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23
Splitting fields of characteristic polynomials of random elements in arithmetic groups, preprint
, 2010
"... In [K1] and earlier in [K2], questions of the following type are considered: suppose a family (gi)i of matrices in some (algebraic) matrix group are given, with rational coefficients. What is the “typical ” Galois group of the splitting field Ki of the characteristic polynomial of gi (defined as the ..."
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Cited by 17 (5 self)
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In [K1] and earlier in [K2], questions of the following type are considered: suppose a family (gi)i of matrices in some (algebraic) matrix group are given, with rational coefficients. What is the “typical ” Galois group of the splitting field Ki of the characteristic polynomial of gi (defined as the field generated over Q by the roots of the characteristic polynomial)? Is this characteristic polynomial typically irreducible? If the elements gi are in GL(n, Q), or in SL(n, Q), there is an obvious “upper bound”, namely the symmetric group Sn. If the elements gi are in a symplectic group (for an alternating form with rational coefficients), or in the group of symplectic similitude, there is also an easy, if slightly less obvious, upper bound: the characteristic polynomial satisfies some relation such as T 2g P (T −1) = P (T) if gi ∈ Sp(2g, Q) for instance, and this leads to relations among the roots which are easily shown to imply that the Galois group of the splitting field is, as subgroup of S2g, isomorphic to a subgroup of W2g, defined as the group of permutations of the g pairs (2i − 1, 2i), 1 � i � g, which also permute the pairs. Now, intrinsically, W2g is also the Weyl group of the algebraic group Sp(2g), or of CSp(2g),
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.
Nonsimple abelian varieties in a family: geometric and analytic approaches
, 2009
"... We consider, in the special case of certain oneparameter families of Jacobians of curves defined over a number field, the problem of how the property that the generic fiber of such a family is absolutely simple ‘spreads’ to other fibers. We show that this question can be approached using arithmeti ..."
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Cited by 6 (1 self)
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We consider, in the special case of certain oneparameter families of Jacobians of curves defined over a number field, the problem of how the property that the generic fiber of such a family is absolutely simple ‘spreads’ to other fibers. We show that this question can be approached using arithmetic geometry or with more analytic methods based on sieve theory. In the first setting, nontrivial grouptheoretic information is needed, while the version of the sieve we use is also of independent interest.
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.
The algebraic principle of the large sieve
"... Pour les soixantes ans de JM. Deshouillers ..."
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Report on the irreducibility of Lfunctions
"... Abstract. In this paper, in honor of the memory of Serge Lang, we apply ideas of Chavdarov and work of Larsen to study the Qirreducibility, or lack thereof, of various orthogonal Lfunctions, especially Lfunctions of elliptic curves over function fields in one variable over finite fields. We also ..."
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Abstract. In this paper, in honor of the memory of Serge Lang, we apply ideas of Chavdarov and work of Larsen to study the Qirreducibility, or lack thereof, of various orthogonal Lfunctions, especially Lfunctions of elliptic curves over function fields in one variable over finite fields. We also discuss two other approaches to these questions, based on work of Matthews, Vaserstein, and Weisfeller, and on work of ZalesskiiSerezkin. 1.