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Connected components of affine Deligne–Lusztig varieties in mixed characteristic Open
We determine the set of connected components of minuscule affine Deligne–Lusztig varieties for hyperspecial maximal compact subgroups of unramified connected reductive groups. Partial results are also obtained for non-minuscule closed affi…
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On the formal arc space of a reductive monoid Open
Let $X$ be a scheme of finite type over a finite field $k$, and let ${\cal L} X$ denote its arc space; in particular, ${\cal L} X(k)=X(k[[t]])$. Using the theory of Grinberg, Kazhdan, and Drinfeld on the finite-dimensionality of singularit…
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Squarefree polynomials and Möbius values in short intervals and arithmetic progressions Open
We calculate the mean and variance of sums of the M\\"obius function and the\nindicator function of the squarefrees, in both short intervals and arithmetic\nprogressions, in the context of the ring of polynomials over a finite field of\n$q…
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Shifted convolution and the Titchmarsh divisor problem over 𝔽<sub><i>q</i></sub>[<i>t</i>] Open
In this paper, we solve a function field analogue of classical problems in analytic number theory, concerning the autocorrelations of divisor functions, in the limit of a large finite field.
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Prime polynomials in short intervals and in arithmetic progressions Open
In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals $(x,x+x^{\\epsilon}]$ is about $x^{\\epsilon}/\\log x$ . The second says that the numbe…
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Counting points on curves using a map to P^1 Open
We introduce a new algorithm to compute the zeta function of a curve over a finite field. This method extends Kedlaya's algorithm to a very general class of curves using a map to the projective line. We develop all the necessary bounds, an…
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Nonabelian Cohen–Lenstra moments Open
In this paper we give a conjecture for the average number of unramified\n$G$-extensions of a quadratic field for any finite group $G$. The Cohen-Lenstra\nheuristics are the specialization of our conjecture to the case that $G$ is\nabelian …
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Random matrices, the Cohen–Lenstra heuristics, and roots of unity Open
The Cohen-Lenstra-Martinet heuristics predict the frequency with which a\nfixed finite abelian group appears as an ideal class group of an extension of\nnumber fields, for certain sets of extensions of a base field. Recently, Malle\nfound …
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The autocorrelation of the Möbius function and Chowla's conjecture for the rational function field in characteristic 2 Open
We prove a function field version of Chowla's conjecture on the autocorrelation of the Möbius function in the limit of a large finite field of characteristic 2, extending previous work in odd characteristic.
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Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number field Open
We prove an integrality result for the value at $s=1$ of the adjoint $L$-function associated to a cohomological cuspidal automorphic representation on ${\rm GL}(n)$ over any number field. We then show that primes (outside an exceptional se…
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Towards a class number formula for Drinfeld modules Open
Let F be the function field of an irreducible, smooth, projective curve over a finite field. Let A be the ring of functions on the curve which are regular away from a fixed closed point ∞. Let F∞ be the completion of F at ∞. Consider an in…
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The geometric Bogomolov conjecture Open
In the following, we prove the geometric Bogomolov conjecture over a function field of characteristic $0$ .
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Mean values of multiplicative functions over function fields Open
We discuss the mean values of multiplicative functions over function fields. In particular, we adapt the authors' new proof of Halász's theorem on mean values to this simpler setting. Several of the technical difficulties that arise over t…
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Local Models For The Moduli Stacks of Global $G$-Shtukas Open
In this article we develop the theory of local models for the moduli stacks of global $G$-shtukas, the function field analogs for Shimura varieties. Here $G$ is a smooth affine group scheme over a smooth projective curve. As the first appr…
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On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic, II Open
Let [math] be an abelian variety over the function field [math] of a curve over a finite field. We describe several mild geometric conditions ensuring that the group [math] is finitely generated and that the [math] -primary torsion subgrou…
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Chow Filtration on Representation Rings of Algebraic Groups Open
We introduce and study a filtration on the representation ring $R(G)$ of an affine algebraic group $G$ over a field. This filtration, which we call Chow filtration, is an analogue of the coniveau filtration on the Grothendieck ring of a sm…
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The geometric average size of Selmer groups over function fields Open
We show, in the large $q$ limit, that the average size of $n$-Selmer groups\nof elliptic curves of bounded height over $\\mathbb F_q(t)$ is the sum of the\ndivisors of $n$. As a corollary, again in the large $q$ limit, we deduce that\n$100…
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Division algebras and maximal orders for given invariants Open
Brauer classes of a global field can be represented by cyclic algebras. Effective constructions of such algebras and a maximal order therein are given for $\mathbb{F}_{q}(t)$ , excluding cases of wild ramification. As part of the construct…
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Reductions of abelian surfaces over global function fields Open
Let $A$ be a non-isotrivial ordinary abelian surface over a global function field of characteristic $p>0$ with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a mu…
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On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic Open
Let K be the function field of a smooth and proper curve S over an algebraically closed field k of characteristic p>0 . Let A be an ordinary abelian variety over K . Suppose that the Néron model \mathcal A of A over S has some closed fibre…
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p-adic estimates of exponential sums oncurves Open
The purpose of this article is to prove a ``Newton over Hodge'' result for exponential sums on curves. Let $X$ be a smooth proper curve over a finite field $\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve…
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Selecting polynomials for the Function Field Sieve Open
The Function Field Sieve algorithm is dedicated to computing discrete\nlogarithms in a finite field GF(q^n), where q is small an prime power. The\nscope of this article is to select good polynomials for this algorithm by\ndefining and meas…
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Galois groups over rational function fields over skew fields Open
Let be a skew field of finite dimension over its center . We solve the Inverse Galois Problem over the field of fractions of the ring of polynomial functions over in the variable , if contains an ample field.
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Hasse principles for higher-dimensional fields Open
For schemes X over global or local fields, or over their rings of integers, K. Kato stated several conjectures on certain complexes of Gersten-Bloch-Ogus type, generalizing the fundamental exact sequence of Brauer groups for a global field…
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On Artin L-functions and Gassmann Equivalence for Global Function Fields Open
In this paper we present an approach to study arithmetical properties of global function fields by working with Artin L-functions. In particular we recall and then extend a criteria of two function fields to be arithmetically equivalent in…
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The $p$-adic monodromy group of abelian varieties over global function fields of characteristic $p$ Open
We prove an analogue of the Tate isogeny conjecture and the semi-simplicity conjecture for overconvergent crystalline Dieudonné modules of abelian varieties defined over global function fields of characteristic p . As a corollary we deduce…
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Traces, high powers and one level density for families of curves over finite fields Open
The zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix Θ C . We develop and present a new technique to compute the expected value of tr(Θ C n ) for various moduli s…
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Local-global principles for zero-cycles on homogeneous spaces over arithmetic function fields Open
We study the existence of zero-cycles of degree one on varieties that are defined over a function field of a curve over a complete discretely valued field. We show that local-global principles hold for such zero-cycles provided that local-…
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Waldspurger formula over function fields Open
In this paper, we derive a function field version of the Waldspurger formula for the central critical values of the Rankin-Selberg $L$-functions. This formula states that the central critical $L$-values in question can be expressed as the …
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Genus fields of abelian extensions of rational congruence function fields, II Open
In this paper, we find the genus field of finite abelian extensions of the global rational function field. We introduce the term conductor of constants for these extensions and determine it in terms of other invariants. We study the partic…