Search

Persistence of nondegeneracy is a phenomenon that appears in the theory of |$\overline{\mathbb{Q}}_l$| -representations of the linear group: every irreducible submodule of the restriction to the mirabolic sub-representation of a nondegenerate irreducible representation is nondegenerate. This is not true anymore in general, if we look at the modulo |$l$| reduction of some stable lattice. As in the Clozel–Harris–Taylor generalization of global Ihara's lemma, we show that this property, called nondegeneracy persistence and related to the notion of essentially absolutely irreducible and generic representations in the work of Emerton and Helm, remains true for lattices given by the cohomology of Lubin–Tate spaces. As a global application, we give a new construction of automorphic congruences in the Ribet spirit. Résumé. La persistence de la non dégénérescence est un phénomène qui apparait dans la théorie des |$\overline{\mathbb{Q}}_l$| -représentations du groupe linéaire: toute sous-représentation irréductible de la restriction au groupe mirabolique d'une représentation irréductible non dégénérée, est non dégénérée. Ce n'est plus le cas en général pour la réduction modulo |$l$| d'un réseau stable. Comme dans la généralisation par Clozel-Harris-Taylor du lemme d'Ihara, nous montrons que cette propriété de non dégénérescence, qui est reliée à la notion de représentation essentiellement absolument générique de Emerton-Helm, reste valide pour les réseaux donnés par la cohomologie des espaces de Lubin-Tate. Nous donnons une application de nature globale en construisant des congruences automorphes dans l'esprit du travail de Ribet. [ABSTRACT FROM AUTHOR]

We introduce a notion of absolute irreducibility for finitary skew linear groups consistent with the existing notions for finite-dimensional linear groups, finite-dimensional skew linear groups and finitary linear groups and then prove some relatively strong results about such absolutely irreducible groups. [ABSTRACT FROM AUTHOR]

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 |$\overline{{\mathbb{Q}}}_{\ell }$| -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. [ABSTRACT FROM AUTHOR]

We study irreducible |$\mathop{\textrm{SL}}\nolimits _2$| -representations of twist knots. We first determine all non-acyclic |$\mathop{\textrm{SL}}\nolimits _2({\mathbb{C}})$| -representations, which turn out to lie on a line denoted as |$x=y$| in |${\mathbb{R}}^2$|⁠. Our main tools are character variety, Reidemeister torsion, and Chebyshev polynomials. We also verify a certain common tangent property, which yields a result on |$L$| -functions, that is, the orders of the knot modules associated to the universal deformations. Secondly, we prove that a representation is on the line |$x=y$| if and only if it factors through the |$-3$| -Dehn surgery, and is non-acyclic if and only if the image of a certain element is of order 3. Finally, we study absolutely irreducible non-acyclic representations |$\overline{\rho }$| over a finite field with characteristic |$p>2$|⁠ , to concretely determine all non-trivial |$L$| -functions |$L_{{\boldsymbol{\rho }}}$| of the universal deformations over complete discrete valuation rings. We show among other things that |$L_{{\boldsymbol{\rho }}}$|   |$\dot{=}$|   |$k_n(x)^2$| holds for a certain series |$k_n(x)$| of polynomials. [ABSTRACT FROM AUTHOR]

We show that if {ρℓ} is a compatible system of absolutely irreducible Galois representations of a number field then the residual representation ¯ρℓ is absolutely irreducible for ℓ in a density 1 set of primes. The key technical result is the following theorem: the image of ρℓ is an open subgroup of a hyperspecial maximal compact subgroup of its Zariski closure with bounded index (as ℓ varies). This result combines a theorem of Larsen on the semi-simple part of the image with an analogous result for the central torus that was recently proved by Barnet-Lamb, Gee, Geraghty, and Taylor, and for which we give a new proof. [ABSTRACT FROM AUTHOR]

We obtain a nontrivial bound on the number of solutions to the equation $$ \begin{align*} &\sum_{i=1}^{\nu} A^{x_i} = \sum_{i=\nu+1}^{2\nu} A^{x_i}, \qquad 1 \leqslant x_i \leqslant \tau, \end{align*}$$ with a fixed |$n\times n$| matrix |$A$| over a finite field |${{\mathbb {F}}}_q$| of |$q$| elements of multiplicative order |$\tau $|⁠. We apply our result to obtain a new bound for additive character sums with a matrix exponential function, nontrivial beyond the square-root threshold. For |$n=2$|⁠ , this equation has been considered by Kurlberg and Rudnick (for |$\nu =2$|⁠) and Bourgain (for large |$\nu $|⁠) in their study of quantum ergodicity for linear maps over residue rings. We use a new approach to improve their results and also obtain a bound on Kloosterman sums over small subgroups, of size below the square-root threshold. [ABSTRACT FROM AUTHOR]

