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Semistable models of elliptic curves over residue characteristic 2May 14 2019Given an elliptic curve $E$ in Legendre form $y^2 = x(x - 1)(x - \lambda)$ over the fraction field of a Henselian ring $R$ of mixed characteristic $(0, 2)$, we present an algorithm for determining a semistable model of $E$ over $R$ which depends only ... More

The Structure of Integral Parabolic Subgroups of Orthogonal GroupsFeb 21 2019We determine the detailed structure of parabolic subgroups of orthogonal groups over $\mathbb{Z}$, and deduce the precise form of canonical boundary components in toroidal compactifications of orthogonal Shimura varieties.

Strongly Non-zero Points and Elliptic PseudoprimesFeb 02 2019We examine the notion of strongly non-zero points and use it as a tool in the study of several types of elliptic pseudoprimes. Moreover, we give give some probabilistic results about the existence of strong elliptic pseudoprimes for a randomly chosen ... More

Tate module and bad reductionSep 26 2018Let C/K be a curve over a local field. We study the natural semilinear action of Galois on the minimal regular model of C over a field F where it becomes semistable. This allows us to describe the Galois action on the l-adic Tate module of the Jacobian ... More

The Picard group of an order and Külshammer reductionJul 13 2018Let $(K,\mathcal O,k)$ be a $p$-modular system and assume $k$ is algebraically closed. We show that if $\Lambda$ is an $\mathcal O$-order in a separable $K$-algebra, then $\textrm{Pic}_{\mathcal O}(\Lambda)$ carries the structure of an algebraic group ... More

Explicit resolution of weak wild quotient singularities on arithmetic surfacesMay 24 2018Jun 15 2018A weak wild arithmetic quotient singularity arises from the quotient of a smooth arithmetic surface by a finite group action, where the inertia group of a point on a closed characteristic p fiber is a p-group acting with smallest possible ramification ... More

On quadratic curves over finite fieldsFeb 28 2018The geometry of algebraic curves over finite fields is a rich area of research. In previous work, the authors investigated a particular aspect of the geometry over finite fields of the classical unit circle, namely how the number of solutions of the circle ... More

A functional logarithmic formula for hypergeometric functions 3F2Jan 19 2018We give a sufficient condition for that the hypergeometric function 3F2 is a linear combination of the logarithmic function. The proof is based on the regulator formula which we proved in another preprint, arXiv:1709.04144.

Regulators of K_1 of Hypergeometric FibrationsSep 13 2017We study a deformation of what we call hypergeometric fibrations. Its periods and K_1-regulators are described in terms of hypergeometric functions 3F2 in a variable given by the deformation parameter.

Inverse Galois problem for ordinary curvesApr 25 2017May 16 2017We consider the inverse Galois problem over function fields of positive characteristic p, for example, the inverse Galois problem over the projective line. We describe a method to construct certain Galois covers of the projective line and other curves, ... More

Regulators of K_2 of Hypergeometric FibrationsMar 30 2017Nov 22 2017We discuss Beilinson's regulator on K_2 of certain fibrations of algebraic varieties which we call the hypergeomtric fibrations. The main result is to describe regulators via the hypergeometric functions 3F2 or 4F3. We also discuss the Beilinson conjecture ... More

Proof of Serge Lang's Heights Conjecture and an almost optimal Bound for the Torsion of Elliptic CurvesJan 25 2017Sep 08 2018This paper focuses on the proof of Serge Lang's Heights Conjecture in a form that is completely effective. As a complementary result the author provides a new proof of Mazur-Merel theorem about a bound for the torsion of elliptic curves in terms of the ... More

Anomalous Primes and the Elliptic Korselt CriterionAug 08 2016Aug 12 2016We introduce the notion of a Bachet anomalous number and we show that, conditional on a special case of the Tijdeman-Zagier conjecture, the Bachet anomalous numbers are exactly the prime powers of the form $3n^2 + 3n + 1$. We then examine Type I elliptic ... More

Anomalous Primes and the Elliptic Korselt CriterionAug 08 2016Aug 21 2018We explore the relationship between elliptic Korselt numbers of Type I, a class of pseudoprimes introduced by Silverman in [20], and anomalous primes. We generalize a result in [20] that gives sufficient conditions for an elliptic Korselt number of Type ... More

