Compositio Mathematica (COMPOS MATH )

Publisher: London Mathematical Society


The aim of Compositio Mathematica is to publish first class mathematical research papers. By tradition the journal focuses on papers in the main stream of pure mathematics. This includes the fields of algebra number theory topology algebraic and analytic geometry and (geometric) analysis. Papers on other topics are welcome if they are of interest to more than specialists alone. All contributions are required to meet high standards of quality and originality and are carefully screened by experts in the field.

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Publications in this journal

  • Compositio Mathematica 07/2014; 150(7).
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    ABSTRACT: We formulate a conjecture which generalizes Darmon's "refined class number formula". We discuss relations between our conjecture and the equivariant leading term conjecture of Burns. As an application, we give another proof of the "except 2-part" of Darmon's conjecture, which was first proved by Mazur and Rubin.
    Compositio Mathematica 06/2014;
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    ABSTRACT: Let X be an algebraic curve. We study the problem of parametrizing geometric structures over X which are only generically defined. For example, parametrizing generically defined maps (rational maps) from X to a fixed target scheme Y. There are three methods for constructing functors of points for such moduli problems (all originally due to Drinfeld), and we show that the resulting functors are equivalent in the fppf Grothendieck topology. As an application, we obtain three presentations for the category of D-modules ‘on’ B(K)∖G(𝔸)/G(𝕆), and we combine results about this category coming from the different presentations.
    Compositio Mathematica 05/2014; 150(5).
  • Compositio Mathematica 04/2014; 150(4).
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    ABSTRACT: We construct new indecomposable elements in the higher Chow group CH 2 (A,1) of a principally polarized Abelian surface over a p-adic local field, which generalize an element constructed by Collino [Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J. Algebraic Geom. 6 (1997), 393-415]. These elements are constructed using a generalization, due to Birkenhake and Wilhelm [HumbertsurfacesandtheKummerplane, Trans. Amer. Math. Soc. 355 (2003), 1819-1841 (electronic)], of a classical construction of Humbert. They can be used to prove a non-Archimedean analogue of the Hodge-D-conjecture – namely, the surjectivity of the boundary map in the localization sequence – in the case where the Abelian surface has good and ordinary reduction.
    Compositio Mathematica 04/2014; 150(4).
  • Compositio Mathematica 01/2014; 150(5).
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    ABSTRACT: Generalizing previous results of Deligne-Serre and Taylor, Galois representations are attached to cuspidal automorphic representations of unitary groups whose Archimedean component is a holomorphic limit of discrete series. The main ingredient is a construction of congruences, using the Hasse invariant, that is independent of q-expansions.
    Compositio Mathematica 01/2014; 150(2).
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    ABSTRACT: We explain how the André-Oort conjecture for a general Shimura variety can be deduced from the hyperbolic Ax-Lindemann conjecture, a good lower bound for Galois orbits of special points and the definability, in the o-minimal structure ℝ an , exp , of the restriction to a fundamental set of the uniformizing map of a Shimura variety. These ingredients are known in some important cases. As a consequence a proof of the André-Oort conjecture for projective special subvarieties of A 6 N for an arbitrary integer N is given.
    Compositio Mathematica 01/2014; 150(2).
  • Compositio Mathematica 01/2014; 150(6).
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    ABSTRACT: In this paper, we construct a generalization of the Kohnen plus space for Hilbert modular forms of half-integral weight. The Kohnen plus space can be characterized by the eigenspace of a certain Hecke operator. It can be also characterized by the behavior of the Fourier coefficients. For example, in the parallel weight case, a modular form of weight κ+(1/2) with ξth Fourier coefficient c(ξ) belongs to the Kohnen plus space if and only if c(ξ)=0 unless (-1) κ ξ is congruent to a square modulo 4. The Kohnen subspace is isomorphic to a certain space of Jacobi forms. We also prove a generalization of the Kohnen-Zagier formula.
