New points on algebraic curves (Joint work with Dino Lorenzini)


09:10 - 10:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

Let K be a field and let L/K be a finite extension. Let X be an algebraic variety over K. A point of X(L) is called a *new point* if it does not belong to any of X(F), when F runs over all proper subextensions of L.

Fix now a separable extension L/K of degree d. We investigate whether there exists a smooth proper geometrically connected curve of genus g > 0 with a new point in X(L). We show that if K is infinite with char(K) different from 2 and if g > [d/4], then there exist infinitely many hyperelliptic curves X of genus g, pairwise non-isomorphic, and with a new point in X(L). When 1 < d < 11, we show that there exist infinitely many elliptic curves X with pairwise distinct j-invariants and with a new point in X(L).