subband energy based reduced complexity spectrum sensing under noise uncertainty and frequency selective spectral characteristics the present work proposes a subband energy detection method that performs efficiently under noise uncertainty nu and frequency selective channels the critical impact of detrimental modeling uncertainties such as nu is analytically quantified and it is shown that the introduced method is robust to both nu and frequency selectivity conditions this is also the case for eigenvalue based sensing techniques in contrast to traditional energy detector based sensing connections of the subband energy based approach and existing eigenvalue based methods are established analytically which leads to a novel reduced complexity processing technique based on the difference between maximum and minimum subband energies the proposed method is capable of providing accurate and robust performance with low signal to noise ratios snr in the presence of nu closed form expressions are derived for the corresponding probability of false alarm and probability of detection under frequency selectivity due to the primary signal spectrum and or the transmission channel the validity of the offered expressions is justified through comparisons with respective results from computer simulations the sensing performance is evaluated in different communication scenarios with different frequency selective channel models and primary user waveforms the offered results indicate that the proposed methods provide quite significant savings in complexity e g 78 reduction in the considered example case while also improving the detection performance at low snrs and in the presence of nu

estimating isogenies on elliptic curves the object of this paper is to prove a new type of estimate for isogenies between elliptic curves this has several diophantine applications effective versions of serre s galois irreducibility theorem and shafarevich s theorem for example which are presented in another paper mw3 later articles will deal with the corresponding problems for abelian varieties of arbitrary dimension right at the beginning we emphasize that we are identifying elliptic curves e with weierstrass equations

perfect powers expressible as sums of two cubes abstract let n 3 this paper is concerned with the equation a 3 b 3 c n which we attack using a combination of the modular approach via frey curves and galois representations with obstructions to the solutions that are of brauer manin type we shall show that there are no solutions in coprime non zero integers a b c for a set of prime exponents n having dirichlet density 28 219 44 928 0 628 and for a set of exponents n having natural density 1

new results on selection diversity the performances of selection diversity receiver structures in a slow flat rayleigh fading environment are assessed a number of new and interesting results are obtained binary digital signaling using noncoherent frequency shift keying ncfsk differential phase shift keying dpsk coherent phase shift keying cpsk and coherent frequency shift keying cfsk is considered the traditional analysis the traditional selection diversity model of a selection diversity system is based on choosing the branch with the largest signal to noise snr power ratio while assuming that the noise power is constant across all branches however many practical selection systems choose the branch based on a largest signal plus noise s n selection sample of a filter output this paper comprises accurate analyses of such s n selection systems results show that s n selection systems perform better than predicted by the traditional selection diversity model this is because the former includes the statistical nature of the noise whereas the latter does not the performance difference between the two models increases as the number of diversity branches increases for each of dpsk and cpsk the dual diversity equal gain eg combining and s n selection systems perform identically for each of ncfsk and cfsk receiver structures which are equivalent when there is no diversity perform differently in a diversity environment certain dual diversity s n selection systems give the same performances as eg combining or square law combining the results are contingent upon perfect cophasing for the eg combining in systems where estimates of the combining carrier phases contain noise s n selection outperforms eg combining for dual diversity

klein forms and the generalized superelliptic equation if f x y 2 z x y is an irreducible binary form of degree k 3 then a theorem of darmon and granville implies that the generalized superelliptic equation f x y z l has given an integer l maxf2 7 kg at most nitely many solutions in coprime integers x y and z in this paper for large classes of forms of degree k 3 4 6 and 12 including heuristically most cubic forms we extend this to prove a like result where the parameter l is now taken to be variable in the case of irreducible cubic forms this provides the rst examples where such a conclusion has been proven the method of proof combines classical invariant theory modular galois representations and properties of elliptic curves with isomorphic mod n galois representations

the l component of the unipotent albanese map we prove a finiteness theorem for the local l p component of the qp unipotent albanese map for curves as an application we refine the non abelian selmer varieties arising in the study of global points and deduce thereby a new proof of siegel s theorem for affine curves over q of genus one with mordell weil rank at most one

