正特征根数和秩
Root numbers and ranks in positive characteristic
摘要：
对整体域K和K（T）上的椭圆曲线E_eta，Silverman特化定理意味着对P^1（K）中除n个t以外的所有t，秩（E_eta（K（T）<=秩（E_t（K））。如果这个不等式对于除了有限多个t之外的所有情况都是严格的，则称椭圆曲线E_eta具有提升的秩。所有已知的K=Q的提升秩的例子都依赖于Q上椭圆曲线的奇偶猜想，并且这些例子都是等平凡的。Q（T）上的一些额外的标准曲线意味着不存在具有提升秩的非等平凡椭圆曲线。在正特征中，这些附加特征之一的类似物是假的。受此启发，对于任意具有奇特征的有限域kappa上的有理函数域K = kappa（u），我们构造了K（T）上非等平凡椭圆曲线的显式2参数族E_{c,d}（依赖于kappa^* 中的任意c，d），使得在奇偶猜想下，每个E_{c,d}都具有提升的秩.
英文摘要：
For a global field K and an elliptic curve E_eta over K(T), Silverman's specialization theorem implies that rank(E_eta(K(T))) <= rank(E_t(K)) for all but finitely many t in P^1(K). If this inequality is strict for all but finitely many t, the elliptic curve E_eta is said to have elevated rank. All known examples of elevated rank for K=Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial. Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K = kappa(u) over any finite field kappa with odd characteristic, we construct an explicit 2-parameter family E_{c,d} of non-isotrivial elliptic curves over K(T) (depending on arbitrary c, d in kappa^*) such that, under the parity conjecture, each E_{c,d} has elevated rank.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