The article discusses the theorem on the irreducible plane curve of a plane having the sides parallel to its coordinate axes. It mentions that the result of the research is to use the bounds for the density of integer points and rational points on higher-dimensional varieties. It also notes on the improvements of the misleading theory of the term using the variant of the elementary mathematics method in searching for the main theorem.

Let |$G$| be a connected linear algebraic group over a number field |$K$|⁠ , let |$\Gamma $| be a finitely generated Zariski dense subgroup of |$G(K)$|⁠ , and let |$Z\subseteq G(K)$| be a thin set, in the sense of Serre. We prove that, if |$G/\textrm {R}_{u}(G)$| is either trivial or semisimple and |$Z$| satisfies certain necessary conditions, then a long random walk on a Cayley graph of |$\Gamma $| hits elements of |$Z$| with negligible probability. We deduce corollaries to Galois covers, characteristic polynomials, and fixed points in group actions. We also prove analogous results in the case where |$K$| is a global function field. [ABSTRACT FROM AUTHOR]

We prove the existence of all companion points on the eigenvariety of definite unitary groups associated with generic crystalline Galois representations with possibly non-regular weights under the Taylor–Wiles hypothesis, based on the previous results of Breuil–Hellmann–Schraen [8] in regular cases and the author in [33] in non-regular cases. [ABSTRACT FROM AUTHOR]

The main goal of this work is to answer a question of Dèbes and Haran by relaxing the condition for Hilbertianity. Namely we prove that for a field K to be Hilbertian it suffices that K has the irreducible specialization property merely for absolutely irreducible polynomials. [ABSTRACT FROM PUBLISHER]

We prove a GL(n)×GL(n-1) local converse theorem for ℓ-adic families of smooth representations of GLn(F) where F is a finite extension of Qp and ℓ≠p. Along the way, we extend the theory of Rankin-Selberg integrals, first introduced in Jacquet et al. [20], to the setting of families, continuing previous work of the author [23] [ABSTRACT FROM AUTHOR]

We prove quantitative versions of Borel and Harish-Chandra's theorems on reduction theory for arithmetic groups. Firstly, we obtain polynomial bounds on the lengths of reduced integral vectors in any rational representation of a reductive group. Secondly, we obtain polynomial bounds in the construction of fundamental sets for arithmetic subgroups of reductive groups, as the latter vary in a real conjugacy class of subgroups of a fixed reductive group. Our results allow us to apply the Pila–Zannier strategy to the Zilber–Pink conjecture for the moduli space of principally polarised abelian surfaces. Building on our previous paper, we prove this conjecture under a Galois orbits hypothesis. Finally, we establish the Galois orbits hypothesis for points corresponding to abelian surfaces with quaternionic multiplication, under certain geometric conditions. [ABSTRACT FROM AUTHOR]

We prove a Galois-equivariant algebraicity result for the ratios of successive critical values of |$L$| -functions for |${\textrm GL}(n)/F,$| where |$F$| is a totally imaginary quadratic extension of a totally real number field |$F^+$|⁠. The proof uses (1) results of Arthur and Clozel on automorphic induction from |${\textrm GL}(n)/F$| to |${\textrm GL}(2n)/F^+$|⁠ , (2) results of my work with Harder on ratios of critical values for |$L$| -functions of |${\textrm GL}(2n)/F^+$|⁠ , and (3) period relations amongst various automorphic and cohomological periods for |${\textrm GL}(2n)/F^+$| using my work with Shahidi. The reciprocity law inherent in the algebraicity result is exactly as predicted by Deligne's conjecture on the special values of motivic |$L$| -functions. [ABSTRACT FROM AUTHOR]

