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## One-parameter families of elliptic curves with non-zero average root number

We investigate in this talk the average root number (i.e. sign of the functional equa- tion) of one-parameter families of elliptic curves (i.e elliptic curves over Q(t), or elliptic surfaces over Q). For most one-parameter families of elliptic curves, the aver- age root number is predicted to be 0. Helfgott showed that under Chowla’s conjecture and the square-free conjecture, the average root number is 0 unless the curve has no place of multiplicative r eduction over Q(t). We then build families of elliptic curves with no place of multiplicative reduction, and compute the average root number of the families. Some families have periodic root number, giving a rational average, and some other families have an average root number which is expressed as an infinite Euler product. We also show several density results for the average root number of families of elliptic curves, and exhibit some surprising examples, for example, non- isotrivial families of elliptic curves with rank r over Q(t) and average root number −(−1)r, which were not found in previous literature.