Factorization of measures and applications to the weak Goldfeld conjecture

Let $E$ be an elliptic curve over $\mathbb{Q}$. The $\mathbb{Q}$-points of $E$ form an abelian group $E(\mathbb{Q}$) called the Mordell-Weil group. Mordell’s theorem states that $E(\mathbb{Q})$ is finitely generated, and thus the rank of $E(\mathbb{Q})$ is a well-defined, nonnegative integer. We call the rank of $E(\mathbb{Q})$ the algebraic rank of $E$, and denote it as $r_{alg}(E)$.

However, the algebraic rank is rather difficult to handle. Instead, we may attach the following $L$-function to the elliptic curve $E/\mathbb{Q}$:

where

and $a_{p}=p+1-|E(\mathbb{F}_{p})|$ is the trace of the Frobenius element associated to $p$. (In the multiplicative reduction case, the type of reduction determines the sign of the plus/minus.) This $L$-function satisfies a functional equation relating its values at $s$ and $2-s$, and thus its order of vanishing at $s=1$ is of interest. We call the order of vanishing of $L(E/\mathbb{Q},s)$ at $s=1$ the analytic rank of $E$, and denote it as $r_{an}(E)$.

Although the notions of analytic and algebraic rank may seem unrelated, they are not. The famous Birch and Swinnerton-Dyer conjecture [6] posits that they are in fact equal.

The BSD conjecture is still wide open, although significant advances have been made. Some of the strongest known results are due to [25], [27], [2], [13], [15], and [16], and they relate the algebraic and analytic ranks in low rank cases.

However, it’s still unproven as to whether $r_{alg}(E)\in\{0,1\}$ implies that $r_{alg}(E)=r_{an}(E)$.

Elliptic curves can be ordered by a property called height. This property is useful when studying statistics of elliptic curves, since it allows us to formalize the notion of an average: to measure the average of a quantity over all elliptic curves, we can calculate the average over the finitely many elliptic curves with height at most $X$, and then take a limit as $X\to\infty$. The analytic rank of an elliptic curve is one particularly important property that can be studied in this way. Originating from [11] and [18], it is widely believed that among all elliptic curves over $\mathbb{Q}$, the elliptic curves with analytic rank $0$ or $1$ should each have density $50\%$, while elliptic curves with analytic rank greater than $1$ should have density $0$. Recent developments by [5], [7], [4], and others have placed increasingly tighter bounds on the average, putting it closer and closer to the conjectured value of $0.5$; for example, the average rank is bounded below by $0.2068$ and bounded above by $0.885$.

Understanding the average rank over all elliptic curves is rather difficult. We can instead look at one particular family of elliptic curves: the quadratic twists $E_{D}$ of a fixed elliptic curve $E$. In [11], Goldfeld postulated that the average rank of a family of quadratic twists should behave in the same way as the set of elliptic curves over $\mathbb{Q}$.

However, Goldfeld’s conjecture is still very open. There is no elliptic curve which has been shown to satisfy Goldfeld’s conjecture. We will instead study the following weaker version of Goldfeld’s conjecture (see, for example, [14, Conjecture 1.2]).

In the last section of this paper, we will prove a result which, given certain conditions on the elliptic curve, guarantee that a positive proportion of its quadratic twists will have rank $0$ and $1$; in addition, we give lower bounds for these proportions.

We follow the exposition in [12, §1]. Let $p\in\mathbb{Z}$ be a prime, $\mathbb{Z}_{p}$ the ring of $p$-adic integers, $\mathbb{Q}_{p}$ the field of $p$-adic numbers, and $\mathbb{C}_{p}$ the algebraic closure of $\mathbb{Q}_{p}$. Now let $\mathbb{D}_{p}$ be the ring of integral elements in $\mathbb{C}_{p}$. For a commutative profinite group $G$, we consider its completed group algebra over $\mathbb{D}_{p}$, $\Lambda_{G}\coloneqq\mathbb{D}_{p}[[G]]=\varprojlim_{H\subset G\text{ open}}\mathbb{D}_{p}[G/H]$. The elements of $\Lambda_{G}$ are called measures on $G$. We also define $\Lambda_{G}^{\prime}$, the total ring of fractions of $\Lambda_{G}$, as the ring whose elements are $\alpha/\beta$ for $\alpha,\beta\in\Lambda_{G}$ and $\beta$ is not a zero-divisor.