solving discrete zero point problems in this paper we present two theorems on the existence of a discrete zero point of a function from the n dimensional integer lattice n to the n dimensional euclidean space n the theorems differ in their boundary conditions for both theorems we give a proof using a combinatorial lemma and present a constructive proof based on a simplicial algorithm that finds a discrete zero point within a finite number of steps

solving fermat type equations via modular ℚ curves over polyquadratic fields we solve the diophantine equations x 4 dy 2 z p for d 2 and d 3 and any prime p 349 and p 131 respectively the method consists in generalizing the ideas applied by frey ribet and wiles in the solution of fermat s last theorem and by ellenberg in the solution of the equation x 4 y 2 z p and we use q curves modular forms and inner twists in principle our method can be applied to solve this type of equations for other values of d

entropy and energy detection based spectrum sensing over mathcal f composite fading channels in this paper we investigate the performance of energy detection based spectrum sensing over mathcal f composite fading channels to this end an analytical expression for the average detection probability is first derived this expression is then extended to account for collaborative spectrum sensing square law selection diversity reception and noise power uncertainty the corresponding receiver operating characteristics roc are analyzed for different conditions of the average signal to noise ratio snr noise power uncertainty time bandwidth product multipath fading shadowing number of diversity branches and number of collaborating users it is shown that the energy detection performance is sensitive to the severity of the multipath fading and the amount of shadowing whereby even small variations in either of these physical phenomena can significantly impact the detection probability as a figure of merit to evaluate the detection performance the area under the roc curve auc is derived and evaluated for different multipath fading and shadowing conditions closed form expressions for the differential entropy and cross entropy are also formulated and assessed for different average snr multipath fading and shadowing conditions then the relationship between the differential entropy of mathcal f composite fading channels and the corresponding roc auc is examined where it is shown that the average number of bits required for encoding a signal becomes small i e low differential entropy when the detection probability is high or when the auc is large the difference between composite fading and traditional small scale fading is emphasized by comparing the cross entropy for rayleigh and nakagami m fading a validation of the analytical results is provided through a careful comparison with the results of some simulations

torsion homology growth and cycle complexity of arithmetic manifolds let m be an arithmetic hyperbolic 3 manifold such as a bianchi manifold we conjecture that there is a basis for the second homology of m where each basis element is represented by a surface of low genus and give evidence for this we explain the relationship between this conjecture and the study of torsion homology growth

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

on the field of definition of p torsion points on elliptic curves over the rationals let s mathbb q d be the set of primes p for which there exists a number field k of degree le d and an elliptic curve e mathbb q such that the order of the torsion subgroup of e k is divisible by p in this article we give bounds for the primes in the set s mathbb q d in particular we show that if p ge 11 p ne 13 37 and p in s mathbb q d then p le 2d 1 moreover we determine s mathbb q d for all d le 42 and give a conjectural formula for all d ge 1 if serre s uniformity problem is answered positively then our conjectural formula is valid for all sufficiently large d under further assumptions on the non cuspidal points on modular curves that parametrize those j invariants associated to cartan subgroups the formula is valid for all d ge 1

one parameter families of elliptic curves over ℚ with maximal galois representations let e be an elliptic curve over q and let q e n be its nth division field in 1972 serre showed that if e is without complex multiplication then the galois group of q e n q is as large as possible that is gl2 z nz for all integers n coprime to a constant integer m e q depending at most on e q serre also showed that the best one can hope for is to have gl2 z nz gal q e n q 2 for all positive integers n we study the frequency of this optimal situation in a one parameter family of elliptic curves over q and show that in essence for almost all one parameter families almost all elliptic curves have this optimal behavior

euler systems and jochnowitz congruences this article relates the gross zagier formula with a simpler formula of gross for special values of l series via the theory of congruences between modular forms given two modular forms f and g of different levels which are congruent but whose functional equations have sign 1 and 1 respectively and an imaginary quadratic field k satisfying certain auxiliary conditions the main result gives a congruence between the algebraic part of l f k 1 expressed in terms of heegner points and the algebraic part of the special value l g k 1 congruences of this type were anticipated by jochnowitz and for this reason are referred to as jochnowitz congruences introduction let e be a modular elliptic curve over q of conductor n and let k be a quadratic imaginary field the l function l e k s has a functional equation of the standard kind relating its values at s and 2 s when all the primes dividing n are split in k then k is said to satisfy the heegner hypothesis relative to e in that case the sign in the functional equation of l e k s is 1 furthermore a construction of birch and heegner based on the theory of com plex multiplication gives a heegner point yk e e k coming from the modular parametrization

