Abstract. We compute the dimension of the tangent space to, and the Krull dimension of the pro-representable hull of two deformation functors. The first one is the "algebraic" deformation functor of an ordinary curve X over a field of positive characteristic with prescribed action of a finite group G, and the data are computed in terms of the ramification behaviour of X → G\X. The second one is the "analytic" deformation functor of a fixed embedding of a finitely generated discrete group N in P GL(2, K) over a non-archimedean valued field K, and the data are computed in terms of the Bass-Serre representation of N via a graph of groups. Finally, if Γ is a free subgroup of N such that N is contained in the normalizer of Γ in P GL(2, K), then the Mumford curve associated to Γ becomes equipped with an action of N/Γ, and we show that the algebraic functor deforming the latter action coincides with the analytic functor deforming the embedding of N .Introduction.
Abstract. We present a method to control gonality of nonarchimedean curves based on graph theory.Let k denote a complete nonarchimedean valued field. We first prove a lower bound for the gonality of a curve over the algebraic closure of k in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve.Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some "volume" of the original graph; this can be seen as a substitute for graphs of the Li-Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs.Finally, we apply the results to give a lower bound for the gonality of arbitrary Drinfeld modular curves over finite fields and for general congruence subgroups Γ of Γ(1) that is linear in the index [Γ(1) : Γ], with a constant that only depends on the residue field degree and the degree of the chosen "infinite" place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian.
Abstract. It is known that two number fields with the same Dedekind zeta function are not necessarily isomorphic. The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical system, which is a C * -algebra with a one parameter group of automorphisms, built from Artin reciprocity. In the first part of this paper, we prove that isomorphism of number fields is the same as isomorphism of these associated systems. Considering the systems as noncommutative analogues of topological spaces, this result can be seen as another version of Grothendieck's "anabelian" program, much like the Neukirch-Uchida theorem characterizes isomorphism of number fields by topological isomorphism of their associated absolute Galois groups.In the second part of the paper, we use these systems to prove the following. If there is a group isomorphism ψ :L between the character groups (viz., Pontrjagin duals) of the abelianized Galois groups of the two number fields that induces an equality of all corresponding L-series s) (not just the zeta function), then the number fields are isomorphic. This is also equivalent to the purely algebraic statement that there exists an isomorphism ψ as a above and a norm-preserving group isomorphism between the ideals of K and L that is compatible with the Artin maps via ψ.
Julia Robinson has given a first-order definition of the rational integers Z in the rational numbers Q by a formula (∀∃∀∃)(F = 0) where the ∀-quantifiers run over a total of 8 variables, and where F is a polynomial. This implies that the Σ 5 -theory of Q is undecidable. We prove that a conjecture about elliptic curves provides an interpretation of Z in Q with quantifier complexity ∀∃, involving only one universally quantified variable. This improves the complexity of defining Z in Q in two ways, and implies that the Σ 3 -theory, and even the Π 2 -theory, of Q is undecidable (recall that Hilbert's Tenth Problem for Q is the question whether the Σ 1 -theory of Q is undecidable).
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