We define a bilinear pairing between continuous functions $G\to\mathbb{C}_{p}$ and measures in $\Lambda_{G}$ by approximating $f$ by locally constant functions and taking a limit, as in [22]:

For $\lambda=\alpha/\beta\in\Lambda_{G}^{\prime}$, we extend this pairing by $\langle f,\lambda\rangle\coloneqq\langle f,\alpha\rangle/\langle f,\beta\rangle$. This construction is well-defined since it does not depend on the representation of $\lambda$, and agrees with our previous definition for $\lambda\in\Lambda_{G}$.

Let $K$ be an imaginary quadratic field. We will primarily consider the case where $G=\text{Gal}(K(\mu_{p^{\infty}})/\mathbb{Q})$ or $G=\text{Gal}(K(\mu_{p^{\infty}})/\mathbb{Q})/\sigma$ where $\sigma$ is complex conjugation, and $f=\chi$ is a (continuous) character from $G$ to $\mathbb{D}_{p}^{\times}$.

Let $p$ be a prime, $K=\mathbb{Q}(\sqrt{-C})$ an imaginary quadratic field where $p$ splits, and $\chi$ a continuous $p$-adic character of $\text{Gal}(K(\mu_{p^{\infty}})/\mathbb{Q})$ which is trivial on complex conjugation. Let $\chi_{K}$ be the restriction of $\chi$ to $\text{Gal}(K(\mu_{p^{\infty}})/K)$, $\epsilon$ the quadratic character modulo $C$, and $\omega$ the Teichmüller character. As in [12], define the measures $\lambda_{1},\lambda_{2},\lambda_{3}$ by $\langle\chi,\lambda_{1}\rangle=L_{p}(0,\chi_{K})$, $\langle\chi,\lambda_{2}\rangle=L_{p}(0,\chi\epsilon\omega)$, and $\langle\chi,\lambda_{3}\rangle=L_{p}(1,\chi^{-1})$. Motivated by the classical factorization of $L$-series $L(s,(\chi_{K})_{\infty})=L(s,\chi_{\infty}\epsilon)L(s,\chi_{\infty})$, Gross [12, Theorem 3.1] derives the factorization of measures

when $p$ is split in $K$ and $\chi$ is a finite even Dirichlet character whose conductor is a power of $p$. In §2, we extend this result in Theorem 2.13 to all $p$ (not just split $p$) and any finite even Dirichlet character $\chi$ (with any conductor).

We then turn our attention to elliptic curves. In §3, we introduce assumptions on the elliptic curve which will hold for the remainder of the paper. We will assume that $E$ is residually reducible modulo $3$ (Assumption 3.1), and we will work with integers $D$ and imaginary quadratic fields $K$ satisfying various divisibility and congruence conditions relating $D$, the conductor of $E$, and the discriminant of $K$ (Assumptions 3.2 and 3.3).

In §4, we discuss general congruences of $L$-series and Eisenstein series, especially those associated with quadratic characters. We obtain some auxiliary results concerning the congruence of certain modular forms with Eisenstein series, and calculate the Euler factor at $p$ after $p$-depleting (see §4.3).

In §5.1, we use the factorization in Theorem 2.13 to arrive at the two key technical results, Theorem 5.6 and Theorem 5.7. Under Assumption 3.3 and the assumption that $D$ satisfies the nonvanishing of a certain class number modulo $3$ (for $D>0$, we need $3\nmid h_{\mathbb{Q}(\sqrt{-3D})}$ and for $D<0$, we need $3\nmid h_{\mathbb{Q}(\sqrt{D})}$), we find a lower bound on the proportion of imaginary quadratic fields $K$ for which $E_{D}/K$ has rank $1$. In §5.2, we vary $D$ instead. Assuming that $\omega(n)$ is sufficiently close to a Gaussian distribution, we find bounds on the proportion of $D$ such that $E_{D}/K$ has rank $1$; these are given in Theorem 5.11.

