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Let be an ordinary (projective, geometrically irreducible, nonsingular) algebraic curve of genus defined over an algebraically closed field of odd characteristic . Let be the group of all automorphisms of which fix elementwise. For any solvable subgroup of we prove that . There are known curves attaining this bound up to the constant . For odd, our result improves the classical Nakajima bound and, for solvable groups , the Gunby–Smith–Yuan bound where for some positive constant .
We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain -functions of degree 2 and higher in , in the limit as . This is achieved by establishing appropriate equidistribution results for the associated Frobenius conjugacy classes. The variances are thus related to matrix integrals, which may be evaluated. Our results differ significantly from those that hold in the case of degree-1 -functions (i.e., situations considered previously using this approach). They correspond to expressions found recently in the number field setting assuming a generalization of the pair correlation conjecture. Our calculations apply, for example, to elliptic curves defined over .
Recently, Andreatta, Iovita and Pilloni constructed spaces of overconvergent modular forms in characteristic , together with a natural extension of the Coleman–Mazur eigencurve over a compactified (adic) weight space. Similar ideas have also been used by Liu, Wan and Xiao to study the boundary of the eigencurve. This all goes back to an idea of Coleman.
In this article, we construct natural extensions of eigenvarieties for arbitrary reductive groups over a number field which are split at all places above . If is , then we obtain a new construction of the extended eigencurve of Andreatta–Iovita–Pilloni. If is an inner form of associated to a definite quaternion algebra, our work gives a new perspective on some of the results of Liu–Wan–Xiao.
We build our extended eigenvarieties following Hansen’s construction using overconvergent cohomology. One key ingredient is a definition of locally analytic distribution modules which permits coefficients of characteristic (and mixed characteristic). When is over a totally real or CM number field, we also construct a family of Galois representations over the reduced extended eigenvariety.
Runge’s method is a tool to figure out integral points on algebraic curves effectively in terms of height. This method has been generalized to varieties of any dimension, but unfortunately the conditions needed to apply it are often too restrictive. We provide a further generalization intended to be more flexible while still effective, and exemplify its applicability by giving finiteness results for integral points on some Siegel modular varieties. As a special case, we obtain an explicit finiteness result for integral points on the Siegel modular variety .
This paper studies universal families of stable genus-2 curves with level structure. Among other things, it is shown that the -part is spanned by divisor classes, and that there are no cycles of type in the third cohomology of the first direct image of under projection to the moduli space of curves. Using this, it shown that the Hodge and Tate conjectures hold for these varieties.
The generalized Riemann hypothesis implies that at least 50% of the central values are nonvanishing as ranges over primitive characters modulo . We show that one may unconditionally go beyond GRH, in the sense that if one averages over primitive characters modulo and averages over an interval, then at least 50.073% of the central values are nonvanishing. The proof utilizes the mollification method with a three-piece mollifier, and relies on estimates for sums of Kloosterman sums due to Deshouillers and Iwaniec.
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