arXiv subjects

Kiran S. Kedlaya

Publications and source records attributed to Kiran S. Kedlaya.

At least 19 recordsLinked to original sources

arXiv2026

On the classification of small cyclotomic integers

We give a general classification theorem for cyclotomic algebraic integers with all complex absolute values bounded by a fixed constant $c$, modeled on the theorem of Cassels which treats the case $c = \sqrt{5}$ up to finitely many exceptions. As a corollary, we establish that the range of the function taking a cyclotomic integer to its maximum complex absolute value is a well-ordered (but not closed) subset of the real numbers. We also formulate analogous statements for algebraic numbers in the maximal cyclotomic extension of a fixed number field. The proofs combine a result of Loxton, which bounds the number of roots of unity in the shortest additive representation of a cyclotomic integer in terms of the maximum complex absolute value, with an equidistribution theorem of Bilu et al. for Galois orbits of torsion points on algebraic tori.

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arXiv2026

Products of point counts of higher genus curves over finite fields

Let $E/\mathbb Q$ be an elliptic curve and for each prime $p$, let $N_p$ denote the number of points of $E$ modulo $p$. The original version of the conjecture of Birch and Swinnerton-Dyer asserts that $\prod \limits _{p \leq x} \frac{N_p}{p} \sim C (\log x) ^{\text{rank}(E(\mathbb Q))}$ as $x \to \infty$. In this paper, we formulate a similar conjectural asymptotic for smooth projective curves of genus at least 2, in which the contributions to the conjectured asymptotic come not only from the rank of the Jacobian but also from the Sato--Tate group of the curve. The key analytic input in formulating our conjecture is a conjecture due to Kurokawa (2012) on the convergence of Euler products of entire $L$-functions on the critical line. We also provide some numerical evidence for our conjecture in various cases.

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arXiv2026

Compatibility of $F$-isocrystals on adjoint Shimura varieties

In this article, we extend past results of the last two authors to include compatibility of canonical $\ell$-adic local systems and canonical $F$-isocrystals on adjoint Shimura varieties in the superrigid regime. Our method relies on the crystallinity of canonical $p$-adic local systems due to Esnault--Groechenig as well as Margulis superrigidity and the crystalline-to-étale companion construction of Drinfeld, Abe--Esnault, and Kedlaya.

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arXiv2026

Etale and crystalline companions, II

Let $X$ be a smooth scheme over a finite field of characteristic $p$. In answer to a conjecture of Deligne, we establish that for any prime $\ell \neq p$, an $\ell$-adic Weil sheaf on $X$ which is algebraic (or irreducible with finite determinant) admits a crystalline companion in the category of overconvergent $F$-isocrystals, for which the Frobenius characteristic polynomials agree at all closed points (with respect to some fixed identification of the algebraic closures of $\mathbb{Q}$ within fixed algebraic closures of $\mathbb{Q}_\ell$ and $\mathbb{Q}_p$). The argument depends heavily on the free passage between $\ell$-adic and $p$-adic coefficients for curves provided by the Langlands correspondence for $\mathrm{GL}_n$ over global function fields (work of L. Lafforgue and T. Abe), and on the construction of Drinfeld (plus adaptations by Abe-Esnault and Kedlaya) giving rise to étale companions of overconvergent $F$-isocrystals. As corollaries, we transfer a number of statements from crystalline to étale coefficient objects, including properties of the Newton polygon stratification (results of Grothendieck-Katz and de Jong-Oort-Yang) and Wan's theorem (previously Dwork's conjecture) on $p$-adic meromorphicity of unit-root $L$-functions.

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arXiv2025

On the relative Nullstellensatz in nonarchimedean geometry

We establish a relative version of the Nullstellensatz for algebras topologically of finite type over a given Banach Tate ring $A$, under the assumption that the corresponding statement holds for rational localizations of $A$. This applies in particular to pseudoaffinoid algebras and to the coordinate rings of affinoid subspaces of a Fargues--Fontaine curve.

