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Small Algebraic Central Values of Twists of Elliptic $L$-Functions

Hershy Kisilevsky, Jungbae Nam

Abstract

We consider heuristic predictions for small non-zero algebraic central values of twists of the $L$-function of an elliptic curve $E/\mathbb{Q}$ by Dirichlet characters. We provide computational evidence for these predictions and consequences of them for instances of an analogue of the Brauer-Siegel theorem associated to $E/\mathbb{Q}$ extended to chosen families of cyclic extensions of fixed degree.

Small Algebraic Central Values of Twists of Elliptic $L$-Functions

Abstract

We consider heuristic predictions for small non-zero algebraic central values of twists of the -function of an elliptic curve by Dirichlet characters. We provide computational evidence for these predictions and consequences of them for instances of an analogue of the Brauer-Siegel theorem associated to extended to chosen families of cyclic extensions of fixed degree.

Paper Structure

This paper contains 23 sections, 2 theorems, 103 equations, 73 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Arithmetic Hypotheses
  3. Statistical Hypotheses
  4. Predictions
  5. Main Prediction
  6. Consequences of the Main Prediction
  7. Brauer-Siegel Quotients
  8. Preliminaries and Notation
  9. Number of primitive characters
  10. Algebraic central values
  11. Probabilities for non-zero values
  12. Some Consequences
  13. For the case that $c = 0$
  14. For the case that $0 < c < \phi(k)/4$
  15. Brauer-Siegel Limits
  16. ...and 8 more sections

Key Result

Proposition 3.1

Let $E/\mathbb{Q}$ be an elliptic curve defined over $\mathbb{Q}$, and let $\chi$ be a primitive Dirichlet character of order $k \geq 3$ and conductor $\mathfrak f_{\chi}.$ Let $\zeta_{\chi}=w_E\chi(-N_E).$ Then where and $\alpha_{\chi} \in \mathcal{O}_{\chi}^+$ are real cyclotomic integers. Also we have for all $\sigma\in\operatorname{Gal}(\mathbb{Q}(\chi)/\mathbb{Q}).$

Figures (73)

  • Figure 1: Ratio \ref{['ratio_A']}: $L =$ 1, 2, 3 for $k =$ 3, 6 and $L =$ 1, 4, 5 for $k =$ 5 from the bottom to the top. Note that for each $E, k$ and a fixed $X$, the ratio in Equation \ref{['ratio_A']} for $L$ is less than or equal to that for $L'$ if $L \le L$ since $0 < |A_{\chi}| \le L \le L'$.
  • Figure 2: Ratio \ref{['ratio_A']}: $L =$ 1, 2, 3 for $k =$ 3, 6 and $L =$ 1, 4, 5 for $k =$ 5 from the bottom to the top. Note that for each $E, k$ and a fixed $X$, the ratio in Equation \ref{['ratio_A']} for $L$ is less than or equal to that for $L'$ if $L \le L$ since $0 < |A_{\chi}| \le L \le L'$.
  • Figure 3: Ratio \ref{['ratio_A']}: $L =$ 1, 2, 3 for $k =$ 3, 6 and $L =$ 1, 4, 5 for $k =$ 5 from the bottom to the top. Note that for each $E, k$ and a fixed $X$, the ratio in Equation \ref{['ratio_A']} for $L$ is less than or equal to that for $L'$ if $L \le L$ since $0 < |A_{\chi}| \le L \le L'$.
  • Figure 4: Ratio \ref{['ratio_c']} for $k = 3, 5, 6$ and $\phi(k)/4 -1 < c \le \phi(k)/4$
  • Figure 5: Ratio \ref{['ratio_c']} for $k = 7, 13$ and $\phi(k)/4 -1 \le c \le \phi(k)/4$
  • ...and 68 more figures

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 3.1
  • Remark 5
  • Lemma 4.1
  • proof
  • Remark 6