**SMS scnews item created by Boris Lishak at Wed 20 Nov 2019 1456**

Type: Seminar

Distribution: World

**Calendar1: 2 Dec 2019 1200-1300**

**CalLoc1: Carslaw 375**

CalTitle1: Bogomolov -- Elliptic curves and unramified correspondences

Auth: borisl@dora.maths.usyd.edu.au

### Geometry and Topology Seminar

# Elliptic curves and unramified correspondences

### Fedor Bogomolov (NYU)

December 2, 12:00-13:00 in Carslaw 375

Seminar schedule

Please join us for lunch after the talk.

**Abstract:**

We define two different ( but related) notions of dominance.
We will mostly consider them for curves defined over number fields of \(\bar F_p\) though they
can be defined for curves over any field.

Definition 1: For a curve \(C\) of genus \(g \geq 2\) we will say that \(C\) is dominant over \(C'\) if
there is an unramified
covering \(\tilde C\) of \(C\) with a surjection onto \(C'\).
In the case of elliptic curves we have a different notion
( assuming \(p\neq 2\) )
There is a involution \(x\to -x\) on elliptic curve \(E\)
if we fix \(0\) and the quotient of this involution is
\(P^1\). Thus we have projection map \(p: E\to P^1\) of degree \(2\)
with \(4\) branch points \((a,b,c,d)\) corresponding to points of order \(2\)
on \(E\). Such a map is unique modulo projective autmorphism of \(P^1\).
Vice versa we can associate to any quadruple of points in \(P^1\)
modulo projective autmorphism of \(P^1\)
unique elliptic curve \(E\) modulo isomorphism.
Moreover since the curve \(E\) is an abelian group we can also
define the subset \(P_E(tors)\subset P^1\) which is the image
of torsion points in \(E\) in \(P^1\).

Definition 2 We will say that \(E\) dominates \(E'\) if \(E'\) corresponds
to a quadruple of points contained in \(P_E(tors)\).
In my talk I will the relation between these two notions and
nontrivial results relating them.
The talk is base on my works with Yuri Tschinkel and our more
recent results with Hang Fu and Jin.

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