linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

algebraic independence of arithmetic gamma values and carlitz zeta values abstract we consider the values at proper fractions of the arithmetic gamma function and the values at positive integers of the zeta function for f q and provide complete algebraic independence results for them

a herbrand ribet theorem for function fields we prove a function field analogue of the herbrand ribet theorem on cyclotomic number fields the herbrand ribet theorem can be interpreted as a result about cohomology with p coefficients over the splitting field of p and in our analogue both occurrences of p are replaced with the mathfrak p torsion scheme of the carlitz module for a prime mathfrak p in f q t

a dirichlet unit theorem for drinfeld modules we show that the module of integral points on a drinfeld module satisfies an analogue of dirichlet s unit theorem despite its failure to be finitely generated as a consequence we obtain a construction of a canonical finitely generated sub module of the module of integral points we use the results to give a precise formulation of a conjectural analogue of the class number formula

special l values of drinfeld modules we state and prove a formula for a certain value of the goss l function of a drinfeld module this gives characteristic p valued function eld analogues of the class number formula and of the birch and swinnerton dyer conjecture the formula and its proof are presented in an entirely selfcontained fashion

arithmetic of characteristic p special l values recently the second author has associated a finite fq t module h to the carlitz module over a finite extension of fq t this module is an analogue of the ideal class group of a number field in this paper we study the galois module structure of this module h for cyclotomic extensions of fq t we obtain function field analogues of some classical results on cyclotomic number fields such as the p adic class number formula and a theorem of mazur and wiles about the fitting ideal of ideal class groups we also relate the galois module h to anderson s module of circular units and give a negative answer to anderson s kummer vandiver type conjecture these results are based on a kind of equivariant class number formula which refines the second author s class number formula for the carlitz module

determination of the algebraic relations among special γ values in positive characteristic we devise a new criterion for linear independence over function fields using this tool in the setting of dual t motives we find that all algebraic relations among special values of the geometric r function over f q t are explained by the standard functional equations

frobenius actions on the de rham cohomology of drinfeld modules we study the action of endomorphisms of a drinfeld a module phi on its de rham cohomology h dr phi l and related modules in the case where phi is defined over a field l of finite a characteristic mathfrak p among others we find that the nilspace h 0 of the total frobenius fr dr on h dr phi l mathfrak p has dimension h height of phi we define and study a pairing between the mathfrak p torsion mathfrak p phi of phi and h dr phi l which becomes perfect after dividing out h 0

stark units in positive characteristic we show that the module of stark units associated to a sign normalized rank one drinfeld module can be obtained from anderson s equivariant a harmonic series we apply this to obtain a class formula a la taelman and to prove a several variable log algebraicity theorem generalizing anderson s log algebraicity theorem we also give another proof of anderson s log algebraicity theorem using shtukas and obtain various results concerning the module of stark units for drinfeld modules of arbitrary rank

algebraic relations among periods and logarithms of rank 2 drinfeld modules for any rank 2 drinfeld module rho defined over an algebraic function field we consider its period matrix p rho which is analogous to the period matrix of an elliptic curve defined over a number field suppose that the characteristic of the finite field bbb f q is odd and that rho does not have complex multiplication we show that the transcendence degree of the field generated by the entries of p rho over bbb f q theta is 4 as a consequence we show also the algebraic independence of drinfeld logarithms of algebraic functions which are linearly independent over bbb f q theta

bounds for the minimum numbers of generators of generalized cohen macaulay ideals throughout this paper r denotes a local ring with maxima1 idea1 m z an ideal of r and s the factor ring r z we call z a generalized cohen macaulay idea1 if s is a generalized cohen macaulay ring by definition 4 3 1 a local ring a with maxima1 ideal is a generalized cohen macaulay ring if the ith local cohomology module h a of a with support b is of finite length for i o dim a 1 a is a cohen macaulay ring iff h a 0 for i o dim a 1 the class of generalized cohen macaulay local rings is rather large for instance if a is a factor ring of a cohen macaulay ring a being a generalized cohen macaulay ring means that a is equidimensional and a is a cohen macaulay ring for all prime ideals y of a 4 2 5 and 381 the aim of this paper is to give bounds for the minimum number u z of generators of a generalized cohen macaulay idea1 z in terms of the following invariants of r and s m u m n v ms r dim r s dim s the multiplicities e r e s and z zyzl s i o s 1 where 1 denotes the length function our interest in such bounds for u z arose from the article 7 of sally there she proved that if r and s are cohen macaulay rings then v z e r e s r s l r s 1 first sally s result may be generalized as

