This paper provides a construction of a quantum statistical mechanical system associated to knots in the 3-sphere and cyclic branched coverings of the 3-sphere, which is an analog, in the sense of arithmetic topology, of the Bost-Connes system, with knots replacing primes, and cyclic branched coverings of the 3-sphere replacing abelian extensions of the field of rational numbers. The operator algebraic properties of this system differ significantly from the Bost-Connes case, due to the properties of the action of the semigroup of knots on a direct limit of knot groups. The resulting algebra of observables is a noncommutative Bernoulli product. We describe the main properties of the associated quantum statistical mechanical system and of the relevant partition functions, which are obtained from simple knot invariants like genus and crossing number.
Let (G, X) be a Shimura datum of Hodge type, and S K (G, X) its integral model with hyperspecial level structure. We prove that S K (G, X) admits a closed embedding, which is compatible with moduli interpretations, into the integral model S K ′ (GSp, S ± ) for a Siegel modular variety. In particular, the normalization step in the construction of S K (G, X) is redundant.In particular, our results apply to the earlier integral models constructed by Rapoport and Kottwitz, as those models agree with the Hodge type integral models for appropriately chosen Shimura data. * (G, X); (2) if Z K ′ * = ∅, then we can take a closed point x ∈ Z K ′ * , hence x ∈ Z K ′ for any K ′ containing K, such that ν is not an isomorphism at x. Therefore, when we are not in situation (1), there exist two points x, x ′ ∈ S K (G, X) that map to x ∈ S K ′ (GSp, S ± ) for all K ′ containing K.
In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-1 subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension g ≥ 3, there are infinitely many abelian varieties over Q with adelic Galois representation having image equal to all of GSp 2g ( Z).
Let f and g be two cuspidal modular forms and let
${\mathcal {F}}$
be a Coleman family passing through f, defined over an open affinoid subdomain V of weight space
$\mathcal {W}$
. Using ideas of Pottharst, under certain hypotheses on f and g, we construct a coherent sheaf over
$V \times \mathcal {W}$
that interpolates the Bloch–Kato Selmer group of the Rankin–Selberg convolution of two modular forms in the critical range (i.e, the range where the p-adic L-function
$L_p$
interpolates critical values of the global L-function). We show that the support of this sheaf is contained in the vanishing locus of
$L_p$
.
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