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arXiv2025

The exceptional set in Cassel's theorem on small cyclotomic integers

In a 1965 paper, R. Robinson made five conjectures about the classification of cyclotomic algebraic integers for which the maximum absolute value in any complex embedding (the house) is small, modulo the equivalence relation generated by Galois conjugation and multiplication by roots of unity. In response to one of these conjectures, Cassels showed in 1969 that when the house is at most $\sqrt{5}$, one obtains three parametric families plus an effectively computable finite set of equivalence classes of exceptions. Building on the work of Jones, Calegari-Morrison-Snyder, and Robinson-Wurtz, we determine this exceptional set. By specializing to the case where the house is strictly less than 2, we resolve the final outstanding conjecture from Robinson's 1965 paper.

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arXiv2025

Monodromy representations of $p$-adic differential equations in families

We derive a relative version of the local monodromy theorem for ordinary differential equations on an annulus over a mixed-characteristic nonarchimedean field, and give several applications in $p$-adic cohomology and $p$-adic Hodge theory. These include a simplified proof of the semistable reduction theorem for overconvergent $F$-isocrystals, a relative version of Berger's theorem that de Rham representations are potentially semistable, and a multivariate version of the local monodromy theorem in the style of Drinfeld's lemma on fundamental groups.

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arXiv2025

Bounds for the relative class number problem for function fields

We establish bounds on a finite separable extension of function fields in terms of the relative class number, thus reducing the problem of classifying extensions with a fixed relative class number to a finite computation. We also solve the relative class number two problem in all cases where the base field has constant field not equal to $\mathbb{F}_2$.

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arXiv2025

Frobenius sign separation for abelian varieties

Let A and A' be nonzero abelian varieties defined over a number field k such that Hom(A,A')=0. Under the Generalized Riemann hypothesis for motivic L-functions attached to A and A', we show that there exists a prime p of k of good reduction for A and A' at which the Frobenius traces of A and A' are nonzero and differ by sign, and such that the norm of p is O_{k,g,g'}(log(2NN')^2), where N and N' respectively denote the absolute conductors of A and A'. We also make the dependence of the big-O constant on k and the dimensions g,g' of A,A' explicit up to an effectively computable absolute constant. Our method extends that of Chen, Park, and Swaminathan who considered the case in which A and A' are elliptic curves.

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arXiv2024

Hypergeometric $L$-functions in average polynomial time, II

For a fixed positive integer $e$, we describe an algorithm for computing, for all primes $p \leq X$, the mod-$p^e$ reduction of the trace of Frobenius at $p$ of a fixed hypergeometric motive over $\mathbb{Q}$ in time quasilinear in $X$. This extends our previous work for the mod-$p$ reduction, again combining the Beukers--Cohen--Mellit trace formula with average polynomial time techniques of Harvey and Harvey--Sutherland; the key new ingredient is an expanded version of Harvey's "generic prime" construction, making it possible to incorporate certain $p$-adic transcendental functions into the computation. One of these is the $p$-adic Gamma function, whose average polynomial time computation is an intermediate step which may be of independent interest. We also provide an implementation in Sage and discuss the remaining computational issues around computing hypergeometric $L$-series.

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arXiv2024

The relative class number one problem for function fields, II

We establish that any finite extension of function fields of genus greater than 1 whose relative class group is trivial is Galois and cyclic. This depends on a result from a preceding paper which establishes a finite list of possible Weil polynomials for both fields. Given this list, we analyze most cases by computing options for the splittings of low-degree places in the extension, then consider the effect of these options on the Weil polynomials of certain isogeny factors of the Jacobian of the Galois closure. In one case, we use instead an analysis based on principal polarizations, modeled on an argument of Howe.

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arXiv2024

Drinfeld's lemma for $F$-isocrystals, I

We prove that in either the convergent or overconvergent setting, an absolutely irreducible $F$-isocrystal on the absolute product of two or more smooth schemes over perfect fields of characteristic $p$, further equipped with actions of the partial Frobenius maps, is an external product of $F$-isocrystals over the multiplicands. The corresponding statement for lisse $\bar{\mathbb{Q}}_\ell$-sheaves, for $\ell \neq p$ a prime, is a consequence of Drinfeld's lemma on the fundamental groups of absolute products of schemes in characteristic $p$. The latter plays a key role in V. Lafforgue's approach to the Langlands correspondence for reductive groups with $\ell$-adic coefficients; the $p$-adic analogue will be considered in subsequent work with Daxin Xu.

