research questions

• RQ1 Can the Shintani lift be extended from cusp forms to arbitrary modular forms, including Eisenstein series, in the non-compact case?
• RQ2 How can the Kudla-Millson theta lift be generalized to cohomology groups that include boundary contributions?
• RQ3 What is the role of non-locally constant local systems in defining cycles and their cohomological periods?
• RQ4 How can a simplicial homology theory be developed to handle non-trivial local coefficients in orbifold quotients?
• RQ5 Can the generating series of periods over capped modular symbols be shown to be a modular form of half-integral weight?

Key findings

• The generating series of cohomological periods over spectacle cycles is a modular form of half-integral weight in $S_{k+3/2}(\Gamma') \otimes H^1(X, \widetilde{E_{2k}})$.
• The map $F: (a,b) \mapsto [a,b]$ induces an isomorphism $\overline{F}: H^\bullet(C) \to H^\bullet_c(X,E)$, establishing a cohomological duality.
• The integral formula $\langle[\eta],[a,b]\rangle = \int_{\overline{X}} \eta \wedge a - \int_{\partial\overline{X}} i^*\eta \wedge b$ provides a geometric pairing for Kronecker pairings.
• The construction allows the extension of the theta lift to full cohomology, including boundary contributions, by capping relative cycles $C_n$ to form absolute cycles $C_n^c$.
• The method recovers the Hirzebruch-Zagier result as a difference of two non-holomorphic modular forms, now interpreted topologically via the mapping cone complex.
• The new simplicial homology theory with non-locally constant coefficients enables the extension of results to orbifold quotients of the upper half-plane.