# A weighted universality theorem for zeta-functions of elliptic curves

Virginija GARBALIAUSKIENĖ (ŠU)

e-mail: virginija@fm.su.lt

Let $E: y^2 = x^3 + ax + b, a, b \in \mathbb{Z}$, be an elliptic curve. Suppose that the discriminant $\Delta = -16(a^3 + 27b^2) \neq 0$. In this case an elliptic curve is non-singular.

Let, for a prime $p$, $\nu(p)$ be the number of solutions of the congruence

$$y^2 \equiv x^3 + ax + b \pmod{p},$$

and $\lambda(p) = p - \nu(p)$. Then the classical result of H. Hasse asserts that

$$|\lambda(p)| \le 2\sqrt{p}$$

for each prime $p$.

Let $s = \sigma + it$ be a complex variable. To the curve $E$ we attach the $L$-function $L_E(s)$ defined, for $\sigma > \frac{3}{2}$, by

$$L_E(s) = \prod_{p|\Delta} \left(1 - \frac{\lambda(p)}{p^s}\right)^{-1} \prod_{p \nmid \Delta} \left(1 - \frac{\lambda(p)}{p^s} + \frac{1}{p^{2s-1}}\right)^{-1}.$$

Now it is known that $L_E(s)$ is analytically continuable to an entire function.

The papers [2], [3], [5] are devoted to the universality of the function $L_E(s)$. For example, in [2], [5] the following statement is given. Let $\nu_T(\dots) = \frac{1}{T}$ meas$\{\tau \in [0, T]: \dots\}$, were in place of dots a condition satisfied by $\tau$ is to be written.

**THEOREM A.** Let $K$ be a compact subset of the strip $D = \{s \in \mathbb{C}: 1 < \sigma < \frac{3}{2}\}$ with connected complement, and let $f(s)$ be a continuous non-vanishing function on $K$ which is analytic in the interior of $K$. Then, for every $\varepsilon > 0$,

$$\liminf_{T \to \infty} \nu_T \left( \sup_{s \in K} \left| L_E(s + i\tau) - f(s) \right| < \varepsilon \right) > 0.$$

The paper [3] contains a discrete version of Theorem A.

The aim of this note is to obtain a weighted universality theorem for the function $L_E(s)$.

Let $T_0$ be a fixed positive number, and let $w(\tau)$ be a positive function of bounded variation on $[T_0, \infty)$. Set

$$U = U(T, w) = \int_{T_0}^{T} w(\tau) \, d\tau,$$
and suppose that $\lim_{T \to \infty} U(T, w) = +\infty$. Moreover, we need some weighted analogue of the Birkhoff-Khinchine theorem. Denote by $E_\xi$ the mean of the random element $\xi$. Let $X(\tau, \omega)$, $\tau \in \mathbb{R}$, be an ergodic process defined on a certain probability space, $E|X(\tau, \omega)| < \infty$, with sample paths almost surely integrable in the Riemann sense over every finite interval. Suppose that the function $w(\tau)$ satisfies

$$ \frac{1}{U} \int_{T_0}^{T} w(\tau) X(t + \tau, \omega) d\tau = EX(0, \omega) + o(1 + |t|)^{\delta} \quad (1) $$

almost surely for all $t \in \mathbb{R}$ with some $\delta > 0$ as $T \to \infty$.

Denote by $I_A$ the indicator function of the set A.

**THEOREM 1.** *Suppose that condition (1) is satisfied. Let K and f(s) be the same as in Theorem A. Then, for every $\varepsilon > 0$,*

$$ \liminf_{T \to \infty} \frac{1}{U} \int_{T_0}^{T} w(\tau) I_{\{\tau: \sup_{s \in K} |L_E(s+i\tau)-f(s)|<\varepsilon\}} d\tau > 0. $$

Clearly, Theorem A is a partial case of Theorem 1.

