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17
Big symplectic or orthogonal monodromy modulo ℓ
 Duke Math J
"... Let k be a field not of characteristic two and Λ be a set consisting of almost all rational primes invertible in k. Suppose we have a variety X/k and strictly compatible system {Mℓ → X: ℓ ∈ Λ} of constructible Fℓsheaves. If the system is orthogonally or symplectically selfdual, then the geometric ..."
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Cited by 15 (1 self)
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Let k be a field not of characteristic two and Λ be a set consisting of almost all rational primes invertible in k. Suppose we have a variety X/k and strictly compatible system {Mℓ → X: ℓ ∈ Λ} of constructible Fℓsheaves. If the system is orthogonally or symplectically selfdual, then the geometric monodromy group of Mℓ is a subgroup of a corresponding isometry group Γℓ over Fℓ, and we say it has big monodromy if it contains the derived subgroup DΓℓ. We prove a theorem which gives sufficient conditions for Mℓ to have big monodromy. We apply the theorem to explicit systems arising from the middle cohomology of families of hyperelliptic curves and elliptic surfaces to show that the monodromy is uniformly big as we vary ℓ and the system. We also show how it leads to new results for the inverse Galois problem. 1
Special Subvarieties Arising from Families of Cyclic Covers of the Projective Line
 DOCUMENTA MATH.
, 2010
"... We consider families of cyclic covers of P 1, where we fix the covering group and the local monodromies and we vary the branch points. We prove that there are precisely twenty such families that give rise to a special subvariety in the moduli space of abelian varieties. Our proof uses techniques i ..."
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Cited by 14 (1 self)
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We consider families of cyclic covers of P 1, where we fix the covering group and the local monodromies and we vary the branch points. We prove that there are precisely twenty such families that give rise to a special subvariety in the moduli space of abelian varieties. Our proof uses techniques in mixed characteristics due to Dwork and Ogus.
Results of CohenLenstra type for quadratic function fields
 CONTEMPORARY MATHEMATICS
"... Consider hyperelliptic curves C of fixed genus over a finite field F. Let L be a finite abelian group of exponent dividng N. We give an asymptotic formula in F, with explicit error term, for the proportion of C for which Jac(C)[N](F) ∼ = L. ..."
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Cited by 10 (1 self)
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Consider hyperelliptic curves C of fixed genus over a finite field F. Let L be a finite abelian group of exponent dividng N. We give an asymptotic formula in F, with explicit error term, for the proportion of C for which Jac(C)[N](F) ∼ = L.
THE pRANK STRATA OF THE MODULI SPACE OF HYPERELLIPTIC CURVES
"... ABSTRACT. We prove results about the intersection of the prank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p ≥ 3. Using this, we prove that the Z/ℓmonodromy of every irreducible component of the stratum H f g of hyperelliptic curves of genus g and prank f ..."
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Cited by 6 (4 self)
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ABSTRACT. We prove results about the intersection of the prank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p ≥ 3. Using this, we prove that the Z/ℓmonodromy of every irreducible component of the stratum H f g of hyperelliptic curves of genus g and prank f is the symplectic group Sp 2g (Z/ℓ) if g ≥ 3, f ≥ 1 and ℓ ̸ = p is an odd prime. These results yield applications about the generic behavior of hyperelliptic curves of given genus and prank. The first application is that a generic hyperelliptic curve of genus g ≥ 3 and prank 0 is not supersingular. Other applications are about absolutely simple Jacobians and the generic behavior of class groups and zeta functions of hyperelliptic curves of given genus and prank over finite fields. 1.
Monodromy of the prank strata of the moduli space of curves
, 2007
"... We compute the Z/ℓmonodromy and Z ℓmonodromy of every irreducible component of the moduli space M f g of curves of genus g and prank f in characteristic p. In particular, we prove that the Z/ℓmonodromy of every component of M f g is the symplectic group Sp 2g (Z/ℓ) if g ≥ 3 and ℓ � = p is prim ..."
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Cited by 4 (3 self)
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We compute the Z/ℓmonodromy and Z ℓmonodromy of every irreducible component of the moduli space M f g of curves of genus g and prank f in characteristic p. In particular, we prove that the Z/ℓmonodromy of every component of M f g is the symplectic group Sp 2g (Z/ℓ) if g ≥ 3 and ℓ � = p is prime. We give applications to the generic behavior of automorphism groups, Jacobians, class groups, and zeta functions of curves of given genus and prank.
CURVES AND ZETA FUNCTIONS OVER FINITE FIELDS  ARIZONA WINTER SCHOOL 2014: ARITHMETIC STATISTICS
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MONODROMY GROUPS OF HURWITZTYPE PROBLEMS
, 2008
"... Abstract. We solve the Hurwitz monodromy problem for degree 4 covers. That is, the Hurwitz space H4,g of all simply branched covers of P1 of degree 4 and genus g is an unramified cover of the space P2g+6 of (2g + 6)tuples of distinct points in P1. We determine the monodromy of π1(P2g+6) on the poin ..."
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Abstract. We solve the Hurwitz monodromy problem for degree 4 covers. That is, the Hurwitz space H4,g of all simply branched covers of P1 of degree 4 and genus g is an unramified cover of the space P2g+6 of (2g + 6)tuples of distinct points in P1. We determine the monodromy of π1(P2g+6) on the points of the fiber. This turns out to be the same problem as the action of π1(P2g+6) on a certain local system of Z/2vector spaces. We generalize our result by treating the analogous local system with Z/N coefficients, 3 ∤ N, in place of Z/2. This in turn allows us to answer a question of Ellenberg concerning families of Galois covers of P1 with deck group (Z/N) 2:S3. A ramified cover C of P 1 of degree d is said to have simple branching if the fiber over every branch point has d−1 distinct points. Another way to say this is that for each branch point p, the permutation of the sheets of the cover induced by a small loop around p is a transposition, i.e., a
Contents To epsilon
, 2009
"... We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over Fq as the curve varies in an irreducible component of the moduli space. We show that for q fixed and g increasing, the limiting distribution of the trace of Frobenius equals the su ..."
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We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over Fq as the curve varies in an irreducible component of the moduli space. We show that for q fixed and g increasing, the limiting distribution of the trace of Frobenius equals the sum of q + 1 independent random variables taking the value 0 with probability 2/(q + 2) and 1, e 2πi/3, e 4πi/3 each with probability q/(3(q + 2)). This extends the work of Kurlberg and Rudnick who considered the same limit for hyperelliptic curves. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution and how to generalize these results