# Compositio Mathematica (COMPOS MATH)

## Journal description

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.

## Journal Impact: 1.26*

## Journal impact history

2017 Journal impact | Available summer 2018 |
---|---|

2015 / 2016 Journal impact | 1.26 |

2014 Journal impact | 1.07 |

2013 Journal impact | 1.38 |

2012 Journal impact | 0.90 |

2011 Journal impact | 1.25 |

2010 Journal impact | 0.73 |

2009 Journal impact | 0.89 |

2008 Journal impact | 0.74 |

2007 Journal impact | 1.57 |

2006 Journal impact | 1.56 |

2005 Journal impact | 1.06 |

2004 Journal impact | 1.00 |

2003 Journal impact | 1.17 |

2002 Journal impact | 1.00 |

2001 Journal impact | 0.81 |

2000 Journal impact | 0.97 |

## Journal impact over time

## Additional details

Cited half-life | >10.0 |
---|---|

Immediacy index | 0.20 |

Eigenfactor | 0.01 |

Article influence | 2.38 |

Website | Compositio Mathematica website |

Other titles | Compositio mathematica |

ISSN | 0010-437X |

OCLC | 1564581 |

Material type | Periodical, Internet resource |

Document type | Journal / Magazine / Newspaper, Internet Resource |

## Publisher details

## Publications in this journal

- [Show abstract] [Hide abstract]
**ABSTRACT:**Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and a positive integer $g$ , there is an integer $m_{0}$ such that for any $m>m_{0}$ there is no principally polarized abelian variety $A/K$ of dimension $g$ with full level- $m$ structure. To this end, we develop a version of Vojta’s conjecture for Deligne–Mumford stacks, which we deduce from Vojta’s conjecture for schemes. - [Show abstract] [Hide abstract]
**ABSTRACT:**In this paper, we propose a conjectural identity between the Fourier–Jacobi periods on symplectic groups and the central value of certain Rankin–Selberg $L$ -functions. This identity can be viewed as a refinement to the global Gan–Gross–Prasad conjecture for $\text{Sp}(2n)\times \text{Mp}(2m)$ . To support this conjectural identity, we show that when $n=m$ and $n=m\pm 1$ , it can be deduced from the Ichino–Ikeda conjecture in some cases via theta correspondences. As a corollary, the conjectural identity holds when $n=m=1$ or when $n=2$ , $m=1$ and the automorphic representation on the bigger group is endoscopic. -
##### Article: Corrigendum: Connected components of affine Deligne–Lusztig varieties in mixed characteristic