The proof of Theorem 1 is based on a limit theorem in the sense of the weak convergence of probability measures in the space of analytic functions for the function $L_E(s)$. Let G be a region in the complex plane. Denote by $H(G)$ the space of functions analytic on G, equipped with the topology of uniform convergence on compacta. Let, for $V > 0$, $D_V = \{s \in \mathbb{C}: 1 < \sigma < \frac{1}{2}, |t| < V\}$. Denote by $\mathcal{B}(S)$ the class of Borel sets of the space S, and define the probability measure

$$ P_T(A) = \frac{1}{U} \int_{T_0}^{T} w(\tau) I_{\{\tau: L_E(s+i\tau) \in A\}} d\tau, \quad A \in \mathcal{B}(H(D_V)). $$

Denote by $\gamma$ the unit circle $\gamma = \{s \in \mathbb{C}: |s| = 1\}$ on the complex plane, and let

$$ \Omega = \prod_{p} \gamma_{p}, $$

where $\gamma_p = \gamma$ for all primes $p$. With the product topology and pointwise multiplication the infinite-dimensional torus $\Omega$ is a compact topological group. Therefore there exists the probability Haar measure $m_H$ on $(\Omega, \mathcal{B}(\Omega))$. Thus we obtain a probability space $(\Omega, \mathcal{B}(\Omega), m_H)$. Let $\omega(p)$ stand for the projection of $\omega \in \Omega$ into the coordinate space $\gamma_p$, and on the probability space $(\Omega, \mathcal{B}(\Omega), m_H)$ define the $H(D_V)$-valued random element $L_E(s, \omega)$ by the formula

$$ L_E(s, \omega) = \prod_{p|\Delta} \left(1 - \frac{\lambda(p)\omega(p)}{p^s}\right)^{-1} \prod_{p \nmid \Delta} \left(1 - \frac{\lambda(p)\omega(p)}{p^s} + \frac{\omega^2(p)}{p^{2s-1}}\right)^{-1}, \quad \omega \in \Omega. \quad (2) $$

**LEMMA 2.** *Under condition (1) the probability measure $P_T$ converges weakly to the distribution of the random element $L_E(s, \omega)$ as $T \to \infty$.*
Proof. It is not difficult to see that, for $\sigma > \frac{3}{2}$,

$$L_E(s) = \prod_{p|\Delta} \left(1 - \frac{\lambda(p)}{p^s}\right)^{-1} \prod_{p \nmid \Delta} \left(1 - \frac{\alpha(p)}{p^s}\right)^{-1} \left(1 - \frac{\beta(p)}{p^s}\right)^{-1},$$

where $\lambda(p) = \alpha(p) + \beta(p)$, and

$$|\alpha(p)| \le \sqrt{p}, \quad |\beta(p)| \le \sqrt{p}.$$

Therefore, $L_E(s)$ is the Matsumoto zeta-function [7], [4] with parameters $\alpha = 0$ and $\beta = \frac{1}{2}$. Moreover, since by [1] every L-function attached to a non-singular elliptic curve over the field of rational numbers is the L-function attached to a certain newform of weight 2 of some congruence subgroup, we have that $L_E(s)$ is an entire function, and, for $\sigma > 1$,

$$L(\sigma + it) = O(|t|^{c_1}), \quad |t| \ge t_0, \ c_1 > 0,$$

and [6]

$$\int_0^T |L(\sigma + it)|^2 dt = O(T), \quad T \to \infty.$$

Therefore, by Theorem 8 of [4] the probability measure

$$\frac{1}{U} \int_{T_0}^{T} w(\tau) I_{\{\tau: L_E(s+i\tau) \in A\}} d\tau, \quad A \in \mathcal{B}(H(D)),$$

where $D_1 = \{s \in \mathbb{C}: \sigma > 1\}$, converges weakly to the distribution of the $H(D)$-valued random element defined by (2) as $T \to \infty$. Since the function $u: H(D_1) \to H(D)$ defined by the coordinatewise restriction is continuous, hence the lemma follows.

For the proof of Theorem 1 the support of the limit measure in Lemma 2 is needed. We recall that the support of a probability measure $P$ defined on $(S, \mathcal{B}(S))$ is a minimal closed set $S_P \subset S$ such that $P(S_P) = 1$. The support $S_P$ consists of all $x \in S$ such that for every neighbourhood $G$ of $x$ the inequality $P(G) > 0$ is satisfied.

Denote the limit measure in Lemma 2 by $P_{L_E}$, i.e., $P_{L_E}$ is the distribution of the random element $L_E(s, \omega)$.

LEMMA 3. The support of the measure $P_{L_E}$ is the set

$$S_V = \{g \in H(D_V): g(s) \ne 0 \text{ or } g(s) = 0\}.$$

Proof. The proof of the lemma coincides with that of Lemma 8 in [5]. A sketch of the proof is also given in [3], Lemma 5. A more general statement for the Matsumoto zeta-function under some additional condition is contained in [4], Lemma 6.

