On the behaviour of root numbers in families of elliptic curves
By Harald Andrés Helfgott. 2004
49 citations recorded by the source corpus.
Let E be a one-parameter family of elliptic curves over Q. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any family E with at least one point of multiplicative reduction over Q(t) has average root number 0, provided that two classical arithmetical conjectures hold for two polynomials constructed explicitly in terms of E. The behaviour of the root number in any family E without multiplicative reduction over Q(t) is shown to be rather regular and non-random; we give expressions for the average root number in this case.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/be606b6944f560bf1cac280788398bc1ea072def
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In Algebra and Number Theory
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Root numbers and parity of ranks of elliptic curves (2009, background): “is a result of Kisin ([13] p. 569), and the root number is a function of VlE = TlE ⊗ Ql. Alternatively, that the root number is locally constant can be proved in a more elementary way: see Helfgott’s [12] Prop. 4.2 when E has potentially good reduction; in the potentially multiplicative case it foll”
- Average Ranks of Elliptic Curves: Tension between Data and Conjecture (2007, background): “See the work of Helfgott [ Hel04 ] for the latest results.”