[Show abstract] [Hide abstract]**ABSTRACT:**We correct an error in our paper ‘Connected components of affine Deligne–Lusztig varieties in mixed characteristic’. - [Show abstract] [Hide abstract]
**ABSTRACT:**We correct the proof of the main result of the paper, Theorem 5.7. Our corrected proof relies on weaker versions of a number of intermediate results from the paper. The original, more general, versions of these statements are not known to be true. - [Show abstract] [Hide abstract]
**ABSTRACT:**We study the $p$ -adic variation of triangulations over $p$ -adic families of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules. In particular, we study certain canonical sub-filtrations of the pointwise triangulations and show that they extend to affinoid neighborhoods of crystalline points. This generalizes results of Kedlaya, Pottharst and Xiao and (independently) Liu in the case where one expects the entire triangulation to extend. We also study the ramification of weight parameters over natural $p$ -adic families. - [Show abstract] [Hide abstract]
**ABSTRACT:**The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on $\mathbb{C}\text{P}^{2}$ with singularities at a collection of complex lines. In the former case we conjecture that quotient spaces always have a $\text{CAT}[0]$ ramification and prove this in several cases. In the latter case we prove that the ramification is $\text{CAT}[0]$ if the metric on $\mathbb{C}\text{P}^{2}$ is non-negatively curved. We deduce that complex line arrangements in $\mathbb{C}\text{P}^{2}$ studied by Hirzebruch have aspherical complement. - [Show abstract] [Hide abstract]
**ABSTRACT:**Let $C$ be a smooth, separated and geometrically connected curve over a finitely generated field $k$ of characteristic $p\geqslant 0$ , ${\it\eta}$ the generic point of $C$ and ${\it\pi}_{1}(C)$ its étale fundamental group. Let $f:X\rightarrow C$ be a smooth proper morphism, and $i\geqslant 0$ , $j$ integers. To the family of continuous $\mathbb{F}_{\ell }$ -linear representations ${\it\pi}_{1}(C)\rightarrow \text{GL}(R^{i}f_{\ast }\mathbb{F}_{\ell }(j)_{\overline{{\it\eta}}})$ (where $\ell$ runs over primes $\neq p$ ), one can attach families of abstract modular curves $C_{0}(\ell )$ and $C_{1}(\ell )$ , which, in this setting, are the analogues of the usual modular curves $Y_{0}(\ell )$ and $Y_{1}(\ell )$ . If $i\not =2j$ , it is conjectured that the geometric and arithmetic gonalities of these abstract modular curves go to infinity with $\ell$ (for the geometric gonality, under a certain necessary condition). We prove the conjecture for the arithmetic gonality of the abstract modular curves $C_{1}(\ell )$ . We also obtain partial results for the growth of the geometric gonality of $C_{0}(\ell )$ and $C_{1}(\ell )$ . The common strategy underlying these results consists in reducing by specialization theory to the case where the base field $k$ is finite in order to apply techniques of counting rational points. - [Show abstract] [Hide abstract]
**ABSTRACT:**In the context of varieties of representations of arbitrary quivers, possibly carrying loops, we define a generalization of Lusztig Lagrangian subvarieties. From the combinatorial study of their irreducible components arises a structure richer than the usual Kashiwara crystals. Along with the geometric study of Nakajima quiver varieties, in the same context, this yields a notion of generalized crystals, coming with a tensor product. As an application, we define the semicanonical basis of the Hopf algebra generalizing quantum groups, which was already equipped with a canonical basis. The irreducible components of the Nakajima varieties provide the family of highest weight crystals associated to dominant weights, as in the classical case. - [Show abstract] [Hide abstract]
**ABSTRACT:**For a prime number $p$ , we show that differentials $d_{n}$ in the motivic cohomology spectral sequence with $p$ -local coefficients vanish unless $p-1$ divides $n-1$ . We obtain an explicit formula for the first non-trivial differential $d_{p}$ , expressing it in terms of motivic Steenrod $p$ -power operations and Bockstein maps. To this end, we compute the algebra of operations of weight $p-1$ with $p$ -local coefficients. Finally, we construct examples of varieties having non-trivial differentials $d_{p}$ in their motivic cohomology spectral sequences. - [Show abstract] [Hide abstract]
**ABSTRACT:**Recently C. Houdayer and Y. Isono have proved among other things that every biexact group $\Gamma$ has the property that for any non-singular strongly ergodic action $\Gamma\curvearrowright (X,\mu)$ on a standard measure space the group measure space von Neumann algebra $\Gamma\ltimes L^\infty(X)$ is full. In this note, we prove the same property for a wider class of groups, notably including $\mathrm{SL}(3,{\mathbb Z})$. We also prove that for any connected simple Lie group $G$ with finite center, any lattice $\Gamma\le G$, and any closed non-amenable subgroup $H\le G$, the non-singular action $\Gamma\curvearrowright G/H$ is strongly ergodic and the von Neumann factor $\Gamma\ltimes L^\infty(G/H)$ is full. - [Show abstract] [Hide abstract]
**ABSTRACT:**In this paper, we study real-dihedral harmonic Maass forms and their Fourier coefficients. The main result expresses the values of Hilbert modular forms at twisted CM 0-cycles in terms of these Fourier coefficients. This is a twisted version of the main theorem in Bruinier and Yang [ CM-values of Hilbert modular functions , Invent. Math. 163 (2006), 229–288] and provides evidence that the individual Fourier coefficients are logarithms of algebraic numbers in the appropriate real-quadratic field. From this result and numerical calculations, we formulate an algebraicity conjecture, which is an analogue of Stark’s conjecture in the setting of harmonic Maass forms. Also, we give a conjectural description of the primes appearing in CM-values of Hilbert modular functions. -
##### Article: Non-rigid quartic -folds

