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Kiran S. Kedlaya

Publications and source records attributed to Kiran S. Kedlaya.

At least 37 records · Page 2Linked to original sources
arXiv2023

Angle ranks of abelian varieties

Using the formalism of Newton hyperplane arrangements, we resolve the open questions regarding angle rank left over from [DKRV20]. As a consequence we end up generalizing theorems of Lenstra--Zarhin and Tankeev proving several new cases of the Tate conjecture for abelian varieties over finite fields. We also obtain an effective version of a recent theorem of Zarhin bounding the heights of coefficients in multiplicative relations among Frobenius eigenvalues.

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arXiv2023

Motivic Serre group and Sato--Tate conjecture

This paper concerns the Algebraic Sato--Tate and Sato--Tate conjectures, based on Serre's original motivic formulation, with an eye towards explicit computations of Sato--Tate groups. We build on the algebraic framework for the Sato--Tate conjecture introduced in a previous paper, which used Deligne's motivic category for absolute Hodge cycles and was restricted to motives of odd weight. Here, we allow general weight and some other motivic categories, notably André's motivic category of motivated cycles; moreover, some results are also new in the odd weight case. The paper consists of two parts; in the first part we work in the framework of strongly compatible families of $l$-adic representations associated with pure, rational, polarized Hodge structures, while in the second part we use the language of motives.

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arXiv2023

A criterion for perfectoid fields

The tilting correspondence is a fundamental property of perfectoid fields. In this note, we show that the tilting construction can also be used to detect perfectoid fields among nonarchimedean fields. In particular, for $K$ a complete subfield of $\mathbb{C}_p$ (a completed algebraic closure of $\mathbb{Q}_p$), $K$ is perfectoid if and only if its tilt is not algebraic over $\mathbb{F}_p$. We also include some conjectures on APF (arithmetically profinite) extensions, perfectoid fields, and their relations.

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arXiv2022

Etale and crystalline companions, I

Let $X$ be a smooth scheme over a finite field of characteristic $p$. Consider the coefficient objects of locally constant rank on $X$ in $\ell$-adic Weil cohomology: these are lisse Weil sheaves in étale cohomology when $\ell \neq p$, and overconvergent $F$-isocrystals in rigid cohomology when $\ell=p$. Using the Langlands correspondence for global function fields in both the étale and crystalline settings (work of Lafforgue and Abe, respectively), one sees that on a curve, any coefficient object in one category has "companions" in the other categories with matching characteristic polynomials of Frobenius at closed points. A similar statement is expected for general $X$; building on work of Deligne, Drinfeld showed that any étale coefficient object has étale companions. We adapt Drinfeld's method to show that any crystalline coefficient object has étale companions; this has been shown independently by Abe--Esnault. We also prove some auxiliary results relevant for the construction of crystalline companions of étale coefficient objects; this subject will be pursued in a subsequent paper.

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arXiv2022

The relative class number one problem for function fields, I

We reduce the classification of finite extensions of function fields (of curves over finite fields) with the same class number to a finite computation; complete this computation in all cases except when both curves have base field $\mathbb{F}_2$ and genus $>1$; and give a conjectural answer in the remaining cases. The conjecture will be resolved in subsequent papers.

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arXiv2022

Geometric decomposition of abelian varieties of order 1

Since the 1970s, the complete classification (up to isogeny) of abelian varieties over finite fields with trivial group of rational points has been known from results of Madan--Pal and Robinson; with two exceptions these are all defined over $\mathbb{F}_2$. We determine the decomposition of these varieties into simple factors over an algebraic closure of $\mathbb{F}_2$; this requires solving a polynomial equation in three roots of unity.

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arXiv2022

Abelian varieties over $\mathbb{F}_2$ of prescribed order

We prove that for every positive integer $m$, there exist infinitely many simple abelian varieties over $\mathbb{F}_2$ of order $m$. The method is constructive, building on the work of Madan--Pal in the case $m=1$ to produce an explicit sequence of Weil polynomials giving rise to abelian varieties over $\mathbb{F}_2$ of order $m$. This sequence itself depends on the choice of a suitable generalized binary representation of $m$; by making careful choices of this representation, we can ensure that the the resulting sequence of polynomials have 2-adic Newton polygons which guarantee the existence of suitable irreducible factors.

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arXiv2022

Computing L-series of hyperelliptic curves

We discuss the computation of coefficients of the L-series associated to a hyperelliptic curve over Q of genus at most 3, using point counting, generic group algorithms, and p-adic methods.

