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Andrew Snowden

  1. Real components of modular curves.

    Ср, 08/17/2011 - 15:01 — SomNambul
    Authors: Andrew Snowden
    Subjects: Number Theory
    Abstract

    We study the real components of modular curves. Our main result is an
    abstract group-theoretic description of the real components of a modular curve
    defined by a congruence subgroup of level N in terms of the corresponding
    subgroup of SL_2(Z/NZ). We apply this result to several families of modular
    curves (such as X_0(N), X_1(N), etc.) to obtain formulas for the number of real
    components.

  2. The ideal of relations for the ring of invariants of n points on the line.

    Пт, 09/18/2009 - 11:36 — SomNambul
    Authors: Andrew Snowden, Ben Howard, John Millson, Ravi Vakil
    Subjects: Algebraic Geometry
    Abstract

    The study of the projective coordinate ring of the (geometric invariant
    theory) moduli space of n ordered points on P^1 up to automorphisms began with
    Kempe in 1894, who proved that the ring is generated in degree one in the main
    (n even, unit weight) case. We describe the relations among the invariants for
    all possible weights. In the main case, we show that up to the symmetric group
    symmetry, there is a single equation. For n not 6, it is a simple quadratic
    binomial relation.

  3. The ideal of relations for the ring of invariants of n points on the line.

    Пт, 09/18/2009 - 11:36 — SomNambul
    Authors: Andrew Snowden, Ben Howard, John Millson, Ravi Vakil
    Subjects: Algebraic Geometry
    Abstract

    The study of the projective coordinate ring of the (geometric invariant
    theory) moduli space of n ordered points on P^1 up to automorphisms began with
    Kempe in 1894, who proved that the ring is generated in degree one in the main
    (n even, unit weight) case. We describe the relations among the invariants for
    all possible weights. In the main case, we show that up to the symmetric group
    symmetry, there is a single equation. For n not 6, it is a simple quadratic
    binomial relation.

  4. The ideal of relations for the ring of invariants of n points on the line: integrality results.

    Пт, 09/18/2009 - 11:36 — SomNambul
    Authors: Andrew Snowden, Ben Howard, John Millson, Ravi Vakil
    Subjects: Algebraic Geometry
    Abstract

    Consider the projective coordinate ring of the GIT quotient (P^1)^n//SL(2),
    with the usual linearization, where n is even. In 1894, Kempe proved that this
    ring is generated in degree one. In [HMSV2] we showed that, over the rationals,
    the relations between degree one invariants are generated by a class of
    quadratic relations -- the simplest binomial relations -- with the exception of
    n=6, where there is a single cubic relation. The purpose of this paper is to
    show that these results hold over Z[1/12!], and to suggest why they may be true
    over Z[1/6].

  5. The ideal of relations for the ring of invariants of n points on the line: integrality results.

    Пт, 09/18/2009 - 11:36 — SomNambul
    Authors: Andrew Snowden, Ben Howard, John Millson, Ravi Vakil
    Subjects: Algebraic Geometry
    Abstract

    Consider the projective coordinate ring of the GIT quotient (P^1)^n//SL(2),
    with the usual linearization, where n is even. In 1894, Kempe proved that this
    ring is generated in degree one. In [HMSV2] we showed that, over the rationals,
    the relations between degree one invariants are generated by a class of
    quadratic relations -- the simplest binomial relations -- with the exception of
    n=6, where there is a single cubic relation. The purpose of this paper is to
    show that these results hold over Z[1/12!], and to suggest why they may be true
    over Z[1/6].

  6. Bigness in compatible systems.

    Пнд, 08/17/2009 - 18:30 — SomNambul
    Authors: Andrew Snowden, Andrew Wiles
    Subjects: Number Theory
    Abstract

    Clozel, Harris and Taylor have recently proved a modularity lifting theorem
    of the following general form: if rho is an l-adic representation of the
    absolute Galois group of a number field for which the residual representation
    rho-bar comes from a modular form then so does rho. This theorem has numerous
    hypotheses; a crucial one is that the image of rho-bar must be "big," a
    technical condition on subgroups of GL(n). In this paper we investigate this
    condition in compatible systems.

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