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Congruent modular forms and anticyclotomic Iwasawa theory

Dac-Nhan-Tam Nguyen

TL;DR

This work extends the congruence methods of Greenberg–Vatsal and Emerton–Pollack–Weston from cyclotomic to anticyclotomic Iwasawa theory by comparing Bertolini–Darmon–Prasanna $p$-adic $L$-functions attached to congruent modular forms. It develops a $p$-adic Gross–Zagier framework via CM points and Serre–Tate coordinates to relate $p$-adic $L$-values to Bloch–Kato logarithms of generalized Heegner classes, and proves that congruences of residual representations force congruences of $L_p$-values and Euler factors, under mild hypotheses. Consequently, the paper shows that the vanishing of the analytic $oldsymbol{ ext μ}$-invariant and equalities of $oldsymbol{ ext λ}$-invariants propagate through families, yielding propagation of the anticyclotomic Iwasawa main conjecture for the BDP Selmer group. By extending to higher weights and generalized Heegner cycles, the results connect residual congruences to arithmetic invariants in a broad anticyclotomic setting with potential applications to explicit arithmetic of generalized Heegner objects.

Abstract

Let $p$ be an odd prime. Consider normalized newforms $f_1,f_2$ that both satisfy the Heegner hypothesis for an imaginary quadratic field $K$ and suppose that they induce isomorphic residual Galois representations. In the work of Greenberg-Vatsal and Emerton-Pollack-Weston, the authors compare the cyclotomic Iwasawa $μ$ and $λ$-invariants of $f_1$ and $f_2$. We extend this to the anticyclotomic indefinite setting by comparing the BDP $p$-adic $L$-functions attached to $f_1$ and $f_2$. Using this comparison, we obtain arithmetic implications for both generalized Heegner cycles and the Iwasawa main conjecture.

Congruent modular forms and anticyclotomic Iwasawa theory

TL;DR

This work extends the congruence methods of Greenberg–Vatsal and Emerton–Pollack–Weston from cyclotomic to anticyclotomic Iwasawa theory by comparing Bertolini–Darmon–Prasanna -adic -functions attached to congruent modular forms. It develops a -adic Gross–Zagier framework via CM points and Serre–Tate coordinates to relate -adic -values to Bloch–Kato logarithms of generalized Heegner classes, and proves that congruences of residual representations force congruences of -values and Euler factors, under mild hypotheses. Consequently, the paper shows that the vanishing of the analytic -invariant and equalities of -invariants propagate through families, yielding propagation of the anticyclotomic Iwasawa main conjecture for the BDP Selmer group. By extending to higher weights and generalized Heegner cycles, the results connect residual congruences to arithmetic invariants in a broad anticyclotomic setting with potential applications to explicit arithmetic of generalized Heegner objects.

Abstract

Let be an odd prime. Consider normalized newforms that both satisfy the Heegner hypothesis for an imaginary quadratic field and suppose that they induce isomorphic residual Galois representations. In the work of Greenberg-Vatsal and Emerton-Pollack-Weston, the authors compare the cyclotomic Iwasawa and -invariants of and . We extend this to the anticyclotomic indefinite setting by comparing the BDP -adic -functions attached to and . Using this comparison, we obtain arithmetic implications for both generalized Heegner cycles and the Iwasawa main conjecture.

Paper Structure

This paper contains 16 sections, 13 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Geometric and $p$-adic modular forms
  3. CM points
  4. Anticyclotomic $p$-adic $L$-functions
  5. $t$-expansion of $p$-adic modular forms
  6. Hecke characters
  7. Anticyclotomic $p$-adic $L$-function
  8. The $\theta$ operator
  9. Congruent modular forms
  10. Hecke operators at $p$ in Serre-Tate coordinates
  11. Hecke operators at $\ell \neq p$ in Serre-Tate coordinates
  12. Applications to generalized Heegner cycles
  13. Generalized Heegner classes
  14. Bloch-Kato logarithm map for a $p$-adic Galois representation
  15. A $p$-adic Gross-Zagier formula
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose that both $f_1, f_2$ satisfy hypothesis Hyp for $K/\mathbb{Q}$. One may write $(N_1) = \frak{N}_1 \overline{\frak{N}_1}$, $(N_2) = \frak{N}_2\overline{\frak{N}_2}$ as ideals in $\mathcal{O}_K$. For each prime $\ell \mid N_1 N_2$, let $v \mid \mathfrak{N}_1 \mathfrak{N}_2$ be the correspondin where $\mathscr{P}_{\overline{v}}(f_1)$ and $\mathscr{P}_{\overline{v}}(f_2)$ are defined in Defini

Theorems & Definitions (33)

  • Theorem : A (Theorem \ref{['thm:main']})
  • Definition 2.1
  • Definition 2.2
  • Definition 4.1
  • Lemma 5.1
  • proof
  • Definition 5.2
  • Definition 5.3
  • Definition 5.4
  • Lemma 5.5
  • ...and 23 more