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arXiv2024

A census of genus 6 curves over $\mathbb{F}_2$

We compile a complete list of isomorphism class representatives of curves of genus 6 over $\mathbb{F}_2$. We use explicit descriptions of canonical curves in each stratum of the Brill--Noether stratification of the moduli space $\mathcal{M}_6$, due to Mukai in the generic case. Our computed value of $\#\mathcal{M}_6(\mathbb{F}_2)$ agrees with the Lefschetz trace formula as recently computed by Bergstrom--Canning--Petersen--Schmitt.

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arXiv2023

The relative class number one problem for function fields, III

We complete the solution of the relative class number one problem for function fields of curves over finite fields. Using work from two earlier papers, this reduces to finding all function fields of genus 6 or 7 over $\mathbb{F}_2$ with one of 40 prescribed Weil polynomials; one may then verify directly that three of these fields admit an everywhere unramified quadratic extension with trivial relative class group. The search is carried out by carefully enumerating curves based on the Brill--Noether stratification of the moduli spaces of curves in these genera, and particularly Mukai's descriptions of the open strata.

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arXiv2023

Effective Sato-Tate conjecture for abelian varieties and applications

From the generalized Riemann hypothesis for motivic L-functions, we derive an effective version of the Sato-Tate conjecture for an abelian variety A defined over a number field k with connected Sato-Tate group. By effective we mean that we give an upper bound on the error term in the count predicted by the Sato-Tate measure that only depends on certain invariants of A. We discuss three applications of this conditional result. First, for an abelian variety defined over k, we consider a variant of Linnik's problem for abelian varieties that asks for an upper bound on the least norm of a prime whose normalized Frobenius trace lies in a given interval. Second, for an elliptic curve defined over k with complex multiplication, we determine (up to multiplication by a nonzero constant) the asymptotic number of primes whose Frobenius trace attain the integral part of the Hasse-Weil bound. Third, for a pair of abelian varieties defined over k with no common factors up to k-isogeny, we find an upper bound on the least norm of a prime at which the respective Frobenius traces have opposite sign.

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arXiv2023

Drinfeld's lemma for $F$-isocrystals, II: Tannakian approach

We prove a Tannakian form of Drinfeld's lemma for isocrystals on a variety over a finite field, equipped with actions of partial Frobenius operators. This provides an intermediate step towards transferring V. Lafforgue's work on the Langlands correspondence over function fields from $\ell$-adic to $p$-adic coefficients. We also discuss a motivic variant and a local variant of Drinfeld's lemma.

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arXiv2023

Sato-Tate groups of abelian threefolds

Given an abelian variety over a number field, its Sato-Tate group is a compact Lie group which conjecturally controls the distribution of Euler factors of the L-function of the abelian variety. It was previously shown by Fité, Kedlaya, Rotger, and Sutherland that there are 52 groups (up to conjugation) that occur as Sato-Tate groups of abelian surfaces over number fields; we show here that for abelian threefolds, there are 410 possible Sato-Tate groups, of which 33 are maximal with respect to inclusions of finite index. We enumerate candidate groups using the Hodge-theoretic construction of Sato-Tate groups, the classification of degree-3 finite linear groups by Blichfeldt, Dickson, and Miller, and a careful analysis of Shimura's theory of CM types that rules out 23 candidate groups; we cross-check this using extensive computations in Gap, SageMath, and Magma. To show that these 410 groups all occur, we exhibit explicit examples of abelian threefolds realizing each of the 33 maximal groups; we also compute moments of the corresponding distributions and numerically confirm that they are consistent with the statistics of the associated L-functions.

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arXiv2023

A note on the cohomology of $p$-adic analytic group actions

We prove that given an analytic action of a compact $p$-adic Lie group on a Banach space over a field of positive characteristic, one can detect either the simultaneous vanishing or the simultaneous finite-dimensionality of all of the continuous cohomology groups from the corresponding statement for the restriction to a pro-$p$ procyclic subgroup. We also formulate a conjecture generalizing this result, in which the base field is allowed to have mixed characteristic and the subgroup is allowed to be nilpotent. Finally, we formulate an analogous conjecture about Lie algebra cohomology and relate this to a theorem of Kostant.

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