Analysis and Probability on Infinite-Dimensional SpacesJul 13 2016Sep 07 2016These lecture notes contain an introduction to some of the fundamental ideas and results in analysis and probability on infinite-dimensional spaces, mainly Gaussian measures on Banach spaces. They originated as the notes for a topics course at Cornell ... More

Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priorsMay 19 2016May 31 2016This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451--559, 2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, ... More

Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priorsMay 19 2016Jun 30 2017This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451--559, 2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, ... More

Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priorsMay 19 2016Nov 18 2016This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451--559, 2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, ... More

Eta quotients, Eisenstein series and Elliptic CurvesApr 26 2016We express all the newforms of weight $2$ and levels $30$, $33$, $35$, $38$, $40$, $42$, $44$, $45$ as linear combinations of eta quotients and Eisenstein series, and list their corresponding strong Weil curves. Let $p$ denote a prime and $E (\zz_p)$ ... More

An algebro-geometric study of special values of hypergeometric functions ${}_3F_2$Mar 15 2016Apr 02 2018For certain class of hypergeometric functions ${}_3F_2$ with rational parameters, we give a sufficient condition for the special value at $1$ to be expressed in terms of logarithms of algebraic numbers. We give two proofs, both of which are algebro-geometric ... More

An algebro-geometric study of the unit arguments ${}_3F_2(1,1,q;a,b;1)$, IMar 15 2016Let $a$, $b$, $q$ be rational numbers such that none of $a$, $b$, $q$, $q-a$, $q-b$, $q-a-b$ is an integer. Then we prove, ${}_3F_2(1,1,q;a,b;1)$ is a $\overline{\mathbb{Q}}$-linear combination of log of algebraic numbers if $\{sq\}+\{s(a-q)\}+\{s(b-q)\}+\{s(q-a-b)\}=2$ ... More

Compactifications of S-arithmetic quotients for the projective general linear groupOct 03 2015Let F be a global field, and let S be a finite set of places of F containing all archimedean places. Consider the product X of the symmetric spaces and Bruhat-Tits buildings for PGL_d of the completions of F at archimedean and non-archimedean places in ... More

Compactifications of S-arithmetic quotients for the projective general linear groupOct 03 2015Dec 09 2016Let F be a global field, and let S be a finite set of places of F containing all archimedean places. Consider the product X of the symmetric spaces and Bruhat-Tits buildings for PGL_d of the completions of F at archimedean and non-archimedean places in ... More

Splitting properties of the reduction of semi-abelian varietiesSep 09 2015Dec 11 2015Let $K$ be a complete discrete valuation field. Let $\mathcal{O}_K$ be its ring of integers. Let $k$ be its residue field which we assume to be algebraically closed of characteristic exponent $p\geq1$. Let $G/K$ be a semi-abelian variety. Let $\mathcal{G}/\mathcal{O}_K$ ... More

Separators of Ideals in Multiplicative Semigroups of Unique Factorization DomainsAug 29 2015In this paper we show that if $I$ is an ideal of a commutative semigroup $C$ such that the separator $SepI$ of $I$ is not empty then the factor semigroup $S=C/P_I$ ($P_I$ is the principal congruence on $C$ defined by $I$) satisfies Condition $(*)$: $S$ ... More

Coherence conditions in flat regular pullbacksJun 17 2015We investigate the behavior of four coherent-like conditions in regular conductor squares. In particular, we find necessary and sufficient conditions in order that a pullback ring be a finite conductor ring, a coherent ring, a generalized GCD ring, or ... More

Adaptive Density Estimation on the Circle by Nearly-Tight FramesApr 02 2015Mar 15 2016This work is concerned with the study of asymptotic properties of nonparametric density estimates in the framework of circular data. The estimation procedure here applied is based on wavelet thresholding methods: the wavelets used are the so-called Mexican ... More

Classification of finite group automorphisms with a large cycleOct 08 2014Mar 31 2015Let $\psi$ be a permutation of a finite set $X$. We define $\lambda(\psi)$ to be the largest fraction of elements of $X$ lying on a single cycle of $\psi$. For a finite group $G$, we define $\lambda(G)$ to be the maximum among the values $\lambda(\alpha)$, ... More

l-Adic Representations and their Associated InvariantsOct 04 2014Mar 17 2015These are notes from a 3-lecture course given by V. Dokchitser at the ICTP in Trieste, Italy, 1st--5th of September 2014, as part of a graduate summer school on "L-functions and modular forms". The course is meant to serve as an introduction to l-adic ... More