We establish |$p$| -adic versions of the Manin–Mumford conjecture, which states that an irreducible subvariety of an abelian variety with dense torsion has to be the translate of a subgroup by a torsion point. We do so in the context of certain rigid analytic spaces and formal groups over a |$p$| -adic field or its ring of integers, respectively. In particular, we show that the underlying rigidity results for algebraic functions generalize to suitable |$p$| -adic analytic functions. This leads us to uncover purely |$p$| -adic Manin–Mumford-type results for formal groups not coming from abelian schemes. Moreover, we observe that a version of the Tate–Voloch conjecture holds: torsion points either lie squarely on a subscheme or are uniformly bounded away from it in the |$p$| -adic distance. [ABSTRACT FROM AUTHOR]

We prove the following function field analog of the Hardy–Littlewood conjecture (which generalizes the twin prime conjecture) over large finite fields. Let n and r be positive integers and q an odd prime power. For a tuple of distinct polynomials of degree

In this paper, we introduce an algorithm for computing |$p$| -adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of |$p$| -adic integration: Berkovich–Coleman integrals, which can be performed locally, and abelian integrals with desirable number-theoretic properties. By covering a bad reduction hyperelliptic curve with basic wide-open sets, we reduce the computation of Berkovich–Coleman integrals to the known algorithms on good reduction hyperelliptic curves. These are due to Balakrishnan, Bradshaw, and Kedlaya and to Balakrishnan and Besser for regular and meromorphic |$1$| -forms, respectively. We then employ tropical geometric techniques due to the 1st-named author with Rabinoff and Zureick-Brown to convert the Berkovich–Coleman integrals into abelian integrals. We provide examples of our algorithm, verifying that certain abelian integrals between torsion points vanish. [ABSTRACT FROM AUTHOR]

Let |$M$| be a compact 3-manifold and |$\Gamma =\pi _1(M)$|⁠. Work by Thurston and Culler–Shalen established the |${\operatorname{\textrm{SL}}}_2({\mathbb{C}})$| character variety |$X(\Gamma)$| as fundamental tool in the study of the geometry and topology of |$M$|⁠. This is particularly the case when |$M$| is the exterior of a hyperbolic knot |$K$| in |$S^3$|⁠. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand algebraic and number theoretic properties of the so-called canonical component of |$X(\Gamma)$|⁠ , as well as distinguished points on the canonical component, when |$\Gamma $| is a knot group. In particular, we study how the theory of quaternion Azumaya algebras can be used to obtain algebraic and arithmetic information about Dehn surgeries, and perhaps of most interest, to construct new knot invariants that lie in the Brauer groups of curves over number fields. [ABSTRACT FROM AUTHOR]

We study various statistics regarding the distribution of the points $$ \left\{\left(\frac{d}{q},\frac{\overline{d}}{q}\right) \in \mathbb{T}^2 : d \in (\mathbb{Z}/q\mathbb{Z})^{\times}\right\} $$ as q tends to infinity. Due to non-trivial bounds for Kloosterman sums, it is known that these points equidistribute on the torus. We prove refinements of this result, including bounds for the discrepancy, small-scale equidistribution, bounds for the covering exponent associated with these points, sparse equidistribution, and mixing. [ABSTRACT FROM AUTHOR]

We compare different notions of convergence for sequences of finite dimensional representations over a valued field. We then consider convergent sequences of Galois representations over ℂp, prove a control result for the component groups of the algebraic envelopes of the representations, and deduce a uniform Cebotarev-type density theorem for the representations in the sequence. [ABSTRACT FROM AUTHOR]

Let $k$ be a perfect field of characteristic $p> 2$ , and let $K$ be a finite totally ramified extension of $W(k)\big[\frac{1}{p}\big]$ of ramification degree $e$. We consider an unramified base ring $R_0$ over $W(k)$ satisfying certain conditions, and let $R = R_0\otimes _{W(k)}\mathcal{O}_K$. Examples of such $R$ include $R = \mathcal{O}_K[\![s_1, \ldots , s_d]\!]$ and $R = \mathcal{O}_K\langle t_1^{\pm 1}, \ldots , t_d^{\pm 1}\rangle $. We show that the generalization of Raynaud's theorem on extending $p$ -divisible groups holds over the base ring $R$ when $e < p-1$ , whereas it does not hold when $R = \mathcal{O}_K[\![s]\!]$ with $e \geq p$. As an application, we prove that if $R$ has Krull dimension $2$ and $e < p-1$ , then the locus of Barsotti–Tate representations of $\textrm{Gal}(\overline{R}\big[\frac{1}{p}\big]/R\big[\frac{1}{p}\big])$ cuts out a closed subscheme of the universal deformation scheme. If $R = \mathcal{O}_K[\![s]\!]$ with $e \geq p$ , we prove that such a locus is not $p$ -adically closed. [ABSTRACT FROM AUTHOR]

