Annales Mathematiques du Quebec最新文献

In this article, we show that Steiner symmetrization ((n-1)) times is sufficient to transform an ellipsoid to a ball in ({mathbb {R}}^n). Specifically, we seek out the ((n-1)) directions in the unit sphere such that the destination of the corresponding Steiner symmetrization is the standard ball.

We show that the character variety for a n-punctured oriented surface has a natural Poisson structure.

Groups of order 4 are isomorphic to either ({mathbb {Z}}/4{mathbb {Z}}) or ({mathbb {Z}}/2{mathbb {Z}} times {mathbb {Z}}/2{mathbb {Z}}). We give certain sufficient conditions permitting to specify the structure of class groups of order 4 in the family of real quadratic fields ({mathbb {Q}}{(sqrt{n^2+1})}) as n varies over positive integers. Further, we compute the values of Dedekind zeta function attached to these quadratic fields at the point (-1). As a side result, we show that the size of the class group of this family could be made as large as possible by increasing the size of the number of distinct odd prime factors of n.

Let (k) be an algebraically closed field of characteristic (0). For a log curve (X/k^{times }) over the standard log point (Kato in Int J Math 11(2):215–232, 2000), we define (algebraically) a combinatorial monodromy operator on its log-de Rham cohomology group. The invariant part of this action has a cohomological description, it is the Du Bois cohomology of (X) (Du Bois in Bull Soc Math Fr 109(1):41–81, 1981). This can be seen as an analogue of the invariant cycles exact sequence for a semistable family (as in the complex, étale and (p)-adic settings). In the specific case in which (k={mathbb {C}}) and (X) is the central fiber of a semistable degeneration over the complex disc, our construction recovers the topological monodromy and the classical local invariant cycles theorem. In particular, our description allows an explicit computation of the monodromy operator in this setting.

Using a cocycle defined by Monod and Shalom (J Differential Geom 67(3):395–455, 2004) we introduce the median quasimorphisms for groups acting on trees. Then we characterise actions on trees that give rise to non-trivial median quasimorphisms. Roughly speaking, either the action is highly transitive on geodesics, or it fixes a point in the boundary, or there exists an infinite family of non-trivial median quasimorphisms. In particular, in the last case the second bounded cohomology of the group is infinite dimensional as a vector space. As an application, we show that a cocompact lattice in the automorphism group of a product of trees has only trivial quasimorphisms if and only if the closures of the projections on each of the two factors are locally (infty )-transitive.

We prove an explicit integral representation—involving the pullback of a suitable Siegel Eisenstein series—for the twisted standard L-function associated to a holomorphic vector-valued Siegel cusp form of degree n and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to L-functions of vector-valued Siegel cusp forms. The key new ingredient in our method is a novel choice of local vectors at the archimedean place which allows us to exactly compute the archimedean local integral. By specializing our integral representation to the case (n=2) we are able to prove a reciprocity law—predicted by Deligne’s conjecture—for the critical values of the twisted standard L-function for vector-valued Siegel cusp forms of degree 2 and arbitrary level. This arithmetic application generalizes previously proved critical-value results for the full level case. By specializing further to the case of Siegel cusp forms obtained via the Ramakrishnan–Shahidi lift, we obtain a reciprocity law for the critical values of the symmetric fourth L-function of a classical newform.

Let H be a uniform pro-p group. Associated to H are rigid analytic affinoid groups ({mathbb {H}}_n), and their “wide open” subgroups ({mathbb {H}}_n^{circ }). Denote by (D^mathrm{la}(H)= C^mathrm{la}(H)'_b) the locally analytic distribution algebra of H and by (D({mathbb {H}}_n^{circ }, H)) Emerton’s ring of ({mathbb {H}}_n^{circ })-rigid analytic distributions on H. If V is an admissible locally analytic representation of H, and if (V_{{mathbb {H}}_n^circ -mathrm{an}}) denotes the subspace of ({mathbb {H}}_n^circ )-rigid analytic vectors (with its intrinsic topology), then we show that the continuous dual of (V_{{mathbb {H}}_n^circ -mathrm{an}}) is canonically isomorphic to (D({mathbb {H}}_n^{circ }, H)otimes _{D^mathrm{la}(H)} V'). From this we deduce the exactness of the functor (V rightsquigarrow V_{{mathbb {H}}_n^circ -mathrm{an}}) on the category of admissible locally analytic representations of H.

Given any non-polynomial G-function (F(z)=sum _{k=0}^infty A_kz^k) of radius of convergence R and in the kernel a G-operator (L_F), we consider the G-functions (F_n^{[s]}(z)=sum _{k=0}^infty frac{A_k}{(k+n)^s}z^k) for every integers (sge 0) and (nge 1). These functions can be analytically continued to a domain ({mathcal {D}}_F) star-shaped at 0 and containing the disk ({vert zvert <R}). Fix any (alpha in {mathcal {D}}_F cap overline{{mathbb {Q}}}^*), not a singularity of (L_F), and any number field ({mathbb {K}}) containing (alpha ) and the (A_k)’s. Let (Phi _{alpha , S}) be the ({mathbb {K}})-vector space spanned by the values (F_n^{[s]}(alpha )), (nge 1) and (0le sle S). We prove that (u_{{mathbb {K}},F}log (S)le dim _{mathbb {K}}(Phi _{alpha , S })le v_FS) for any S, for some constants (u_{{mathbb {K}},F}>0) and (v_F>0). This appears to be the first general Diophantine result for values of G-functions evaluated outside their disk of convergence. This theorem encompasses a previous result of the authors in [Linear independence of values of G-functions, J. Europ. Math. Soc. 22(5), 1531–1576 (2020)], where (alpha in overline{{mathbb {Q}}}^*) was assumed to be such that (vert alpha vert <R). Its proof relies on an explicit construction of a Padé approximation problem adapted to certain non-holomorphic functions associated to F, and it is quite different of that in the above mentioned paper. It makes use of results of André, Chudnovsky and Katz on G-operators, of a linear independence criterion à la Siegel over number fields, and of a far reaching generalization of Shidlovsky’s lemma built upon the approach of Bertrand–Beukers and Bertrand.