On Barnes Beta Distributions, Selberg Integral and Riemann XiFeb 25 2014The theory of Barnes beta probability distributions is advanced and related to the Riemann xi function. The scaling invariance, multiplication formula, and Shintani factorization of Barnes multiple gamma functions are reviewed using the approach of Ruijsenaars ... More

Foundations of Rigid Geometry IAug 21 2013Feb 07 2014In this research oriented manuscript, foundational aspects of rigid geometry are discussed, putting emphasis on birational side of formal schemes and topological feature of rigid spaces. Besides the rigid geometry itself, topics include the general theory ... More

Foundations of Rigid Geometry IAug 21 2013Feb 28 2017In this research oriented manuscript, foundational aspects of rigid geometry are discussed, putting emphasis on birational side of formal schemes and topological feature of rigid spaces. Besides the rigid geometry itself, topics include the general theory ... More

Elliptic ReciprocityDec 10 2012Sep 13 2016The paper introduces the notions of an elliptic pair, an elliptic cycle and an elliptic list over a square free positive integer d. These concepts are related to the notions of amicable pairs of primes and aliquot cycles that were introduced by Silverman ... More

Generic Phenomena in Groups -- Some Answers and Many QuestionsNov 28 2012We give a survey of some known results and of the many open questions in the study of generic phenomena in geometrically interesting groups.

Topologies and structures of the Cremona groupsOct 25 2012Mar 27 2013We study the algebraic structure of the $n$-dimensional Cremona group and show that it is not an algebraic group of infinite dimension (ind-group) if $n\ge 2$. We describe the obstruction to this, which is of a topological nature. By contrast, we show ... More

Supercongruences and Complex MultiplicationOct 16 2012Nov 20 2012We study congruences involving truncated hypergeometric series of the form_rF_{r-1}(1/2,...,1/2;1,...,1;\lambda)_{(mp^s-1)/2} = \sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r \lambda^k where p is a prime and m, s, r are positive integers. These truncated hypergeometric ... More

Isogeny volcanoesAug 27 2012May 07 2013The remarkable structure and computationally explicit form of isogeny graphs of elliptic curves over a finite field has made them an important tool for computational number theorists and practitioners of elliptic curve cryptography. This expository paper ... More

Congruences of models of elliptic curvesJul 03 2012Jul 02 2013Let O_K be a discrete valuation ring with field of fractions K and perfect residue field. Let E be an elliptic curve over K, let L/K be a finite Galois extension and let O_L be the integral closure of O_K in L. Denote by X' the minimal regular model of ... More

Euler factors determine local Weil representationsDec 20 2011We show that a Frobenius-semisimple Weil representation over a local field K is determined by its Euler factors over the extensions of K. The construction is explicit, and we illustrate it for l-adic representations attached to elliptic and genus 2 curves. ... More

Identifying supersingular elliptic curvesJul 06 2011Nov 09 2016Given an elliptic curve E over a field of positive characteristic p, we consider how to efficiently determine whether E is ordinary or supersingular. We analyze the complexity of several existing algorithms and then present a new approach that exploits ... More

Identifying supersingular elliptic curvesJul 06 2011Sep 05 2012Given an elliptic curve E over a field of positive characteristic p, we consider how to efficiently determine whether E is ordinary or supersingular. We analyze the complexity of several existing algorithms and then present a new approach that exploits ... More

Colourings of lattices and coincidence site latticesNov 03 2010The relationship between the coincidence indices of a lattice $\Gamma_1$ and a sublattice $\Gamma_2$ of $\Gamma_1$ is examined via the colouring of $\Gamma_1$ that is obtained by assigning a unique colour to each coset of $\Gamma_2$. In addition, the ... More

Elliptic SurfacesJul 02 2009Jul 09 2010This survey paper concerns elliptic surfaces with section. We give a detailed overview of the theory including many examples. Emphasis is placed on rational elliptic surfaces and elliptic K3 surfaces. To this end, we particularly review the theory of ... More

When do nonlinear filters achieve maximal accuracy?Jan 08 2009Jul 01 2009The nonlinear filter for an ergodic signal observed in white noise is said to achieve maximal accuracy if the stationary filtering error vanishes as the signal to noise ratio diverges. We give a general characterization of the maximal accuracy property ... More