We give new instances where Chabauty–Kim sets can be proved to be finite, by developing a notion of "generalised height functions" on Selmer varieties. We also explain how to compute these generalised heights in terms of iterated integrals and give the 1st explicit nonabelian Chabauty result for a curve |$X/\mathbb{Q}$| whose Jacobian has Mordell–Weil rank larger than its genus. [ABSTRACT FROM AUTHOR]

For a projective curve |$C\subset \textbf{P} ^n$| defined over a finite field |$\textbf{F} _q$| we study the statistics of the |$\textbf{F} _q$| -structure of a section of |$C$| by a random hyperplane defined over |$\textbf{F} _q$| in the |$q\to \infty $| limit. We obtain a very general equidistribution result for this problem. We deduce many old and new results about decomposition statistics over finite fields in this limit. Our main tool will be the calculation of the monodromy of transversal hyperplane sections of a projective curve. [ABSTRACT FROM AUTHOR]

This article is a continuation of our previous work [ 7 ] on the Iwasawa theory of an elliptic modular form over an imaginary quadratic field |$K$|⁠ , where the modular form in question was assumed to be ordinary at a fixed odd prime |$p$|⁠. We formulate integral Iwasawa main conjectures at non-ordinary primes |$p$| for suitable twists of the base change of a newform |$f$| to an imaginary quadratic field |$K$| where |$p$| splits, over the cyclotomic |${\mathbb{Z}}_p$| -extension, the anticyclotomic |${\mathbb{Z}}_p$| -extensions (in both the definite and the indefinite cases) as well as the |${\mathbb{Z}}_p^2$| -extension of |$K$|⁠. In order to do so, we define Kobayashi–Sprung-style signed Coleman maps, which we use to introduce doubly signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson–Flach elements (out of the collection of unbounded Beilinson–Flach elements of Loeffler–Zerbes), which we use to define doubly signed |$p$| -adic |$L$| -functions. The main conjecture then relates these two sets of objects. Furthermore, we show that the integral Beilinson–Flach elements form a locally restricted Euler system, which in turn allow us to deduce (under certain technical assumptions) one inclusion in each one of the four main conjectures we formulate here (which may be turned into equalities in favorable circumstances). [ABSTRACT FROM AUTHOR]

Let |$L$| be a finite extension of |${\mathbb{Q}}_p$| and |$n\geq 2$|⁠. We associate to a crystabelline |$n$| -dimensional representation of |${\operatorname{Gal}}(\overline L/L)$| satisfying mild genericity assumptions a finite length locally |${\mathbb{Q}}_p$| -analytic representation of |${\operatorname{GL}}_n(L)$|⁠. In the crystalline case and in a global context, using the recent results on the locally analytic socle from [ 6 ], we prove that this representation indeed occurs in spaces of |$p$| -adic automorphic forms. We then use this latter result in the ordinary case to show that certain "ordinary" |$p$| -adic Banach space representations constructed in our previous work appear in spaces of |$p$| -adic automorphic forms. This gives strong new evidence to our previous conjecture in the |$p$| -adic case. [ABSTRACT FROM AUTHOR]

We study the rigid generic fiber |$\mathcal{X}^\square _{\overline \rho }$| of the framed deformation space of the trivial representation |$\overline \rho : G_K \to \textrm{GL}_n(k)$| where |$k$| is a finite field of characteristic |$p>0$| and |$G_K$| is the absolute Galois group of a finite extension |$K/\textbf{Q}_p$|⁠. Under some mild conditions on |$K$| we prove that |$\mathcal{X}^\square _{\overline \rho }$| is normal. When |$p> n$| we describe its irreducible components and show Zariski density of its crystalline points. [ABSTRACT FROM AUTHOR]

