Rank of elliptic curves and exponential sums
It is usually supposed true that the existence of elliptic curves \(E\) defined over \(\mathbb{Q}\) with high rank is rare. So there is a natural question of giving an average bound for the number of such elliptic curves. This was done by the first author unconditionally for the curves \(E(0,k)\) with Weierstrass equation \(y^ 2=x^ 3+k\), where \(k\) is a nonzero integer [\textit{E. Fouvry}, Sur le comportement en moyenne du rang des courbes \(y^ 2=x^ 3+k\), in Sém. Théorie Nombres, Paris 1990- 91, Prog. Math. 61-84 (1993)]. More precisely \[ \sum_{| k | \leq X} (\sqrt 3)^{\text{rk} \bigl( E(0,k) \bigr)} = O(X), \] for \(X \geq 1\), which shows that for a strong type of average the rank of the elliptic curves of this family is bounded. This result involves an argument of 3- descent which cannot be done so easily when one considers the elliptic curves \(E(a,b)\) with equation \(y^ 2 = x^ 3 + ax + b\), with \(a,b \in \mathbb{Z}\) and \(4a^ 3 + 27b^ 2 \neq 0\). To deal with these curves the authors suppose that they satisfy the conjectures of Birch and Swinnerton-Dyer and Weil-Taniyama and that their Hasse-Weil \(L\)-function \(L(E,s)\) satisfies the generalized Riemann's hypothesis. In this set-up there is a result due to \textit{A. Brumer} [Invent. Math. 109, No. 3, 445-472 (1992; Zbl 0783.14019)] where under the above hypotheses he shows that \[ \sum_{a,b} \text{rank} \bigl( E(a,b) \bigr) \leq \bigl( 2,3 + o(1) \bigr)\;(4T^{5/6})\;\zeta (10), \] as \(T \to \infty\), where the sum is taken over the \(a\) and \(b\)'s such that if \(n^ 4 | a\), \(n^ 6 | b\), then \(n = \pm 1\), \(| a | \leq T^{1/3}\), \(| b | \leq T^{1/2}\) and \(\Delta (a,b)= -16(4a^ 3+27b^ 2) \neq 0\). In easier terms his result shows that the average rank of an elliptic curve in this family is less than 2,3. In his paper Brumer also obtains unconditional results for elliptic curves of this family but defined over \(\mathbb{F}_ q (t)\). In the paper under review the authors consider elliptic curves \(E(a(t), b(t))\) with equation \(y^ 2 = x^ 3 + a(t)x + b(t)\), where \(a(t)\), \(b(t) \in \mathbb{Z} [t]\) such that \(\deg (b(t)) \geq 1\), \(b(t)\) has a nonzero discriminant, \(\deg (\Delta (a(t), b(t))) = \max (3 \deg (a),2 \deg (b))\) and \(\Delta (a(t), b(t)) \neq 0\). These are quite general hypotheses. They show that if these curves satisfy the above conjectures then \[ \sum_{| t | \leq T} \text{rank} \biggl( E \bigl( a(t), b(t) \bigr) \biggr) \leq 2 \max \bigl( 3 \deg (a), 2 \deg (b) \bigr)\;(1 + o(1) 2T. \] As in Brumer's paper the authors' strategy is to start with an explicit formula which links the zeros of \(L(E,s)\) with \(a_ p (E)\), where \(n_ p (E)\) denotes the number of \(\mathbb{F}_ p\)-rational points of \(E(a,b) \pmod p\) and \(a_ p(E) = p+1-n_ p (E)\). One chooses a test function and bounds \(\text{ord}_{s=1} (L(E,s))\) in terms of a sum of involving the \(a_ p(E)\)'s. This method allowed Mestre to obtain the bound \(\text{rank} (E) = O(\log (N_ E)/ \log \log (N_ E))\), where \(N_ E\) denotes the conductor of \(E\) [\textit{J. F. Mestre}, in Sém. Théorie Nombres, Paris 1981/82, Prog. Math. 38, 179-187 (1983; Zbl 0556.14015)]. Since \(a_ p \bigl( E(a,b) \bigr) = \sum_{x \pmod p} (x^ 3 + ax + b)/p\) an inversion in the summation shows that the study of \(\sum_{a,b} \text{rank} (E(a,b))\) is reduced to the sum \(\sum ({x^ 3 + ax + b \over p})\), where the sum is taken over \(a\), \(b\) and \(x\) modulo \(p\).
scientific article; zbMATH DE number 490682
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