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arXiv2022

A cut-by-curves criterion for overconvergent of $F$-isocrystals

Let $X$ be a smooth scheme over a finite field. It is conjectured that a convergent $F$-isocrystal on $X$ is overconvergent if its restriction to every curve contained in $X$ is overconvergent. Using the theory of étale and crystalline companions, we establish a weaker version of this criterion in which we also assume that the wild local monodromy of the restrictions to curves is trivialized by pullback along a single dominant morphism to $X$.

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arXiv2022

Notes on isocrystals

For varieties over a perfect field of characteristic p, etale cohomology with Q_l-coefficients is a Weil cohomology theory only when l is not equal to p; the corresponding role for l = p is played by Berthelot's rigid cohomology. In that theory, the coefficient objects analogous to lisse l-adic sheaves are the overconvergent F-isocrystals. This expository article is a brief user's guide for these objects, including some features shared with l-adic cohomology (purity, weights) and some features exclusive to the p-adic case (Newton polygons, convergence and overconvergence). The relationship between the two cases, via the theory of companions, will be treated in a sequel paper.

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arXiv2021

Frobenius structures on hypergeometric equations

We give an exposition of Dwork's construction of Frobenius structures associated to generalized hypergeometric equations via the interpretation of the latter due to Gelfand-Kapranov-Zelevinsky in the language of A-hypergeometric systems. As a consequence, we extract some explicit formulas for the degeneration at 0 in terms of the Morita p-adic gamma function.

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arXiv2021

Tamely ramified morphisms of curves and Belyi's theorem in positive characteristic

We show that every smooth projective curve over a finite field k admits a finite tame morphism to the projective line over k. Furthermore, we construct a curve with no such map when k is an infinite perfect field of characteristic two. Our work leads to a refinement of the tame Belyi theorem in positive characteristic, building on results of Saïdi, Sugiyama-Yasuda, and Anbar-Tutdere.

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arXiv2021

Drinfeld's lemma for perfectoid spaces and overconvergence of multivariate $(φ, Γ)$-modules

Let $p$ be a prime, let $K$ be a finite extension of $\mathbb{Q}_p$, and let $n$ be a positive integer. We construct equivalences of categories between continuous $p$-adic representations of the $n$-fold product of the absolute Galois group $G_K$ and $(φ, Γ)$-modules over one of several rings of $n$-variable power series. The case $n=1$ recovers the original construction of Fontaine and the subsequent refinement by Cherbonnier--Colmez; for general $n$, the case $K = \mathbb{Q}_p$ had been previously treated by the third author. To handle general $K$ uniformly, we use a form of Drinfeld's lemma on profinite fundamental groups of products of spaces in characteristic $p$, but for perfectoid spaces instead of schemes. We also construct the multivariate analogue of the Herr complex to compute Galois cohomology; the case $K = \mathbb{Q}_p$ had been previously treated by Pal and the third author, and we reduce to this case using a form of Shapiro's lemma.

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arXiv2021

Sato-Tate groups of abelian threefolds: a preview of the classification

We announce the classification of Sato-Tate groups of abelian threefolds over number fields; there are 410 possible conjugacy classes of closed subgroups of USp(6) that occur. We summarize the key points of the "upper bound" aspect of the classification, and give a more rigorous treatment of the "lower bound" by realizing 33 groups that appear in the classification as maximal cases with respect to inclusions of finite index. Further details will be provided in a subsequent paper.

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arXiv2021

Simple connectivity of Fargues--Fontaine curves

We show that the Fargues--Fontaine curve associated to an algebraically closed field of characteristic p is geometrically simply connected; that is, its base extension from Q_p to any complete algebraically closed overfield admits no nontrivial connected finite etale covering. We then deduce from this an analogue for perfectoid spaces (and some related objects) of Drinfeld's lemma on the fundamental group of a product of schemes in characteristic p.

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arXiv2020

Space vectors forming rational angles

We classify all sets of nonzero vectors in $\mathbb{R}^3$ such that the angle formed by each pair is a rational multiple of $π$. The special case of four-element subsets lets us classify all tetrahedra whose dihedral angles are multiples of $π$, solving a 1976 problem of Conway and Jones: there are $2$ one-parameter families and $59$ sporadic tetrahedra, all but three of which are related to either the icosidodecahedron or the $B_3$ root lattice. The proof requires the solution in roots of unity of a $W(D_6)$-symmetric polynomial equation with $105$ monomials (the previous record was $12$ monomials).

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arXiv2020

Hypergeometric L-functions in average polynomial time

We describe an algorithm for computing, for all primes $p \leq X$, the mod-$p$ reduction of the trace of Frobenius at $p$ of a fixed hypergeometric motive in time quasilinear in $X$. This combines the Beukers--Cohen--Mellit trace formula with average polynomial time techniques of Harvey et al.

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