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Abstract 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 primep, every Newton polygon whosep-rank is at least$$g-4$$occurs for a smooth curve of genusgin characteristicp. In addition, this method resolves some cases of Oort’s conjecture about Newton polygons of curves. 

Suppose C is a cyclic Galois cover of the projective line branched at the three points 0, 1, and ∞. Under a mild condition on the ramification, we determine the structure of the graded Lie algebra of the lower central series of the fundamental group of C in terms of a basis which is well-suited to studying the action of the absolute Galois group of Q. 

Abstract. Information about the absolute Galois group G K of a number field K is encoded in how it acts on the ´etale fundamental group π of a curve X defined over K. In the case that K = Q ( ζ n ) is the cyclotomic field and X is the Fermat curve of degree n ≥ 3, Anderson determined the action of G K on the ´etale homology with coefficients in Z/nZ. The ´etale homology is the first quotient in the lower central series of the ´etale fundamental group. In this paper, we determine the Galois module structure of the graded Lie algebra for π. As a consequence, this determines the action of G K on all degrees of the associated graded quotient of the lower central series of the ´etale fundamental group of the Fermat curve of degree n, with coefficients in Z/nZ. 

Abstract. We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose C and C ′ are curves over a finite field K, with K-rational base points P and P ′ , and let D and D ′ be the pullbacks (via the Abel–Jacobi map) of the multiplication-by-2 maps on their Jacobians. We say that (C, P) and (C ′ , P ′ ) are doubly isogenous if Jac(C) and Jac(C ′ ) are isogenous over K and Jac(D) and Jac(D ′ ) are isogenous over K. For curves of genus 2 whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than na¨ıve heuristics predict, and we provide an explanation for this phenomenon. 

Abstract. We study the p-rank stratification of the moduli space of cyclic degree ! covers of the projective line in characteristic p for distinct primes p and !. The main result is about the intersection of the p-rank 0 stratum with the boundary of the moduli space of curves. When ! = 3 and p ≡ 2 mod 3 is an odd prime, we prove that there exists a smooth trielliptic curve in characteristic p, for every genus g, signature type (r,s), and p-rank f satisfying the clear necessary conditions. 

Abstract We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $$p$$ reduction of certain Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus with some Newton polygon strata is non-empty. This allows us to give a positive answer, under some compatibility conditions, to a question of Oort about smooth curves in characteristic $$p$$ whose Newton polygons are an amalgamate sum. As an application, we produce infinitely many new examples of Newton polygons that occur for smooth curves that are cyclic covers of the projective line. Most of these arise in inductive systems that demonstrate unlikely intersections of the open Torelli locus with the Newton polygon stratification in Siegel modular varieties. In addition, for the 20 special Shimura varieties found in Moonen’s work, we prove that all Newton polygon strata intersect the open Torelli locus (if $p>>0$ in the supersingular cases).