Newton polygons for L-functions of generalized Kloosterman sums

In the present paper, we study the Newton polygons for the L-functions of n-variable generalized Kloosterman sums. Generally, the Newton polygon has a topological lower bound, called the Hodge polygon. In order to determine the Hodge polygon, we explicitly construct a basis of the top-dimensional Dwork cohomology. Using Wan’s decomposition theorem and diagonal local theory, we obtain when the Newton polygon coincides with the Hodge polygon. In particular, we concretely get the slope sequence for the L-function of F ¯ ⁢ ( λ ¯ , x ) := ∑ i = 1 n x i a i + λ ¯ ⁢ ∏ i = 1 n x i - 1 , \bar{F}(\bar{\lambda},x):=\sum_{i=1}^{n}x_{i}^{a_{i}}+\bar{\lambda}\prod_{i=1}% ^{n}x_{i}^{-1}, with a 1 , … , a n {a_{1},\ldots,a_{n}} being pairwise coprime for n ≥ 2 {n\geq 2} .

Logarithmic growth filtrations for -modules over the bounded Robba ring

In the 1970s, Dwork defined the logarithmic growth (log-growth for short) filtrations for $p$ -adic differential equations $Dx=0$ on the $p$ -adic open unit disc $|t|<1$ , which measure the asymptotic behavior of solutions $x$ as $|t|\to 1^{-}$ . Then, Dwork calculated the log-growth filtration for $p$ -adic Gaussian hypergeometric differential equation. In the late 2000s, Chiarellotto and Tsuzuki proposed a fundamental conjecture on the log-growth filtrations for $(\varphi ,\nabla )$ -modules over $K[\![t]\!]_0$ , which can be regarded as a generalization of Dwork's calculation. In this paper, we prove a generalization of the conjecture to $(\varphi ,\nabla )$ -modules over the bounded Robba ring. As an application, we prove a generalization of Dwork's conjecture proposed by Chiarellotto and Tsuzuki on the specialization property for log-growth Newton polygons.

Universal axion backreaction in flux compactifications

We study the backreaction effect of a large axion field excursion on the saxion partner residing in the same $$ \mathcal{N} $$ N = 1 multiplet. Such configurations are relevant in attempts to realize axion monodromy inflation in string compactifications. We work in the complex structure moduli sector of Calabi-Yau fourfold compactifications of F-theory with four-form fluxes, which covers many of the known Type II orientifold flux compactifications. Noting that axions can only arise near the boundary of the moduli space, the powerful results of asymptotic Hodge theory provide an ideal set of tools to draw general conclusions without the need to focus on specific geometric examples. We find that the boundary structure engraves a remarkable pattern in all possible scalar potentials generated by background fluxes. By studying the Newton polygons of the extremization conditions of all allowed scalar potentials and realizing the backreaction effects as Puiseux expansions, we find that this pattern forces a universal backreaction behavior of the large axion field on its saxion partner.

Classifying Galois groups of an orthogonal family of quartic polynomials

We consider the quartic generalized Laguerre polynomials $L_{4}^{(\alpha)}(x)$ for $\alpha \in \mathbb Q$. It is shown that except $\mathbb Z/4\mathbb Z$, every transitive subgroup of $S_{4}$ appears as the Galois group of $L_{4}^{(\alpha)}(x)$ for infinitely many $\alpha \in \mathbb Q$. A precise characterization of $\alpha\in \mathbb Q$ is obtained for each of these occurrences. Our methods involve the standard use of resolvent cubics and the theory of p-adic Newton polygons. Using these, the Galois group computations are reduced to Diophantine problem of finding integer and rational points on certain curves.

Solution of tetrahedron equation and cluster algebras

We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

Newton Polygon Stratification of the Torelli Locus in Unitary Shimura Varieties

We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $p$ reduction of certain Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus with some Newton polygon strata is non-empty. This allows us to give a positive answer, under some compatibility conditions, to a question of Oort about smooth curves in characteristic $p$ whose Newton polygons are an amalgamate sum. As an application, we produce infinitely many new examples of Newton polygons that occur for smooth curves that are cyclic covers of the projective line. Most of these arise in inductive systems that demonstrate unlikely intersections of the open Torelli locus with the Newton polygon stratification in Siegel modular varieties. In addition, for the 20 special Shimura varieties found in Moonen’s work, we prove that all Newton polygon strata intersect the open Torelli locus (if $p&gt;&gt;0$ in the supersingular cases).