Logarithmic good reduction of abelian varieties

Good reduction of abelian varieties

Abelian varieties over cyclotomic fields with good reduction everywhere

For every conductor f ∈ {1, 3, 4, 5, 7, 8, 9, 11, 12, 15} there exist non-zero abelian varieties over the cyclotomic field Q(ζf ) with good reduction everywhere. Suitable isogeny factors of the Jacobian variety of the modular curve X1(f ) are examples of such abelian varieties. In the other direction we show that for all f in the above set there do not exist any non-zero abelian varieties over ...

Semi-stable abelian varieties with good reduction outside 15

We show that there are no non-zero semi-stable abelian varieties over Q( √ 5) with good reduction outside 3 and we show that the only semi-stable abelian varieties over Q with good reduction outside 15 are, up to isogeny over Q, powers of the Jacobian of the modular curve X0(15).

Semistable Reduction for Abelian Varieties

The semistable reduction theorem for curves was discussed in Christian’s notes. In these notes, we will use that result to prove an analogous theorem for abelian varieties. After some preliminaries on semi-abelian varieties (to convince us that the notion is a robust one), we will review the notion of a semi-abelian scheme (introduced in Christian’s lecture), recall the statement of the semista...

Abelian varieties over the field of the 20th roots of unity that have good reduction everywhere

The elliptic curve E given by Y 2 + (i+1)XY + iY = X + iX acquires good reduction everywhere over the cyclotomic field Q(ζ20). We show, under assumption of GRH, that every abelian variety over Q(ζ20) with good reduction everywhere is isogenous to E for some g ≥ 0.