heegner points on mumford tate curves 1 shimura curves 417 2 heegner points 429 3 the regulator term 441 4 the conjecture 445 5 applications and re nements 447

diagonal cycles and euler systems ii the birch and swinnerton dyer conjecture for hasse weil artin l functions this article establishes new cases of the birch and swinnerton dyer conjecture in analytic rank 0 for elliptic curves over mathbb q viewed over the fields cut out by certain self dual artin representations of dimension at most 4 when the associated l function vanishes to even order ge 2 at its central point two canonical classes in the corresponding selmer group are constructed and shown to be linearly independent assuming the non vanishing of a garrett hida p adic l function at a point lying outside its range of classical interpolation the key tool for both results is the study of certain p adic families of global galois cohomology classes arising from gross kudla schoen diagonal cycles in a tower of triple products of modular curves

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

non vanishing of high derivatives of automorphic l functions at the center of the critical strip keywords modular l functions mollification nonvanishing results central value of high derivatives analytic rank jacobian variety reference tan article 2000 004doi 10 1515 crll 2000 074 record created on 2008 11 14 modified on 2017 05 12

a rigid analytic gross zagier formula and arithmetic applications let f be a newform of weight 2 and squarefree level n its fourier coefficients generate a ring of whose fraction field kf has finite degree over q fix an imaginary quadratic field k of discriminant prime to n corresponding to a dirichlet character e the l series l f k s l f s l f 0 e s of f over k has an analytic continuation to the whole complex plane and a functional equation relating l f k s to l f k 2 s assume that the sign of this functional equation is 1 so that l f k s vanishes to even order at s 1 this is equivalent to saying that the number of prime factors of n which are inert in k is odd fix any such prime say p the field k determines a factorization n n nof n by taking n resp nto be the product of all the prime factors of n which are split resp

surjectivity of galois representations associated with quadratic mathbb q curves we prove in this paper an uniform surjectivity result for galois representations associated with non cm mathbb q curves over imaginary quadratic fields using various tools for the proof such as mazur s method isogeny theorems runge s method and analytic estimates of sums of l functions

p adic l functions and p adic periods of modular forms let e be an elliptic curve which is defined over q and has stable reduction modulo a given prime p assuming that e is modular one can associate to e a p adic l function lp e s see mz swd a v vi mz t t for its construction in various cases this function is defined by a certain interpolation property and is analytic for sezp in this paper we will assume that e has split multiplicative reduction at p under this assumption the interpolation property implies that lp e 1 0 we will prove a formula for ep e 1 which was discovered experimentally by mazur tate and teitelbaum mz t t by tate s p adic urfiformization theory there is a p adic integer q epzp which we refer to as the tate period for e and a p adic analytic isomorphism

on the existence of finite state supervisors under partial observations we consider discrete event systems under partial observations we show the equivalence of the m spl sigma sub c l g observability of a control specification and the m spl sigma sub c spl sigma observability of its corresponding augmented language we derive necessary and sufficient conditions for the existence of finite state supervisors under partial observations

explicit surjectivity of galois representations for abelian surfaces and gl 2 varieties abstract let a be an absolutely simple abelian variety without potential complex multiplication defined over the number field k suppose that either dim a 2 or a is of gl 2 type we give an explicit bound l 0 a k such that for every prime l l 0 a k the image of gal k k in aut t l a is as large as it is allowed to be by endomorphisms and polarizations

proving the triviality of rational points on atkin lehner quotients of shimura curves in this paper we give a method for studying global rational points on certain quotients of shimura curves by atkin lehner involutions we obtain explicit conditions on such quotients for rational points to be trivial coming from cm points only and exhibit an explicit infinite family of such quotients satisfying these conditions