We describe a novel method for bounding the dimension |$d$| of the largest simple Hecke submodule of |$S_{2}(\Gamma _{0}(N);\mathbb{Q})$| from below. Such bounds are of interest because of their relevance to the structure of |$J_{0}(N)$|⁠ , for instance. In contrast with previous results of this kind, our bound does not rely on the equidistribution of Hecke eigenvalues. Instead, it is obtained via a Hecke-compatible congruence between the target space and a space of modular forms whose Hecke eigenvalues are easily controlled. We prove conditional bounds, the strongest of which is |$d\gg _{\epsilon } N^{1/2-\epsilon }$| over a large set of primes |$N$|⁠ , contingent on Soundararajan's heuristics for the class number problem and Artin's conjecture on primitive roots. For prime levels |$N\equiv 7\mod 8,$| our method yields an unconditional bound of |$d\geq \log _{2}\log _{2}(\frac{N}{8})$|⁠ , which is larger than the known bound of |$d\gg \sqrt{\log \log N}$| due to Murty–Sinha and Royer. A stronger unconditional bound of |$d\gg \log N$| can be obtained in more specialized (but infinitely many) cases. We also propose a number of Maeda-style conjectures based on our data, and we outline a possible congruence-based approach toward the conjectural Hecke simplicity of |$S_{k}(\textrm{SL}_{2}(\mathbb{Z});\mathbb{Q})$|⁠. [ABSTRACT FROM AUTHOR]

Mathematical physicists have studied degenerations of Lie groups and their representations, which they call contractions. In this paper we study these contractions, and also other families, within the framework of algebraic families of Harish-Chandra modules. We construct a family that incorporates both a real reductive group and its compact form, separate parts of which have been studied individually as contractions. We give a complete classification of generically irreducible families of Harish-Chandra modules in the case of the family associated to |$SL(2,\mathbb{R})$|⁠. [ABSTRACT FROM AUTHOR]

We examine from an algebraic point of view some families of unitary group representations that arise in mathematical physics and are associated to contraction families of Lie groups. The contraction families of groups relate different real forms of a reductive group and are continuously parametrized, but the unitary representations are defined over a parameter subspace that includes both discrete and continuous parts. Both finite- and infinite-dimensional representations can occur, even within the same family. We shall study the simplest nontrivial examples and use the concepts of algebraic families of Harish-Chandra pairs and Harish-Chandra modules, introduced in a previous paper, together with the Jantzen filtration, to construct these families of unitary representations algebraically. [ABSTRACT FROM AUTHOR]

Let D be the ring of Grothendieck differential operators of the ring R of polynomials in d ≥ 3 variables with coefficients in a perfect field of characteristic p. We compute the D -module length of the 1st local cohomology module |${H^{1}_{f}}(R)$| with respect to a polynomial f with an isolated singularity, for p large enough. The expression we give is in terms of the Frobenius action on the top coherent cohomology of the exceptional fibre of a resolution of the singularity. Our proof rests on a tight closure computation of Hara. Since the above length is quite different from that of the corresponding local cohomology module in characteristic zero, we also consider a characteristic zero D -module whose length is expected to equal that above, for ordinary primes. [ABSTRACT FROM AUTHOR]

We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein's theory of analytic continuation of holonomic distributions. We use this to construct H -equivariant functionals on principal series representations of G , where G is a real reductive group and H is an algebraic subgroup. We also deduce the existence of generalized Whittaker models for degenerate principal series representations. As a special case, this gives short proofs of existence of Whittaker models on principal series representations and of analytic continuation of standard intertwining operators. Finally, we extend our constructions to the p -adic case using a recent result of Hong and Sun. [ABSTRACT FROM AUTHOR]

We study some closed rigid subspaces of the eigenvarieties, constructed by using the Jacquet–Emerton functor for parabolic non-Borel subgroups. As an application (and motivation), we prove some new results on Breuil's locally analytic socle conjecture for GLn(Qp) and show the existence of companion points for simple reflections. [ABSTRACT FROM AUTHOR]

Samit Dasgupta has proved a formula factoring a certain restriction of a three-variable Rankin–Selberg p-adic L-function as a product of a two-variable p-adic L-function related to the adjoint representation of a Hida family and a Kubota–Leopoldt p-adic L-function. We prove a result involving Selmer groups that along with Dasgupta's result is consistent with the main conjectures associated to the four-dimensional representation (to which the three-variable p-adic L-function is associated), the three-dimensional representation (to which the two-variable p-adic L-function is associated) and the one-dimensional representation (to which the Kubota–Leopoldt p-adic L-function is associated). Under certain additional hypotheses, we indicate how one can use work of Urban to deduce main conjectures for the three-dimensional representation and the four-dimensional representation. One key technical input to our methods is studying the behavior of Selmer groups under specialization. [ABSTRACT FROM AUTHOR]