In §6, we address cubic and sextic twists. In §6.1, we obtain results similar to §4 but for cubic twists. Since sextic twists are a composition of a cubic twist and a quadratic twist, we apply our results from §5.1 to obtain similar results on sextic twists in §6.2. The results, paralleling Theorem 5.6 and Theorem 5.7, are given by Theorem 6.6 and Theorem 6.7.

Finally, in §7, we positive lower bounds on the proportion of $D$ for which $E_{D}/\mathbb{Q}$ has rank $0$ and $1$, under similar assumptions. Given the assumptions before, plus the additional assumption that $3\nmid N\coloneqq\text{cond}(E)$, in Theorem 7.1 we find that $E_{D}/\mathbb{Q}$ has rank $0$ for at least $\frac{\phi(N)}{4N}$ of all such $D$, and rank $1$ for at least $\frac{\phi(N)}{4N}$ of all such $D$. As an easy corollary, we conclude Conjecture 1.4 for certain elliptic curves.

The author would like to thank Daniel Kriz for supervising this project, mentoring the author, and providing much needed guidance. The author also thanks Jonathan Love and Professor Andrew Sutherland for many helpful discussions and feedback.

We start with the classical factorization $L(s,(\chi_{K})_{\infty})=L(s,\chi_{\infty}\epsilon)L(s,\chi_{\infty})$ and the functional equation for $L(s,\chi)$:

where $\Gamma$ is the gamma function, $k$ is the conductor of $\chi$, $\tau=\sum_{n=1}^{k}\chi(n)e^{2\pi in/k}$ is the Gauss sum, and $a=0$ if $\chi(-1)=1$ while $a=1$ if $\chi(-1)=-1$.

The explicit formulas of Dirichlet and Kronecker are as follows:

• •

  $L^{\prime}(0,(\chi_{K})_{\infty})=-\frac{1}{6p^{r}}\sum_{A}\chi_{\infty}(a)\log F^{+}(a)_{\infty}$ [12, p. 91],

• •

  $L(0,\chi)=-\sum_{a=1}^{f}\frac{a}{f}\chi(a)=-B_{1,\chi}$ for $f$ the conductor of $\chi$ [12, p. 88],

• •

  $L(1,\chi_{\infty})=-g(\chi_{\infty})\sum_{A}\chi_{\infty}^{-1}(a)\log C^{+}(a)_{\infty}$ [12, p. 91] where $g(\chi)=\frac{1}{f}\sum_{a=1}^{f}\chi(a)e^{2\pi ia/f}$ for $f$ the conductor of $\chi$ [12, p. 88].

Using the identity $L(1,\chi_{\infty})=-g(\chi_{\infty})\sum_{A}\overline{\chi_{\infty}}(a)\log C^{+}(a)_{\infty}$, and the fact that $\tau(\overline{\chi_{\infty}})g(\chi_{\infty})=\frac{\overline{\tau(\chi_{\infty})}\tau(\chi_{\infty})}{f}=\frac{|\sqrt{f}|^{2}}{f}=1$, we find that

Now, combining these with the fact that $L(0,\chi_{\infty}\epsilon)=-B_{1,\chi_{\infty}\epsilon}$, we find that

or equivalently

Now recall that $C^{+}(a)_{\infty}$ and $F^{+}(a)_{\infty}$ are $p$-units in the field $M_{p^{r}}=\mathbb{Q}\left(\cos\frac{2\pi}{p^{r}}\right)$. Let $E(M_{p^{r}})$ denote the group of all $p$-units. It is a finitely generated subgroup of $\mathbb{R}^{\times}$. Now consider the complex vector space $\mathbb{C}\otimes_{\mathbb{Z}}E(M_{p^{r}})$. This is isomorphic to the regular representation of $A=\text{Gal}(M_{p^{r}}/\mathbb{Q})$. Now note that for all $\sigma\in A$, due to transport of structure, we have that

