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Elliptic curves and Fourier coefficients of meromorphic modular forms

Pengcheng Zhang

TL;DR

This work investigates meromorphic modular forms of level $1$ with a single pole at a non-cuspidal point, focusing on forms $F_{k,C}=\frac{E_k}{j-j(C)}$ and their $p$-adic behavior under the $U_p$ operator. It demonstrates Atkin–Swinnerton-Dyer type congruences and magnetic properties, drawing deep connections to symmetric powers of the associated elliptic curve and via hypergeometric reinterpretations and the Borcherds–Shimura lift. The CM case is developed through magnetic modular forms $G_{k,D}^{(r)}$ and Shimura-lift techniques, establishing both ASD-type congruences and a robust $U_p$-Frobenius correspondence, including the $(k-2)/2$-magnetic phenomenon. The results illuminate a motivic link between meromorphic modular forms and the symmetric powers of elliptic curves, with generalizations to higher pole orders and broader weight ranges, and they provide concrete computational frameworks (hypergeometric sums, trace formulas) to verify these relations.

Abstract

We discuss several congruences satisfied by the coefficients of meromorphic modular forms, or equivalently, $p$-adic behaviors of meromorphic modular forms under the $U_p$ operator, that are summarized from numerical experiments, connecting meromorphic modular forms to symmetric powers of elliptic curves. We also provide heuristic explanations for these congruences as well as prove some of them using hypergeometric functions and the Borcherds--Shimura lift.

Elliptic curves and Fourier coefficients of meromorphic modular forms

TL;DR

This work investigates meromorphic modular forms of level with a single pole at a non-cuspidal point, focusing on forms and their -adic behavior under the operator. It demonstrates Atkin–Swinnerton-Dyer type congruences and magnetic properties, drawing deep connections to symmetric powers of the associated elliptic curve and via hypergeometric reinterpretations and the Borcherds–Shimura lift. The CM case is developed through magnetic modular forms and Shimura-lift techniques, establishing both ASD-type congruences and a robust -Frobenius correspondence, including the -magnetic phenomenon. The results illuminate a motivic link between meromorphic modular forms and the symmetric powers of elliptic curves, with generalizations to higher pole orders and broader weight ranges, and they provide concrete computational frameworks (hypergeometric sums, trace formulas) to verify these relations.

Abstract

We discuss several congruences satisfied by the coefficients of meromorphic modular forms, or equivalently, -adic behaviors of meromorphic modular forms under the operator, that are summarized from numerical experiments, connecting meromorphic modular forms to symmetric powers of elliptic curves. We also provide heuristic explanations for these congruences as well as prove some of them using hypergeometric functions and the Borcherds--Shimura lift.
Paper Structure (17 sections, 22 theorems, 179 equations, 1 table)

This paper contains 17 sections, 22 theorems, 179 equations, 1 table.

Table of Contents

  1. Introduction
  2. Simple poles
  3. Poles of higher order
  4. The CM case and magnetic modular forms
  5. Proof of \ref{['good-prime-conjecture']}
  6. A first reinterpretation
  7. Reducing to $k=4$
  8. Hypergeometric reinterpretation of $a_{p^l}(F_{4,c})$
  9. Proof of \ref{['good-prime-theorem']}
  10. Proof of \ref{['cm-middle-supercongruence']} and the $\frac{k-2}{2}$-magnetic property
  11. The Shimura lift
  12. The $U_p$ action
  13. Proof of \ref{['magnetic-theorem']} when $k/2$ is even
  14. Proof of \ref{['magnetic-theorem']} when $k/2$ is odd
  15. Other remarks
  16. ...and 2 more sections

Key Result

Theorem 1.1

Both $\frac{E_4}{j}$ and $\frac{E_4}{j-1728}$ are $1$-magnetic.

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • Conjecture 2.1
  • Remark 2.2
  • Conjecture 2.3
  • Conjecture 2.4
  • ...and 58 more