Laurence Sterne's Tristram Shandy (1759–67) appears to resist adaptation. Its verbal density, narrative complexity, and self-conscious bookishness mark it out as intensely medium-specific. However, its richly allusive style, scepticism about conventions of representation, and involvement of readers may make it a model for contemporary approaches to adaptation. This essay analyses two re-mediations: Martin Rowson's 1996 graphic novel and Michael Winterbottom's 2005 movie. It argues that these artists use Sterne to fashion virtuosic originality within a culture of mimicry and pastiche. To this end, they produce self-conscious adaptations that reveal and reflect on their own creative processes. The essay proposes that moving beyond axiological and teleological views of adaptation will be facilitated by recognizing literary classics as themselves acts of appropriation which in turn solicit acts of adaptation. [ABSTRACT FROM AUTHOR]

We study whether there exist infinitely many surfaces with given discrete invariants for which the $$H^2$$ is of CM type. This is a surface analogue of a conjecture of Coleman about curves. We construct a large number of examples of families of surfaces with $$p_{\text{g}} = 1$$ that in the moduli space cut out a special subvariety; these provide a positive answer to our question. Most of these are families of K3 surfaces but we also obtain some families of surfaces of general type. As input for our construction we use the work of Deligne and Mostow. Finally we prove that a very general K3 surface cannot be dominated by a product of curves of small genus. [ABSTRACT FROM AUTHOR]

Let $$X$$ be a scheme of finite type over $$\mathbf{Z}$$. For $$p \in \mathcal{P}$$ , the set of prime numbers, let $$N_{X}(p)$$ be the number of $$\mathbf{F}_{p}$$ -points of $$X/\mathbf{F}_{p}$$. For fixed $$n\geq 1$$ and $$a_{1}, \ldots, a_{n} \in \mathbf{Z}$$ , we study the set $$\bigcap_{i=1}^{n}\lbrace p\in\mathcal{P}: N_{X}(p)\not\equiv a_{i}\ [\bmod\ p]\rbrace$$. In case $$\dim X\leq 3$$ (under an additional condition in the case $$\dim X=3$$), we show the set is either finite or has positive lower density. We also address the question of the size of the smallest prime in that set. Using sieve methods, we obtain for example an upper bound for the size of the least prime of $$\bigcap_{i=1}^{n}\lbrace p\in\mathcal{P}: N_{X}(p)\not\equiv a_{i}\ [\bmod\ p]\rbrace$$ , on average in particular families of hyperelliptic curves. [ABSTRACT FROM AUTHOR]

We study the problem of lifting curves from finite fields to number fields in a genus and gonality preserving way. More precisely, we sketch how this can be done efficiently for curves of gonality at most four, with an in-depth treatment of curves of genus at most five over finite fields of odd characteristic, including an implementation in Magma. We then use such a lift as input to an algorithm due to the second author for computing zeta functions of curves over finite fields using p-adic cohomology. [ABSTRACT FROM AUTHOR]

Let k be a number field, Ω be a finite symmetric subset of GLn0 (k), and Γ = Ω. Let C(Γ):= {p ∈ Vf (k) ∣ Γ is a bounded subgroup of GLn0 (kp)}, and Γp be the closure of Γ in GLn0 (kp). Assuming that the Zariski-closure of Γ is semisimple, we prove that the family of left translation actions {Γ → Γp}p∈C(Γ) has uniform spectral gap. As a corollary we get that the left translation action Γ → G has local spectral gap if Γ is a countable dense subgroup of a semisimple p-adic analytic group G and Ad(Γ) consists of matrices with algebraic entries in some ℚp-basis of Lie(G). This can be viewed as a (stronger) p-adic version of [10, Theorem A], which enables us to give applications to the Banach-Ruziewicz problem and orbit equivalence rigidity. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic p > 0 and let G be a subgroup of K* x K*with dimℚ(G ⊗z ℚ) = r finite. Then Voloch proved that the equation ax + by = 1 in (x, y) ∈ G for given a, b ∈ K* has at most pr (pr + p - 2)/(p - 1) solutions (x, y) ∈ G, unless (a, b)n ∈ G for some n ≥ 1. Voloch also conjectured that this upper bound can be replaced by one depending only on r. Our main theorem answers this conjecture positively. We prove that there are at most 31 x 19r+1 solutions (x, y) unless (a, b)n ∈ G for some n ≥ 1 with (n, p) = 1. During the proof of our main theorem, we generalize the work of Beukers and Schlickewei to positive characteristic, which heavily relies on diophantine approximation methods. This is a surprising feat on its own, since usually these methods cannot be transferred to positive characteristic. [ABSTRACT FROM AUTHOR]