quadratic chabauty for modular curves and modular forms of rank one in this paper we provide refined sufficient conditions for the quadratic chabauty method on a curve x to produce an effective finite set of points containing the rational points x mathbb q with the condition on the rank of the jacobian of x replaced by condition on the rank of a quotient of the jacobian plus an associated space of chow heegner points we then apply this condition to prove the effective finiteness of x mathbb q for any modular curve x x 0 n or x mathrm ns n of genus at least 2 with n prime the proof relies on the existence of a quotient of their jacobians whose mordell weil rank is equal to its dimension and at least 2 which is proven via analytic estimates for orders of vanishing of l functions of modular forms thanks to a kolyvagin logachev type result

integration on cal h _p times cal h and arithmetic applications this article describes a conjectural p adic analytic construction of global points on modular elliptic curves points which are defined over the ring class fields of real quadratic fields the resulting conjectures suggest that the classical heegner point construction and the theory of complex multiplication on which it is based should extend to a variety of contexts in which the underlying field is not a cm field

some tractable supervisory control problems for discrete event systems modeled by buchi automata discrete event systems des are modeled by buchi automata together with a means of online control in this setting the concept of a controllable language is extended to infinite strings and conditions for the existence of a supervisor controller to implement a prescribed closed loop behavior are derived the focus is on a class of des called product systems these are des composed of a finite set of asynchronous components a control problem for such a system typically requires the synthesis of an online controller so as to achieve some prescribed coordinated behavior of the component subsystems one of the principal difficulties in this task is that the size of the state space increases exponentially with the number of components it is shown that despite this fact several interesting control synthesis problems for such systems are computationally feasible and algorithms are developed for solution

on the modularity of ℚ curves a q curve is an elliptic curve over a number field k which is geometrically isogenous to each of its galois conjugates ribet 16 asked whether every q curve is modular and showed that a positive answer would follow from serre s conjecture on mod p galois representations we answer ribet s question in the affirmative subject to certain local conditions at 3 msc classification 11g18 14g35 14h52

the fermat type equations 2 or 3 solved through ℚ curves we solve the diophantine equations x 5 y 5 dz p with d 2 3 for a set of prime numbers of density 1 4 1 2 respectively the method consists in relating a possible solution to another diophantine equation and solving the later by using q curves and a generalized modular technique as in work of ellenberg and dieulefait jimenez along with some new techniques for eliminating newforms

unlikely intersections with isogeny orbits in a product of elliptic schemes fix an elliptic curve e 0 without cm and a non isotrivial elliptic scheme over a smooth irreducible curve both defined over the algebraic numbers consider the union of all images of a fixed finite rank subgroup of arbitrary rank of e 0 g also defined over the algebraic numbers under all isogenies between e 0 g and some fiber of the g th fibered power mathcal a of the elliptic scheme where g is a fixed natural number as a special case of a slightly more general result we characterize the subvarieties of arbitrary dimension inside mathcal a that have potentially zariski dense intersection with this set in the proof we combine a generalized vojta remond inequality with the pila zannier strategy

special values of anticyclotomic l functions the object of this paper is to extend the results and methods of vat01 where it was shown how cases of a conjecture of mazur on the behavior of l functions in an anticyclotomic zp extension could be deduced by studying the distribution of heegner points associated to definite quaternion algebras the main result of vat01 showed that when the sign in the functional equation is 1 then the special values of the l functions in question are generically nonzero in this paper we propose to study the special values modulo a prime of q and can offer three new theorems in this direction the first result theorem 1 1 determines the iwasawa invariant of the p adic l functions of bertolini and darmon we show that is usually zero but not always when 0 we give a precise formula for the value and an interpretation of the positivity in terms of congruences the second result theorem 1 2 pertains to the case where has residue characteristic p in this case we show that the l values are typically units but again that this is not always the case for technical reasons this theorem applies only to a restricted class of but it seems rather likely that the restrictions can be lifted finally we use the result of theorem 1 2 to transfer our results from the case where the sign in the functional equation is 1 to the case of sign 1 and to derivatives of l functions this is achieved by using congruences and the sign change phenomenon exploited by bertolini darmon namely we prove a jochnowitz congruence which relates the nontriviality of a special value modulo to the nontriviality of a classical heegner point on a modular curve the result is stated in theorem 1 4 to describe these results more precisely let g denote a newform on 0 n let k denote an imaginary quadratic field of discriminant d such that d is prime to n write n n n where n is divisible only by those primes which are split in k whereas n is divisible only by primes