    Compositio Mathematica 12/2013; 149(12).
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    ABSTRACT: The author extends a previous joint work with B. C. Ngô [J. Inst. Math. Jussieu 7, No. 1, 181–203 (2008; Zbl 1141.22005)] and gives a fixed point formula for the elliptic part of moduli spaces of G-shtukas with arbitrary modifications. The formula is similar to the fixed point formula of Kottwitz for certain Shimura varieties. The method is inspired by that of Kottwitz and simpler than that of Lafforgue for the fixed point formula of the moduli space of Drinfeld GL (r)-shtukas.
    Compositio Mathematica 12/2013; 149(12).
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    ABSTRACT: Let k be a number field and S a set of primes containing all the infinite ones. Let A/k be a semi-abelian variety, Γ 0 a finitely generated subgroup of A(k ¯) and Γ⊆A(k ¯) be the division group attached to Γ 0 , i.e the set of points P∈A(k ¯) such that there exists a integer n such that nP∈Γ 0 . If X/k is any variety and X ¯ its completion, define ∂X:=X ¯-X. Let T be any subset of X ¯, and let T ¯ be its Zariski closure in X ¯. Any P∈X(k ¯) is said to be S-integral relative to T if it is (T ¯∪∂X,S)-integral on X ¯. The authors pose the following conjecture: Conjecture Let k and S be as above and let A/k be a semi-abelian variety and Γ a division group in A(k ¯). Suppose that D is a non-zero effective divisor on A which is not the translate of any torsion divisor by any point of Γ. Then the set {P∈Γ:PisS-integralrelativetoD} is not Zariski dense in A. The authors then prove the conjecture for 1-dimensional semi-abelian varieties, i.e. for elliptic curves and 1-dimensional tori.
    Compositio Mathematica 12/2013; 149(12).
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    ABSTRACT: Let $(X,B)$ be a projective log canonical pair such that $B$ is a $\Q$-divisor, and that there is a surjective morphism $f\colon X\to Z$ onto a normal variety $Z$ satisfying: $K_X+B\sim_\Q f^*M$ for some $\Q$-divisor $M$, and the augmented base locus ${\bf{B_+}}(M)$ does not contain the image of any log canonical centre of $(X,B)$. We will show that $(X,B)$ has a good log minimal model. An interesting special case is when $f$ is the identity morphism.
    Compositio Mathematica 05/2013;
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    ABSTRACT: Let K and F be complete discrete valuation fields of residue characteristic p>0. Let m be a positive integer no more than their absolute ramification indices. Let s and t be their uniformizers. Let L/K and E/F be finite extensions such that the modulo s^m of the extension O_L/O_K and modulo t^m of O_E/O_F are isomorphic. Let j=<m be a positive rational number. In this paper, we prove that the ramification of L/K is bounded by j if and only if the ramification of E/F is bounded by j. As an application, we prove that the categories of finite separable extensions of K and F whose ramifications are bounded by j are equivalent to each other, which generalizes a theorem of Deligne to the case of imperfect residue fields. We also show the compatibility of Scholl's theory of higher fields of norms with the ramification theory of Abbes-Saito, and the integrality of small Artin and Swan conductors of abelian extensions of mixed characteristic.
    Compositio Mathematica 04/2013;
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    ABSTRACT: Schinzel's Hypothesis (H) was used by Colliot-Th\'el\`ene and Sansuc, and later by Serre, Swinnerton-Dyer and others, to prove that the Brauer-Manin obstruction controls the Hasse principle and weak approximation on pencils of conics and similar varieties. We show that when the ground field is Q and the degenerate geometric fibres of the pencil are all defined over Q, one can use these methods to obtain unconditional results by replacing Hypothesis (H) with the finite complexity case of the generalised Hardy-Littlewood conjecture recently established by Green, Tao and Ziegler.
    Compositio Mathematica 04/2013;