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Register a new userInfinitesimal deformation of p-adic differential equations on Berkovich curves
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Authors
Andrea Pulita
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We show that if a differential equations $mathscr{F}$ over a quasi-smooth Berkovich curve $X$ has a certain compatibility condition with respect to an automorphism $sigma$ of $X$, and if the automorphism is sufficiently close to the identity, then $mathscr{F}$ acquires a semi-linear action of $sigma$ (i.e. lifting that on $X$). This generalizes the previous works of Yves Andre, Lucia Di Vizio, and the author about $p$-adic $q$-difference equations. We also obtain an application to Moritas $p$-adic Gamma function, and to related values of $p$-adic $L$-functions.
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We study the variation of the convergence Newton polygon of a differential equation along a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Relying on work of the second author who investigated its properties on affinoid domains of the affine line, we prove that its slopes give rise to continuous functions that factorize by the retraction through a locally finite subgraph of the curve.
We prove that the radii of convergence of the solutions of a $p$-adic differential equation $mathcal{F}$ over an affinoid domain $X$ of the Berkovich affine line are continuous functions on $X$ that factorize through the retraction of $XtoGamma$ of $X$ onto a finite graph $Gammasubseteq X$. We also prove their super-harmonicity properties. Roughly speaking, this finiteness result means that the behavior of the radii as functions on $X$ is controlled by a finite family of data.
We obtain an algorithm computing explicitly the values of the non solvable spectral radii of convergence of the solutions of a differential module over a point of type 2, 3 or 4 of the Berkovich affine line.
In 1922, Mordell conjectured the striking statement that for a polynomial equation $f(x,y)=0$, if the topology of the set of complex number solutions is complicated enough, then the set of rational number solutions is finite. This was proved by Faltings in 1983, and again by a different method by Vojta in 1991, but neither proof provided a way to provably find all the rational solutions, so the search for other proofs has continued. Recently, Lawrence and Venkatesh found a third proof, relying on variation in families of $p$-adic Galois representations; this is the subject of the present exposition.
In this paper, we focus on a family of generalized Kloosterman sums over the torus. With a few changes to Haessig and Sperbers construction, we derive some relative $p$-adic cohomologies corresponding to the $L$-functions. We present explicit forms of bases of top dimensional cohomology spaces, so to obtain a concrete method to compute lower bounds of Newton polygons of the $L$-functions. Using the theory of GKZ system, we derive the Dworks deformation equation for our family. Furthermore, with the help of Dworks dual theory and deformation theory, the strong Frobenius structure of this equation is established. Our work adds some new evidences for Dworks conjecture.
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