teitelbaum s exceptional zero conjecture in the anticyclotomic setting teitelbaum formulated a conjecture relating first derivatives of the mazur swinnerton dyer adic l functions attached to modular forms of even weight k 2 to certain invariants arising from shimura curve parametrizations this article formulates an analogue of teitelbaum s conjecture in which the cyclotomic zp extension of q is replaced by the anticyclotomic zp extension of an imaginary quadratic field this analogue is then proved by using the cerednik drinfeld theory of 7 adic uniformisation of shimura curves introduction let 0 y an 2 on to n the classical l function l cf s admits an analytic continuation to the entire complex plane and a functional equation which relates its values at s and k s of special arithmetic interest for the present work is the central value l j k 2 for example when k 2 the birch and swinnerton dyer conjecture relates the behavior of l j s at s 1 to the arithmetic of the abelian variety a associated to j by the eichler shimura construction in mtt a adic variant of the birch and swinnerton dyer conjecture is formulated with l s replaced by a 7 adic analogue lp f s attached to the cyclotomic zp extension of q when p divides n exactly and ap 1 which implies that a has split multiplicative reduction at p the function lp f s vanishes at s 1 in this case the conjectures of mtt imply the following relationship between the first

rankin selberg l functions in the level aspect keywords moments rankin selberg convolution level aspect convexity breaking reference tan article 2002 003doi 10 1215 s0012 7094 02 11416 1 record created on 2008 11 14 modified on 2017 05 12

on relations between jacobians of certain modular curves abstract we confirm a conjecture of l merel h darmon and l merel j reine angew math 490 1997 81 100 describing a certain relation between the jacobians of various quotients of x p in terms of specific correspondences the method of proof involves reducing this conjecture to a question about certain z gl 2 f p module homomorphisms which is in turn answered by exhibiting some peculiar relations in a double coset algebra

elliptic curves with abelian division fields let e be an elliptic curve over mathbb q and let n ge 1 the central object of study of this article is the division field mathbb q e n that results by adjoining to mathbb q the coordinates of all n torsion points on e overline mathbb q in particular we classify all curves e mathbb q such that mathbb q e n is as small as possible that is when mathbb q e n mathbb q zeta n and we prove that this is only possible for n 2 3 4 or 5 more generally we classify all curves such that mathbb q e n is contained in a cyclotomic extension of mathbb q or equivalently by the kronecker weber theorem when mathbb q e n mathbb q is an abelian extension in particular we prove that this only happens for n 2 3 4 5 6 or 8 and we classify the possible galois groups that occur for each value of n

fermat type equations of signature 13 13 p via hilbert cuspforms in this paper we prove that for p 13649 equations of the form x 13 y 13 cz p have no non trivial primitive solutions a b c such that 13 not mid c for an infinite family of values for c our method consists on relating a solution a b c to the previous equation to a solution a b c 1 of another diophantine equation with coefficients in mathbb q sqrt 13 then we attach to a b c 1 a frey curve e a b defined over mathbb q sqrt 13 that is not a mathbb q curve we prove a modularity result of independent interest for certain elliptic curves over totally real abelian number fields satisfying some local conditions at 3 this theorem in particular implies modularity of e a b this enables us to use level lowering results and apply the modular approach via hilbert cuspforms over mathbb q sqrt 13 to prove the non existence of a b c 1 and consequently of a b c

