How to force abstract class childs to implement a static method?

A possible approach of combining a static behavior with inheritance or interface implementation is to use the singleton pattern. The access to a singleton object is static, but the object is created with new like a "normal" object

public interface ISomeInterface { ... }

public class SomeClass : ISomeInterface
{ 
    public static readonly SomeClass Instance = new SomeClass();

    private SomeClass()
    { 
    }

    // TODO: Implement ISomeInterface
    // and/or override members from a base class.
}

---

Accessing a method of the singleton

ISomeInterface obj = SomeClass.Instance; // Static access to interface.
var y = obj.SomeMethod(x);

You cannot. Static methods are not subject to polymorphic behavior. You cannot even override static methods voluntarily, let alone force a class to override them.

Sorry, but it is impossible. Abstract and/or base classes are all about the object inheritance.

static methods are specific to one and only one class and are NOT inheritable.