Annales Mathematiques du Quebec最新文献

In this article, we study the growth of (fine) Selmer groups of elliptic curves in certain infinite Galois extensions where the Galois group G is a uniform, pro-p, p-adic Lie group. By comparing the growth of (fine) Selmer groups with that of class groups, we show that it is possible for the (mu )-invariant of the (fine) Selmer group to become arbitrarily large in a certain class of nilpotent, uniform, pro-p Lie extension. We also study the growth of fine Selmer groups in false Tate curve extensions.

In this computational paper we verify a truncated version of the Buzzard–Calegari conjecture on the Newton polygon of the Hecke operator (T_2) for all large enough weights. We first develop a formula for computing p-adic valuations of exponential sums, which we then implement to compute 2-adic valuations of traces of Hecke operators acting on spaces of cusp forms. Finally, we verify that if Newton polygon of the Buzzard–Calegari polynomial has a vertex at (nle 15), then it agrees with the Newton polygon of (T_2) up to n.

Recently in symplectic geometry there arose an interest in bounding various functionals on spaces of matrices. It appears that Grothendieck’s theorem about factorization is a useful tool for proving such bounds. In this note we present two such applications.

We examine the spectrum of a family of Sturm–Liouville operators with regularly spaced delta function potentials parametrized by increasing strength. The limiting behavior of the eigenvalues under this spectral flow was described by Berkolaiko et al. (Lett Math Phys 109(7):1611–1623, 2019), where it was used to study the nodal deficiency of Laplacian eigenfunctions. Here we consider the eigenfunctions of these operators. In particular, we give explicit formulas for the limiting eigenfunctions, and also characterize the eigenfunctions and eigenvalues for all values for the spectral flow parameter (not just in the limit). We also develop spectrally accurate numerical tools for comparison and visualization.

In this note we prove that for each positive integer m there exists a bi-Lipschitz embedding ({mathbf{Z}}^mrightarrow {text {Ham}}(S^2)), where ({text {Ham}}(S^2)) is equipped with the entropy metric. In particular, the same result holds when the entropy metric is replaced with the autonomous metric.

In this article we define an elliptic double shuffle Lie algebra (scriptstyle {{mathfrak {ds}}_{ell}}) that generalizes the well-known double shuffle Lie algebra (scriptstyle {{mathfrak {ds}}}) to the elliptic situation. The double shuffle, or dimorphic, relations satisfied by elements of the Lie algebra (scriptstyle {{mathfrak {ds}}}) express two families of algebraic relations between multiple zeta values that conjecturally generate all relations. In analogy with this, elements of the elliptic double shuffle Lie algebra (scriptstyle {{mathfrak {ds}}_{ell}}) are Lie polynomials having a dimorphic property called (scriptstyle {Delta })-bialternality that conjecturally describes the (dual of the) set of algebraic relations between elliptic multiple zeta values, which arise as coefficients of a certain elliptic generating series (constructed explicitly in Lochak et al.[15]) in On elliptic multiple zeta values 2016, in preparation) and closely related to the elliptic associator defined by Enriquez[10]. We show that one of Ecalle’s major results in mould theory can be reinterpreted as yielding the existence of an injective Lie algebra morphism (scriptstyle {{mathfrak {ds}}rightarrow {mathfrak {ds}}_{ell}}). Our main result is the compatibility of this map with the tangential-base-point section (scriptstyle {mathrm{Lie},pi _1(MTM)rightarrow mathrm{Lie},pi _1(MEM)}) constructed by Hain and Matsumoto[14] and with the section (scriptstyle {{mathfrak {grt}}rightarrow {mathfrak {grt}}_{ell}}) mapping the Grothendieck–Teichmüller Lie algebra (scriptstyle {{mathfrak {grt}}}) into the elliptic Grothendieck–Teichmüller Lie algebra (scriptstyle {{mathfrak {grt}}_{ell}}) constructed by Enriquez. This compatibility is expressed by the commutativity of the following diagram (excluding the dotted arrow, which is conjectural).