on l functions of elliptic curves and cyclotomic towers in this paper we shall strengthen a recent theorem of ralph greenberg 1 concerning l functions of elliptic curves and anticyclotomic towers let k be an imaginary quadratic field and m an abelian extension of k possibly of infinite degree we say that the extension m k is anticyclotomic if m is galois over q and if the nontrivial element of gal k q acts on gal m k by inversion now fix a finite set p of rational primes and consider the compositum in an algebraic closure of k of all anticyclotomic extensions of k which are unramified outside p we denote this compositum by l and refer to it as the maximal anticyclotomic extension of k unramified outside p the field l may also be described as the union of all ring class fields of k with conductor divisible only by primes in p class field theory shows that gal l k is isomorphic to the product of a finite group and the group

uniform distribution of heegner points let e be a modular elliptic curve over q of conductor n let k denote an imaginary quadratic field of discriminant d with n d 1 if p is a prime then there exists a unique zp extension k k such that gal k q acts nontrivially on gal k k the field k is called the anticyclotomic zp extension of k let e k denote the mordell weil group of e over k then a fundamental conjecture of mazur maz predicts that the size of e k is controlled by the prime factorization of n in k equivalently mazur s conjecture relates the size of the mordell weil group to the sign in the functional equation of certain l series the conjecture was verified by greenberg rohrlich and rubin in what mazur calls the exceptional case when e has complex multiplication by k more generally they settled the conjecture for certain abelian varieties with complex multiplication for a discussion of this cm case we refer the reader to gre roh and rub our goal in this paper is to treat the generic case which occurs either when e has no cm or when the field of complexmultiplications is distinct from k under certain conditions on e and k mazur s conjecture predicts that the group e k is finitely generated our main result asserts that this is in fact the case at least when p is an ordinary prime for e or when the class number of k is prime to p the main new ingredient we introduce is that of equidistribution following ideas used by ferrero andwashington to study the cyclotomic invariant more precisely we show that theheegner points associated to definite quaternion algebras are uniformly distributed on the components of a certain curve x and that the elements of a certain galois group act independently in a suitable sense this combined with a special value formula due to gross allows us to conclude that the special values of anticyclotomic l functions are almost always nonzero so that the statement about the mordell weil groups follows from the machinery of euler systems as developed by bertolini and darmon bd

serre s uniformity problem in the split cartan case we prove that there exists an integer p 0 such that x split p q is made of cusps and cm points for any prime p p 0 equivalently for any non cm elliptic curve e over q and any prime p p 0 the image of the galois representation induced by the galois action on the p division points of e is not contained in the normalizer of a split cartan subgroup this gives a partial answer to an old question of serre

recipes to fermat type equations of the form x r y r cz p we describe a strategy to attack infinitely many fermat type equations of signature r r p where r ge 7 is a fixed prime and p is a prime allowed to vary indeed to a solution a b c of x r y r cz p we will attach several frey curves e e a b defined over totally real subfields of mathbb q zeta r we prove modularity of all the frey curves and the exsitence of a constant constant m r depending only on r such that for all p m r the representations bar rho e p are absolutely irreducible along the way we also prove modularity of certain elliptic curves that are semistable at all v mid 3 finally we illustrate our methods by proving arithmetic statements about equations of signature 7 7 p among which we emphasize that using a multi frey technique we show there is some constant m such that if p m then the equation x 7 y 7 3z p has no non trivial primitive solutions

comparison of generalized q function algorithms corresp several methods have been proposed for calculating the generalized q function q m alpha beta but only isolated comparisons of the various techniques have been reported the accuracy range and execution time of prominent methods for computing the q function are compared and limitations of the algorithms are identified

the infimal prefix closed and observable superlanguage of given language abstract the supervisory control of discrete event systems under partial observation is examined in particular the condition of observability needed to solve the two main existing problems is investigated a fixed point characterization of observability is given and is shown to yield an effective test for observability where the system under consideration is specified by regular languages a formula for the infimal prefix closed observable superlanguage of a given language is presented where the system is specified by regular languages the formula for the infimal observable language is effectively computable

