[Paper Review] Vanishing of L-functions of elliptic curves over number fields

Chantal David, Jack Fearnley+1 more|ArXiv.org|Jun 1, 2004

Analytic Number Theory Research13 references8 citations

TL;DR

This paper uses random matrix theory to predict the vanishing of twisted L-functions $L_E(1,\chi)$ for elliptic curves over $\mathbb{Q}$, where $\chi$ is a Dirichlet character of prime order $k$. It conjectures that for $k=3$, there are infinitely many such characters with vanishing central value, while for $k \geq 7$, only finitely many exist, with $k=5$ lying on the threshold of divergence. These predictions are linked to the rank growth of $E$ over cyclic number fields via the Birch and Swinnerton-Dyer conjecture.

ABSTRACT

Let $E$ be an elliptic curve over $\mathbb{Q}$, with L-function $L_E(s)$. For any primitive Dirichlet character $χ$, let $L_E(s, χ)$ be the L-function of $E$ twisted by $χ$. In this paper, we use random matrix theory to study vanishing of the twisted L-functions $L_E(s, χ)$ at the central value $s=1$. In particular, random matrix theory predicts that there are infinitely many characters of order 3 and 5 such that $L_E(1, χ)=0$, but that for any fixed prime $k \geq 7$, there are only finitely many character of order $k$ such that $L_E(1, χ)$ vanishes. With the Birch and Swinnerton-Dyer Conjecture, those conjectures can be restated to predict the number of cyclic extensions $K/\mathbb{Q}$ of prime degree such that $E$ acquires new rank over $K$.

Motivation & Objective

• To understand how the rank of an elliptic curve $E$ over $\mathbb{Q}$ grows over abelian extensions $K/\mathbb{Q}$ of prime degree.
• To investigate the frequency of vanishing of twisted L-functions $L_E(1,\chi)$ at the central critical point.
• To apply random matrix theory to predict the asymptotic distribution of such vanishing twists for characters of prime order $k$.
• To reframe the rank growth problem in terms of the number of cyclic extensions $K/\mathbb{Q}$ of degree $k$ where $E$ acquires new rank.

Proposed method

• Use modular symbols to express the special value $L_E(1,\chi)$ as a product involving an algebraic integer $n_E(\chi)$ dependent on the character $\chi$.
• Discretize the algebraic integers $n_E(\chi)$ via embedding into $\mathbb{C}$, modeling them as lattice points.
• Apply the Keating-Snaith random matrix model to estimate the probability that $|L_E(1,\chi)|$ is small, using the characteristic polynomial of random matrices.
• Estimate the probability of vanishing by bounding the joint probability that multiple Galois conjugates of $L_E(1,\chi)$ are simultaneously small.
• Use the resulting probability estimates to sum over all characters of conductor $\leq X$, leading to asymptotic conjectures for $N_{E,k}(X)$, the number of vanishing twists.
• Relate the conjectured number of vanishing twists to the number of cyclic extensions $K/\mathbb{Q}$ of degree $k$ where $E$ gains rank, under the Birch and Swinnerton-Dyer conjecture.

Experimental results

Research questions

• RQ1 For which orders $k$ of Dirichlet characters are there infinitely many twists $\chi$ such that $L_E(1,\chi) = 0$?
• RQ2 What is the asymptotic growth rate of the number of such vanishing twists as a function of conductor size $X$?
• RQ3 How does the vanishing of $L_E(1,\chi)$ relate to the rank growth of $E$ over cyclic extensions $K/\mathbb{Q}$ of degree $k$?
• RQ4 Why does the behavior change at $k=7$, where only finitely many vanishing twists are expected?

Key findings

• For $k=3$, the number of vanishing twists $N_{E,3}(X)$ satisfies $\log N_{E,3}(X) \sim \frac{1}{2}\log X$ as $X \to \infty$, indicating infinitely many such twists.
• For $k=5$, $N_{E,5}(X)$ is unbounded but grows slower than any positive power of $X$, i.e., $N_{E,5}(X) \ll X^\epsilon$ for any $\epsilon > 0$, suggesting infinitely many, but sparse, vanishing twists.
• For $k \geq 7$, $N_{E,k}(X)$ is bounded as $X \to \infty$, implying only finitely many such vanishing twists exist.
• The random matrix model predicts that the probability of vanishing decays as $\frac{\log^{(k-1)/8} m}{m^{(k-1)/4}}$ for order-$k$ characters, leading to convergent sums for $k \geq 7$, which supports boundedness.
• Numerical evidence for $k=3,5,7$ over Cremona curves with conductor up to two million supports the conjectures, showing increasing scarcity of vanishing as $k$ increases.
• No vanishing twists were found for $k=11$ among curves with conductor up to two million, consistent with the conjecture that $N_{E,k}(X)$ is bounded for $k \geq 7$.