The distribution of natural numbers divisible by 2,3,5,11,13 and 17 on the Square Root SpiralJan 29 2008The natural numbers divisible by the Prime Factors 2, 3, 5, 11, 13 and 17 lie on defined spiral graphs, which run through the Square Root Spiral. A mathematical analysis shows, that these spiral graphs are defined by specific quadratic polynomials. Basically ... More

The ordered distribution of natural numbers on the square root spiralDec 13 2007Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the Square Root Spiral (also named as the Spiral of Theodorus or Wurzel Spirale or Einstein Spiral). Prime Numbers also clearly accumulate on such ... More

Regulator constants and the parity conjectureSep 18 2007Mar 21 2009The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p-infinity Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/Q is semistable at ... More

The Fundamental Theorem of prehomogeneous vector spaces modulo $p^m$. With an appendix "L-functions of prehomogeneous vector spaces" by Fumihiro SatoAug 10 2004Aug 03 2005For a number field $K$ with ring of integers ${\mathcal O}_K$, we prove an analogue over finite rings of the form ${\mathcal O}_K/{\mathcal P}^m$ of the Fundamental Theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, ... More

p-adic properties of division polynomials and elliptic divisibility sequencesApr 22 2004For a given point P in the group of K-rational points E(K) of an elliptic curve, we consider the sequence of values (F_1(P),F_2(P),F_3(P),...) of the division polynomials of E at P. If K is a finite field, we prove that the sequence is periodic. If K/Q_p ... More

Weight-monodromy conjecture over equal characteristic local fieldsAug 14 2003Jan 26 2005The aim of this paper is to study certain properties of the weight spectral sequences of Rapoport-Zink by a specialization argument. By reducing to the case over finite fields previously treated by Deligne, we prove that the weight filtration and the ... More

Weight-monodromy conjecture for certain threefolds in mixed characteristicDec 09 2002Jul 31 2003The weight-monodromy conjecture claims the coincidence of the weight filtration and the monodromy filtration, up to shift, on the $l$-adic \'etale cohomology of a proper smooth variety over a complete discrete valuation field. Although it has been proved ... More

Hodge modules on Shimura varieties and their higher direct images in the Baily-Borel compactificationSep 26 2002Jan 07 2004We prove an analogue for Hodge modules of Pink's theorem on the degeneration of l-adic sheaves (Math. Ann. 292). Let j be the open immersion of a Shimura variety M into its Baily-Borel compactification. Its boundary has a natural stratification into locally ... More

On the Eisenstein symbolJun 22 2000The main purpose of this paper is the geometric construction, and the analysis of the formalism of elliptic Bloch groups. In the setting of absolute cohomology, we obtain a generalization of Beilinson's Eisenstein symbol to divisors of an elliptic curve, ... More

Dynamical systems arising from elliptic curvesJul 02 1999We exhibit a family of dynamical systems arising from rational points on elliptic curves in an attempt to mimic the familiar toral automorphisms. At the non-archimedean primes, a continuous map is constructed on the local elliptic curve whose topological ... More

Galois Theory for a Class of Modular LatticesMar 03 1999Aug 12 1999We construct Galois theory for sublattices of certain complete modular lattices and their automorphism groups. A well-known description of the intermediate subgroups of the general linear group over an Artinian ring containing the group of diagonal matrices, ... More

Galois Theory for a Class of Complete Modular LatticesMar 01 1999Aug 11 1999We construct Galois theory for sublattices of certain complete modular lattices and their automorphism groups. A well-known description of the intermediate subgroups of the general linear group over a semilocal ring containing the group of diagonal matrices, ... More

A Note on the Arrangement of Subgroups in the Automorphism Groups of Submodule Lattices of Free ModulesFeb 18 1999Aug 12 1999A complete description of subgroups in the general linear group over a semilocal ring containing the group of diagonal matrices was obtained by Z.I.Borewicz and N.A.Vavilov. It is shown in the present paper that a similar description holds for the intermediate ... More

Mixed sheaves on Shimura varieties and their higher direct images in toroidal compactificationsNov 04 1998Oct 19 1999Let (P,X) be Shimura data, M=M(P,X,K) the Shimura variety of level K. To an algebraic representation of P, one can associate a mixed sheaf (variation of Hodge structure, l-adic sheaf) on M. In the paper, we study the degeneration of such sheaves along ... More