This implies that both $\sum_{A}\chi_{\infty}(a)\otimes_{\mathbb{Z}}C^{+}(a)_{\infty}$ and $\sum_{A}\chi_{\infty}(a)\otimes_{\mathbb{Z}}F^{+}(a)_{\infty}$ lie in the $\chi_{\infty}^{-1}$-eigenspace of $\mathbb{C}\otimes_{\mathbb{Z}}E(M_{p^{r}})$, which is one-dimensional. Therefore

for some $\tilde{c}\in\mathbb{C}$. Consider the map

defined by $\gamma(c\otimes a)=c\log a$. This map is clearly $\mathbb{C}$-linear, so

Applying $\gamma$ to both sides of $\tilde{c}\sum_{A}\chi_{\infty}(a)\otimes_{\mathbb{Z}}C^{+}(a)_{\infty}=\sum_{A}\chi_{\infty}(a)\otimes_{\mathbb{Z}}F^{+}(a)_{\infty}$ yields that

In particular, note that $E(M_{p^{r}})\subset\overline{\mathbb{Q}}$. We can actually say that

Now take some $\varphi:\overline{\mathbb{Q}}\xhookrightarrow{}\mathbb{C}_{p}$. Applying $(1\otimes_{\mathbb{Z}}\log_{p})\circ\varphi$ to both sides, we find the following equality in $\mathbb{C}_{p}$:

Now consider the explicit formulas provided by [12, p. 93]:

Putting these together yields the $p$-adic identity

Now, following [12, p. 93], there exist measures $\lambda_{2},\lambda_{3}$ such that for any finite even Dirichlet character $\chi$ of conductor $p^{r}$, we have

Now define a measure $\lambda_{1}$ given by

Then we have the equality

for all finite even Dirichlet characters $\chi$ with conductor $p^{r}$. Thus we have that

We will now work more generally and extend to the remaining cases. We fix $f$ to be some positive integer with at least two distinct prime divisors. We again start with the classical factorization $L(s,(\chi_{K})_{\infty})=L(s,\chi_{\infty}\epsilon)L(s,\chi_{\infty})$ and the functional equation for $L(s,\chi)$:

where $\Gamma$ is the gamma function, $f$ is the conductor of $\chi$, $\tau=\sum_{n=1}^{f}\chi(n)e^{2\pi in/f}$ is the Gauss sum, and $a=0$ if $\chi(-1)=1$ while $a=1$ if $\chi(-1)=-1$.

We have the following explicit formulas:

• •

  $L(0,\chi)=-\sum_{a=1}^{f}\frac{a}{f}\chi(a)=-B_{1,\chi}$ [12, p. 88],

• •

  $L(1,\chi_{\infty})=-\frac{\tau(\chi_{\infty})}{f}\sum_{A}\chi_{\infty}^{-1}(a)\log C^{+}(a)_{\infty}$ [12, p. 88] (note that this is for general conductors $f$),

• •

  $L^{\prime}(0,\chi)=-\frac{1}{6fw(\mathfrak{f})}\sum_{c\in\mathcal{C}}\chi(c)\log|E(c)|$ [24, p. 281], where $\chi$ is a ray class character modulo $\mathfrak{f}$, $w(\mathfrak{f})$ is the number of roots of unity equivalent to $1$ mod $\mathfrak{f}$, and $f$ is the conductor of $\chi$.

Note that $Cl(f)=(\mathbb{Z}/f\mathbb{Z})^{\times}$, but $A=\text{Gal}(\mathbb{Q}(\cos\frac{2\pi}{f})/\mathbb{Q})=(\mathbb{Z}/f\mathbb{Z})^{\times}/\pm 1$.

Now, combining these with the fact that $L(0,\chi_{\infty}\epsilon)=-B_{1,\chi_{\infty}\epsilon}$, we find that

or equivalently