Proof of Theorem 1. Let $K$ be a compact subset of the strip $D$ with connected complement. Then there exists $V > 0$ such that $K \subset D_V$.
First we suppose that the function $f(s)$ has a non-vanishing continuation to the rectangle $D_V$. Let $G$ be the set of functions $g \in H(D_V)$ such that

$$ \sup_{s \in K} |g(s) - f(s)| < \varepsilon. $$

Clearly, the set $G$ is open. Moreover, by Lemma 3 we have that $G \subset S_V$. It is well known that the probability measure $P_n$ converges weakly to $P$ ($P_n$ and $P$ are given on $(S, \mathcal{B}(S))$ if and only if

$$ \liminf_{n \to \infty} P_n(G) \geq P(G) \quad (3) $$

for all open sets $G$ of $S$. Therefore, in view of Lemmas 2 and 3

$$ \liminf_{T \to \infty} \frac{1}{U} \int_{T_0}^{T} w(\tau) I_{\{\tau: \sup_{s \in K} |L_E(s+i\tau)-f(s)|<\varepsilon\}} d\tau \geq P(G) > 0, $$

and in this case the theorem is proved.

The general case is reduced to the above proved case. Let $f(s)$ be the same as in the statement of Theorem 1. Then by the Mergelyan theorem, see, for example, [8], there exists a sequence of polynomials $\{p_n(s)\}$ such that $p_n(s) \to f(s)$ as $n \to \infty$ uniformly on $K$. Since $f(s) \neq 0$ on $K$, there exists a sufficiently large $n_0$ such that $p_{n_0}(s) \neq 0$ on $K$ and

$$ \sup_{s \in K} |f(s) - p_{n_0}(s)| < \frac{\varepsilon}{4}. $$

However, the polynomial $p_{n_0}(s)$ has only finitely many zeros, and therefore we can find a region $G$ with connected complement such that $K \subset G$ and $p_{n_0}(s) \neq 0$ on $G$. Hence, we can choose a continuous version of $\log p_{n_0}(s)$ on $G$ which is analytic in the interior of $G$. By the Mergelyan theorem again we can find a polynomial $q_n(s)$ such that

$$ \sup_{s \in K} |p_{n_0}(s) - e^{q_n(s)}| < \frac{\varepsilon}{4}. $$

The latter two inequalities show that

$$ \sup_{s \in K} |f(s) - e^{q_n(s)}| < \frac{\varepsilon}{2}. \quad (4) $$

Since, clearly, $e^{q_n(s)} \neq 0$, the first part of the proof implies

$$ \liminf_{T \to \infty} \frac{1}{U} \int_{T_0}^{T} w(\tau) I_{\{\tau: \sup_{s \in K} |L_E(s+i\tau)-e^{q_n(s)}|<\frac{\varepsilon}{2}\}} d\tau > 0. $$

This and inequality (3) yield the assertion of the theorem.

For example, we can take $w(\tau) = \tau^{-1}$.
References

1. C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over $\mathbb{Q}$: wild 3-adic exercises, *J. Amer. Math. Soc.*, **14**, 843–939 (2001).

2. V. Garbaliauskienė, A.Laurinčikas, *Some Analytic Properties for L-functions of Elliptic Curves*, Department of Math. and Inform., Vilnius University, Preprint 2003–16 (2003).

3. V. Garbaliauskienė, A.Laurinčikas, *Discrete Value-Distribution of L-functions of Elliptic Curves*, Department of Mathematics and Inform., Vilnius University, Preprint 2004–01 (2004).

4. A. Laurinčikas, On the Matsumoto zeta-function, *Acta Arith.*, **84**(1) 1–16 (1998).

5. A. Laurinčikas, K. Matsumoto, J. Steuding, The universality of L-functions associated to newforms, *Izv. Math.*, **67**, 77–90 (2003).

6. K. Matsumoto, Value-distribution of zeta-functions, *Lecture Notes in Math.*, **1434**, 178–187 (1990).

7. K. Matsumoto, A probabilistic study on the value-distribution of Dirichlet series attached to certain cusp forms. *Nagoya Math. J.*, **116**, 123–138 (1989).

8. J.L. Walsh, Interpolation and approximation by rational functions in the complex domain, *Amer. Math. Soc. Collog. Publ.*, **20** (1960).

REZIUMĖ

V. Garbaliauskienė. Ribinė teorema su svoriu elipsinių kreivių dzeta funkcijoms

Gauta universalumo teorema su svoriu elipsinės kreivės L-funkcijai.