Computing L-Invariants for the Symmetric Square of an Elliptic Curve

Abstract

Let E be an elliptic curve over $\mathbb{Q}$, and $p\ne 2$ a prime of good ordinary reduction. The p-adic L-function for ${\text{Sym}}^{2}E$ always vanishes at s = 1, even though the complex L-function does not have a zero there. The $\mathcal{L}$-invariant itself appears on the right-hand side of the formula $$\frac{\mathrm{d}}{\mathrm{d}s}{\mathbf{L}}_p({\text{Sym}}^{2}E,s){|}_{s=1}={\mathcal{L}}_p({\text{Sym}}^{2}E)\times (1-{\alpha }_p^{-2})(1-p{\alpha }_p^{-2})\times \frac{L_{\infty }({\text{Sym}}^{2}E,1)}{{(2\pi i)}^{-1}{\Omega }_E^{+}{\Omega }_E^{-}}$$ where $X^2-a_p(E)X+p=(X-{\alpha }_p)(X-{\beta }_p)$ with ${\alpha }_p\in {\mathbb{Z}}_p^{\times }$. We first devise a method to calculate ${\mathcal{L}}_p({\text{Sym}}^{2}E)$ effectively, then show it is non-trivial for all elliptic curves E of conductor $N_E\le 300$ with $4|N_E$, and almost all ordinary primes p < 17. Hence, in these cases at least, the order of the zero in ${\mathbf{L}}_p({\text{Sym}}^{2}E,s)$ at s = 1 is exactly one.

Keywords:

• Elliptic curves
• Iwasawa theory
• L-functions
• deformation theory
• modular forms

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 11R23
• Secondary 11E95
• 11G40

Acknowledgments

The first author warmly thanks John Coates and Ralph Greenberg for originally suggesting this problem. Both authors are also grateful to Antonio Lei, Max Flander, and Denis Benois for their helpful suggestions on computing symmetric square $\mathcal{L}$-invariants. They are also grateful to the referee for their insights, in particular the theoretical approach suggested in Section 2.5 (although we have been unable to implement this approach in practice).

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 In the case of split multiplicative reduction the $\mathcal{L}$-invariant for Sym${}^{2}E$ is the same as the $\mathcal{L}$-invariant for E, and it is further conjectured (by Greenberg) that the $\mathcal{L}$-invariants for Sym${}^{m}E$ should be independent of m > 0.

2 We also computed ${\mathcal{L}}_p({\text{Sym}}^{2}E)$ for $E=304e1$ at the good ordinary prime p = 5, using an identical method. In fact ${\mathcal{L}}_5({\text{Sym}}^2(304e1))={\mathcal{L}}_5({\text{Sym}}^2(19a1))$ because $E\otimes {\varpi }_2$ is $\mathbb{Q}$-isogenous to $19a1$; thankfully, the value we obtained numerically agreed with the 5-adic expansion for ${\mathcal{L}}_5(19a1)$ given in [Dummit et al. 16, p. 52], at the weight $k+2=2$.

Additional information

Funding

The second named author is financially supported by a University of Waikato PhD scholarship.