We give an improved estimate for the density of k-free values of integral binary forms with no fixed kth power divisor. Further, we give the corresponding improvement to a theorem of Stewart and Top on the number of power-free values in an interval that may be assumed by a binary form. The approach we use involves a generalization of the global determinant method of Salberger. [ABSTRACT FROM AUTHOR]

The linear independence (LI) hypothesis, which states roughly that the imaginary parts of the critical zeros of Dirichlet L-functions are linearly independent over the rationals, is known to have interesting consequences in the study of prime number races, as pointed out by Rubinstein and Sarnak. In this article, we prove that a function field analog of LI holds generically within certain families of elliptic curve L-functions and their symmetric powers. More precisely, for certain algebro-geometric families of elliptic curves defined over the function field of a fixed curve over a finite field, we give strong quantitative bounds for the number of elements in the family for which the relevant L-functions have their zeros as linearly independent over the rationals as possible. [ABSTRACT FROM AUTHOR]

The feasibility of a classification-by-rank program for modular categories follows from the Rank-Finiteness Theorem. We develop arithmetic, representation theoretic and algebraic methods for classifying modular categories by rank. As an application, we determine all possible fusion rules for all rank=5 modular categories and describe the corresponding monoidal equivalence classes. [ABSTRACT FROM AUTHOR]

Let p be a prime number, and F be a nonarchimedean local field of residual characteristic p. We explore the interaction between the pro-p-Iwahori-Hecke algebras of the group GLn(F) and its derived subgroup SLn(F). Using the interplay between these two algebras, we deduce two main results. The first is an equivalence of categories between Hecke modules in characteristic p over the pro-p-Iwahori-Hecke algebra of SL2(Qp) and smooth mod-p representations of SL2(Qp) generated by their pro-p-Iwahori-invariants. The second is a "numerical correspondence" between packets of supersingular Hecke modules in characteristic p over the pro-p-Iwahori-Hecke algebra of SLn(F), and irreducible, n-dimensional projective Galois representations. [ABSTRACT FROM AUTHOR]

The distribution of the unipotent modules (in nondefining prime characteristic) of the finite unitary groups into Harish-Chandra series is investigated. We formulate a series of conjectures relating this distribution with the crystal graph of an integrable module for a certain quantum group. Evidence for our conjectures is presented, as well as proofs for some of their consequences for the crystal graphs involved. In the course of our work we also generalize Harish-Chandra theory for some of the finite classical groups, and we introduce their Harish-Chandra branching graphs. [ABSTRACT FROM AUTHOR]

In this paper, we study deformations of mod p Galois representations τ (over animaginary quadratic field F)of dimension 2 whose semi-simplification is the direct sum of two characters τ1 and τ2.As opposed to [7],we do not impose any restrictions on the dimension of the crystalline Selmer group H1(F, Hom(τ2,τ1)) Φ Ext1 (τ2,τ1). We establish that 1/Σ there exists a basis B of H1(F, Hom(τ2,τ1)) arising from automorphic representations 1/Σ over F (Theorem 8.1). Assuming among other things that the elements of B admit only finitely many crystalline characteristic 0 deformations, we prove a modularity lifting theorem asserting that if τ itself is modular then so is its every crystalline characteristic zero deformation (Theorems 8.2 and 8.5). [ABSTRACT FROM AUTHOR]

Let T be a split torus over local or global function field. The theory of Brylinski-Deligne gives rise to the metaplectic central extensions of T by a finite cyclic group. The representation theory of these metaplectic tori has been developed to some extent in the works of M. Weissman, G. Savin, W. T. Gan, P. McNamara, and others. In this paper we propose a geometrization of this theory in the framework of the geometric Langlands program (in the everywhere nonramified case). [ABSTRACT FROM AUTHOR]

This article shows that on a generic principally polarized abelian variety of dimension 5 the Q-vector space spanned by the classes of the twenty-seven Abel-Prym curves modulo algebraic equivalence has dimension 7. [ABSTRACT FROM AUTHOR]

We establish the uniform estimate ≪dN2/d for the number of rational points of height at most N on an irreducible curve of degree d. We deduce this from a result for general hypersurfaces that is sensitive to the coefficients of the corresponding form. [ABSTRACT FROM AUTHOR]