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This paper contains a method to prove the existence of smooth curves in positive characteristic whose Jacobians have unusual Newton polygons. Using this method, I give a new proof that there exist supersingular curves of genus 
$4$ in every prime characteristic. More generally, the main result of the paper is that, for every 
$g \geq 4$ and prime p, every Newton polygon whose p-rank is at least 
$g-4$ occurs for a smooth curve of genus g in characteristic p. In addition, this method resolves some cases of Oort’s conjecture about Newton polygons of curves.
Let F be a non-Archimedean local field and let p be the residual characteristic of F. Let G=GL2(F) and let P be a Borel subgroup of G. In this paper we study the restriction of irreducible smooth representations of G on 
-vector spaces to P. We show that in a certain sense P controls the representation theory of G. We then extend our results to smooth 
-modules of finite length and unitary K-Banach space representations of G, where 
is the ring of integers of a complete discretely valued field K with residue field .
