Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algebraic cycles and values of Green's functions I- Products of Elliptic Curves

Ramesh Sreekantan

TL;DR

This work connects the algebraicity of values of higher Green's functions at CM points to regulator calculations of indecomposable motivic cycles on the Kummer K3 surface arising from the universal family of products of elliptic curves. By constructing an explicit infinite family of indecomposable motivic cycles on Ṽ_{E_1×E_2}, it derives algebraicity of degree-2 Green's function values via regulator-period relations, echoing Li’s Borcherds-lift perspective. The paper also outlines a precise conjectural link between motivic cycles and Borcherds lifts, and situates these results within a broader modular-complex framework, with detailed Appendix computations for a concrete n_1=1, n_2=3 case. Overall, it advances a concrete, algebraic route to Gross–Zagier–type algebraicity phenomena and lays groundwork for higher-dimensional generalizations and connections to automorphic lifts.

Abstract

Gross and Zagier defined certain `higher Green's functions' on products of modular curves and conjectured that the value of these functions at complex multiplication points should be logarithms of algebraic numbers. This is now a theorem of Li. We relate this question to the existence of motivic cycles in the universal family of products of elliptic curves along the lines of Mellit and Zhang. We then construct infinitely many such cycles. In the appendix we work out an example of algebraicity. The work of Li, Bruinier-Ehlen-Yang, Viazovska and others relate this conjecture to Borcherds' lifts of weakly holomorphic modular forms. This suggests that there should be a link between motivic cycles in the universal family on the one hand and Borcherds' lifts on the other. We formulate a precise conjecture relating the two objects.

Algebraic cycles and values of Green's functions I- Products of Elliptic Curves

TL;DR

This work connects the algebraicity of values of higher Green's functions at CM points to regulator calculations of indecomposable motivic cycles on the Kummer K3 surface arising from the universal family of products of elliptic curves. By constructing an explicit infinite family of indecomposable motivic cycles on Ṽ_{E_1×E_2}, it derives algebraicity of degree-2 Green's function values via regulator-period relations, echoing Li’s Borcherds-lift perspective. The paper also outlines a precise conjectural link between motivic cycles and Borcherds lifts, and situates these results within a broader modular-complex framework, with detailed Appendix computations for a concrete n_1=1, n_2=3 case. Overall, it advances a concrete, algebraic route to Gross–Zagier–type algebraicity phenomena and lays groundwork for higher-dimensional generalizations and connections to automorphic lifts.

Abstract

Gross and Zagier defined certain `higher Green's functions' on products of modular curves and conjectured that the value of these functions at complex multiplication points should be logarithms of algebraic numbers. This is now a theorem of Li. We relate this question to the existence of motivic cycles in the universal family of products of elliptic curves along the lines of Mellit and Zhang. We then construct infinitely many such cycles. In the appendix we work out an example of algebraicity. The work of Li, Bruinier-Ehlen-Yang, Viazovska and others relate this conjecture to Borcherds' lifts of weakly holomorphic modular forms. This suggests that there should be a link between motivic cycles in the universal family on the one hand and Borcherds' lifts on the other. We formulate a precise conjecture relating the two objects.

Paper Structure

This paper contains 36 sections, 14 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Products of Elliptic Curves
  3. Elliptic curves with full level $2$ structure.
  4. The Kummer surface of a product of elliptic curves
  5. The Kummer curve of an elliptic curve
  6. The Kummer surface of a product of elliptic curves
  7. Motivic Cycles
  8. Motivic cycles on a surface
  9. Decomposable and Indecomposable Cycles
  10. Regulators
  11. Archimedean regulator
  12. 'Non-Archimedean' regulator
  13. Applications to indecomposability
  14. Motivic cycles on the Kummer Surface
  15. Geometry of ${\mathds P}^1 \times {\mathds P}^1$
  16. ...and 21 more sections

Key Result

Proposition 3.1

Let $Q$ be a nodal rational curve on $X$ with node $P$. Let $\nu:\tilde{Q}\rightarrow Q$ be its strict transform in the blow up $\tilde{X}$ of $X$ at $P$. $\tilde{Q}$ meets the exceptional fibre ${{\mathrm E}}_P$ at two points $P_1$ and $P_2$. Both $\tilde{Q}$ and ${{\mathrm E}}_P$ are rational cur is an element of $H^3_{{\mathcal{M}}}(\tilde{X},{\mathds Q}(2))$.

Theorems & Definitions (34)

  • Conjecture 1.1
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • proof
  • ...and 24 more