ring theoretic properties of certain hecke algebras the purpose of this article is to provide a key ingredient of w2 by establishing that certain minimal hecke algebras considered there are complete intersections as is recorded in w2 a method going back to mazur m allows one to show that these algebras are gorenstein but for the complete intersection property a new approach is required the methods of this paper are related to those of chapter 3 of w2 the methods of section 3 of this paper are based on a previous approach of one of us a w we would like to thank henri darmon fred diamond and gerd faltings for carefully reading the first version of this article gerd faltings has also suggested a simplification of our argument as well as of the argument of chapter 3 of w2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper r t would like to thank a w for his invitation to collaborate and for sharing his many insights into the questions considered r t would also like to thank princeton university universite de paris 7 and harvard university for their hospitality during this collaboration a w was supported by an nsf grant

equations of hyperelliptic shimura curves we describe a method for computing explicit models of hyperelliptic shimura curves attached to an indenite quaternion algebra over q and atkin lehner quotients of them it exploits cerednik drinfeld s non archimedean uniformisation of shimura curves a formula of gross and zagier for the endomorphism ring of heegner points over artinian rings and the connection between ribet s bimodules and the specialization of heegner points as introduced in 22 as an application we provide a list of equations of shimura curves and quotients of them obtained by our method that had been conjectured by kurihara

some cases of serre s uniformity problem we show that if e mathbb q is an elliptic curve without complex multiplication and for which there is a prime q such that the image of bar rho e q is contained in the normaliser of a split cartan subgroup of mathrm gl 2 mathbb f q then bar rho e p surjects onto mathrm gl 2 mathbb f p for every prime p 37 this result complements a previous result by the author we also prove analogue results for certain families of mathbb q curves building on results of ellenberg am j math 126 4 763 787 2004 and le fourn math ann 365 1 2 173 214 2016

galois representations attached to q curves and the generalized fermat equation a 4 b 2 c p we prove that the equation a 4 b 2 c p has no solutions in coprime positive integers when p 211 the main step is to show that for all sufficiently large primes p every q curve over an imaginary quadratic field k with a prime of potentially multiplicative reduction greater than 6 has a surjective mod p galois representation the bound on p depends on k and the degree of the isogeny between e and its galois conjugate but is independent of the choice of e the proof of this theorem combines geometric arguments due to mazur momose darmon and merel with an analytic estimate of the average special values of certain l functions

a necessary and sufficient condition for the existence of finite state supervisors in discrete event systems a supervisor realized by finite automaton is called a finite state supervisor it is clear that a finite state supervisor exists for every regular controllable language however an example of a finite state supervisor for a nonregular one has been reported a necessary and sufficient condition is proved here for the existence of a finite state supervisor and an example of a simple manufacturing system that has a finite state supervisor for a nonregular controllable language is given

galois actions on q curves and winding quotients we prove two large images results for the galois representations attached to a degree d q curve e over a quadratic field k if k is arbitrary we prove maximality of the image for every prime p 13 not dividing d provided that d is divisible by q but d q with q 2 or 3 or 5 or 7 or 13 if k is real we prove maximality of the image for every odd prime p not dividing d d where d disc k provided that e is a semistable q curve in both cases we make the standard assumptions that e does not have potentially good reduction at all primes p 6 and that d is square free

on the existence of finite state supervisors for arbitrary supervisory control problems given two prefix closed languages k l spl sube spl sigma where k spl sube l represents the desired closed loop behavior and l is the open loop behavior there exists a finite state supervisor that enforces k in the closed loop if and only if there is a regular prefix closed language m spl sube spl sigma such that 1 m spl sigma sub u spl cup l spl sube m and 2 m spl cup l k in this paper we show that this is equivalent to 1 the controllability of sup spl lcub p spl sube k spl cup l spl macr spl verbar pr p p spl rcub with respect to spl sigma and 2 the regularity of sup spl lcub p spl sube k spl cup l spl macr spl verbar pr p p spl rcub where l spl macr spl sigma spl minus l and pr spl middot is the set of prefixes of strings in the language argument we use this property to investigate the issue of deciding the existence of a finite state supervisor for different representations we also present some properties of the language sup spl lcub p spl sube k spl cup l spl macr spl verbar pr p p spl rcub along with implications to the synthesis of solutions to the supervisory control problem with the fewest states