[Show abstract] [Hide abstract]**ABSTRACT:**Let $X\subset \mathbb{P}^{4}$ be a terminal factorial quartic $3$ -fold. If $X$ is non-singular, $X$ is birationally rigid , i.e. the classical minimal model program on any terminal $\mathbb{Q}$ -factorial projective variety $Z$ birational to $X$ always terminates with $X$ . This no longer holds when $X$ is singular, but very few examples of non-rigid factorial quartics are known. In this article, we first bound the local analytic type of singularities that may occur on a terminal factorial quartic hypersurface $X\subset \mathbb{P}^{4}$ . A singular point on such a hypersurface is of type $cA_{n}$ ( $n\geqslant 1$ ), or of type $cD_{m}$ ( $m\geqslant 4$ ) or of type $cE_{6},cE_{7}$ or $cE_{8}$ . We first show that if $(P\in X)$ is of type $cA_{n}$ , $n$ is at most $7$ and, if $(P\in X)$ is of type $cD_{m}$ , $m$ is at most $8$ . We then construct examples of non-rigid factorial quartic hypersurfaces whose singular loci consist (a) of a single point of type $cA_{n}$ for $2\leqslant n\leqslant 7$ , (b) of a single point of type $cD_{m}$ for $m=4$ or $5$ and (c) of a single point of type $cE_{k}$ for $k=6,7$ or $8$ . - [Show abstract] [Hide abstract]
**ABSTRACT:**We present a conjectural formula describing the cokernel of the Albanese map of zero-cycles of smooth projective varieties $X$ over $p$-adic fields in terms of the N\'eron-Severi group and provide a proof under additional assumptions on an integral model of $X$. The proof depends on a non-degeneracy result of Brauer-Manin pairing due to Saito-Sato and on Gabber-de Jong's comparison result of cohomological- and Azumaya-Brauer groups. We can consider the local-global problem of the Albanese-cokernel; the abelian group on the "local side" turns out to be a finite group. - [Show abstract] [Hide abstract]
**ABSTRACT:**Let $C$ be a curve and $L$ a very ample line bundle. The Green-Lazarsfeld Secant conjecture predicts that if the degree of $L$ is at least $2g+p+1-2h^1(C,L)-Cliff(C)$ and if, in addition, $L$ is $p+1$ very ample, then the Koszul group $K_{p,2}(C,L)$ vanishes. In this article, we establish the conjecture in the extremal case, i.e.\ the case where the degree is exactly $2g+p+1-2h^1(C,L)-Cliff(C)$, subject to explicit genericity assumptions on $C$ and $L$. In particular, the gonality of $C$ is allowed to be arbitrary (in our cases $gon(C)=Cliff(C)+2$). - [Show abstract] [Hide abstract]
**ABSTRACT:**The notion of Berman-Gibbs stability was originally introduced by Robert Berman for $\mathbb{Q}$-Fano varieties $X$. We show that the pair $(X, -K_X)$ is K-stable (resp. K-semistable) provided that $X$ is Berman-Gibbs stable (resp. semistable). - [Show abstract] [Hide abstract]
**ABSTRACT:**Let $R$ be a commutative ring, let $F$ be a locally compact non-archimedean field of finite residual field $k$ of characteristic $p$ , and let $\mathbf{G}$ be a connected reductive $F$ -group. We show that the pro- $p$ -Iwahori Hecke $R$ -algebra of $G=\mathbf{G}(F)$ admits a presentation similar to the Iwahori–Matsumoto presentation of the Iwahori Hecke algebra of a Chevalley group, and alcove walk bases satisfying Bernstein relations. This was previously known only for a $F$ -split group $\mathbf{G}$ . - [Show abstract] [Hide abstract]
**ABSTRACT:**Given $k \geq 2$, we show that there are at most finitely many rational numbers $x$ and $y \neq 0$ and integers $\ell \geq 2$ (with $(k,\ell) \neq (2,2)$) for which $$ x (x+1) \cdots (x+k-1) = y^\ell. $$ In particular, if we assume that $\ell$ is prime, then all such triples $(x,y,\ell)$ satisfy either $y=0$ or $\log \ell < 3^k$.

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