Annales Mathematiques du Quebec最新文献

In this paper we continue the study of symplectically self-polar convex bodies started in [3]. We construct symplectically self-polar convex bodies of the minimal Ekeland–Hofer–Zehnder capacity. This in turn proves that the lower bound for the Ekeland–Hofer–Zehnder capacity for centrally symmetric convex bodies obtained in [1] cannot be improved. We also make some numerical experiments and speculations regarding the minimal volume of symplectically self-polar convex bodies.

We consider the Steklov problem on differential (ptext {-})forms defined by Karpukhin and present geometric eigenvalue bounds in the setting of warped product manifolds in various scenarios. In particular, we obtain Escobar type lower bounds for warped product manifolds with non-negative Ricci curvature and strictly convex boundary, and certain sharp bounds for hypersurfaces of revolution, among others. We compare and contrast the behaviour with known results in the case of functions (i.e., (0text {-})forms), highlighting the influence of the underlying topology on the spectrum for (ptext {-})forms in general.

The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the corresponding eigenvalue of a disjoint union of two equal disks, for Robin parameters in ([-4pi ,4pi ].) This sharp inequality was known previously only for negative parameters in ([-4pi ,0],) by Girouard and Laugesen. Their proof fails for positive Robin parameters because the second eigenfunction on a disk has non-monotonic radial part. This difficulty is overcome for parameters in ((0,4pi ]) by means of a degree-theoretic approach suggested by Karpukhin and Stern that yields suitably orthogonal trial functions.

Let M and N be simply-connected n-dimensional Poincaré Duality complexes. A condition is given on M and N which allows for the based loops on the connected sum (M#N) to be expressed as a product of the based loops on M, the based loops on N, and an explicitly identified complementary factor. This is analogous to Ganea’s decomposition of the based loops on a wedge. A generalization is given for a connected sum of k Poincaré Duality complexes for any (kge 2). The required condition holds, for instance, for products of spheres. Examples are given that are of particular interest in toric topology.

For (S_k), the space of cusp forms of weight k for the full modular group, we first introduce periods on (S_k) associated to symmetric square L-functions. We then prove that for a fixed natural number n, if k is sufficiently large relative to n, then any n such periods are linearly independent. With some extra assumption, we also prove that for (kge e^{12}), we can always pick up to (frac{log k}{4}) arbitrary linearly independent periods.

Let (Gamma ={mathbb {Q}}(root 3 of {n})) be a pure cubic field with normal closure (k={mathbb {Q}}(root 3 of {n},zeta ),) where (n>1) denotes a cube free integer, and (zeta ) is a primitive cube root of unity. Suppose k possesses an elementary bicyclic 3-class group ({textrm{Cl}}_3(k),) and the conductor of (k/{mathbb {Q}}(zeta )) has the shape (fin lbrace pq_1q_2,3pq,9pqrbrace ) where (pequiv 1,({textrm{mod}},9)) and (q,q_1,q_2equiv 2,5,({textrm{mod}},9)) are primes. It is disproved that there are only two possible capitulation types (varkappa (k),) either type ({textrm{a}}.1,) (0000),  or type ({textrm{a}}.2,) (1000). Evidence is provided, theoretically and experimentally, of two further types, ({text {b}}.10,) (0320),  and ({textrm{d}}.23,) (1320).

Let N be a positive integer and K be a number field. Suppose that (f_1, f_2in S_k(Gamma _0(N))) are two newforms such that their residual Galois representations at 2 are isomorphic. Let (omega _2:G_{mathbb {Q}}rightarrow {mathbb {Z}}_2^*) be the 2-adic cyclotomic character. Then, under suitable hypotheses, we have shown that for every quadratic character (chi ) of K and each critical twist j,  the residual Greenberg 2-Selmer groups of (f_1chi omega _p^{-j}) and (f_2chi omega _p^{-j}) over K are isomorphic. This generalizes the corresponding result of Mazur–Rubin on 2-Selmer companion elliptic curves. Conversely, if the difference of the residual Greenberg (respectively Bloch–Kato) 2-Selmer ranks of (f_1chi ) and (f_2chi ) is bounded independent of every quadratic character (chi ) of K,  then under suitable hypotheses we have shown that the residual Galois representations at 2 of (f_1) and (f_2) are isomorphic as (G_K)-modules. The corresponding result for elliptic curves was a conjecture of Mazur–Rubin, which was proved by M. Yu.

In this article, we construct a family of elliptic curves (E^A: y^2 = (x + A)(x^2 + A^2)) with one (2)-torsion point over (mathbb {Q}) and prove that there exist infinitely many square-free integers ( d ) such that the rank of the quadratic twists of ( E^A ) by ( d ) is zero. This work is a generalization of the result of M. Xiong: [On positive proportion of rank-zero twists of elliptic curves over ({mathbb {Q}}), J Aust Math Soc 98:281–288, (2015)].

We provide explicit bounds for the number of integral ideals of norms at most X in (mathbb {Q}[sqrt{d}]) when (d <0) is a fundamental discriminant with an error term of size (mathcal {O}(X^{1/3})). In particular, we prove that, when (chi ) is the non-principal character modulo 3 and (Xge 1), we have (sum _{nle X}(1!!!1star chi )(n) = frac{pi X}{3sqrt{3}} +mathcal {O}^*(1.94,X^{1/3})), and that, when (chi ) is the non-principal character modulo 4 and (Xge 1), we have (sum _{nle X}(1!!!1star chi )(n) = frac{pi X}{4} +mathcal {O}^*(1.4,X^{1/3})). Résumé. Nous dénombrons de façon explicite avec un terme d’erreur (mathcal {O}(X^{1/3})) le nombre d’idéaux entiers de norme au plus X du corps (mathbb {Q}[sqrt{d}]) lorsque (d <0) est un discriminant fondamental. Nous montrons en particulier que, lorsque (chi ) est le caractère non principal modulo 3 et (Xge 1), nous avons (sum _{nle X}(1!!!1star chi )(n) = frac{pi X}{3sqrt{3}} +mathcal {O}^*(1.94,X^{1/3})), et que , lorsque (chi ) est le caractère non principal modulo 4 et (Xge 1), nous avons (sum _{nle X}(1!!!1star chi )(n) = frac{pi X}{4} +mathcal {O}^*(1.4,X^{1/3} )).

We provide examples of contractible complexes which fail to have non-positive immersions and weak non-positive immersions, answering a conjecture of Wise in the negative.