pure anderson motives and abelian τ sheaves pure t motives were introduced by g anderson as higher dimensional generalizations of drinfeld modules and as the appropriate analogs of abelian varieties in the arithmetic of function fields in order to construct moduli spaces for pure t motives the second author has previously introduced the concept of abelian sheaf in this article we clarify the relation between pure t motives and abelian sheaves we obtain an equivalence of the respective quasi isogeny categories furthermore we develop the elementary theory of both structures regarding morphisms isogenies tate modules and local shtukas the later are the analogs of p divisible groups

the de rham isomorphism for drinfeld modules over tate algebras abstract introduced by angles pellarin and tavares ribeiro drinfeld modules over tate algebras are closely connected to anderson log algebraicity identities pellarin l series and taelman class modules in the present paper we define the de rham map for drinfeld modules over tate algebras and we prove that it is an isomorphism under natural hypotheses as part of this investigation we determine further criteria for the uniformizability and rigid analytic triviality of drinfeld modules over tate algebras

a difference version of nori s theorem we consider frobenius difference equations over mathbb f q s t phi q where phi q fixes t and acts on mathbb f q s as the frobenius endomorphism we prove that every semisimple simply connected linear algebraic group mathcal g defined over mathbb f q can be realized as a difference galois group over mathbb f q i s t phi q i for some i in mathbb n the proof uses upper and lower bounds on the galois group scheme of a frobenius difference equation that are developed in this paper the result can be seen as a difference analogue of nori s theorem which states that mathcal g mathbb f q occurs as a finite galois group over mathbb f q s

arithmetic of function field units we introduce the module of stark units associated to a drinfeld module defined over the ring of integers of a global function field we investigate the arithmetic properties of these objects and as an application we prove a discrete analogue for taelman s class modules of certain conjectures formulated by greenberg for cyclotomic fields

on identities for zeta values in tate algebras zeta values in tate algebras were introduced by pellarin in 2012 they are generalizations of carlitz s zeta values and play an increasingly important role in function field arithmetic in this paper we prove a conjecture of pellarin on identities for these zeta values the proof is based on arithmetic properties of carlitz s zeta values and an explicit formula for bernoulli type polynomials attached to pellarin s zeta values

fitting ideals of l adic realizations of picard 1 motives and class groups of global function fields in previous work we proved a result on the equivariant fitting ideal of the adic realization of the picard 1 motive attached to an abelian covering of curves defined over a finite field in this paper we build upon this work to deduce results on the equivariant fitting ideal of the tate modules of the jacobian of the top curve and on the equivariant fitting ideal of the dual of the degree zero class group of the top curve at the end we discuss refinements of brumer s conjecture and consider examples

on liaison arithmetical buchsbaum curves and monomial curves in p3 in the context of liaison by 181 and 191 for curves in ip3 the property of being the zero scheme of a section of a rank two bundle on ip3 is connected to that of being ideally the intersection of three surfaces an immediate consequence of this property is that the homogeneous ideal i c of a curve c c p3 is generated by precisely three elements if and only if c is ideally the intersection of three suraces and c is arithmetically cohen macaulay non complete intersection furthermore the characterization of the equivalence classes defined by liaison among curves in p3 yields the well known statement that a curve c is in the liaison class of a complete intersection if and only if c is arithmetically cohen macaulay see e g 2 10 13 or 141 it seems that a cayley see 7 p 1521 in 1847 was the first who posed the problem to describe this liaison class of a complete intersection now in this paper we will begin to investigate the next simple case that is the liaison classes which are characterized by a finite dimensional vector space of dimension l using the theory of buchsbaum rings see section 2 this means we will study liaison among arithmetical buchsbaum curves in p3 first from the point of view of local algebra we get the following more general result

stickelberger elements fitting ideals of class groups of cm fields and dualisation in this paper we systematically construct abelian extensions of cm fields over a totally real field whose stickelberger elements are not in the fitting ideals of the class groups our evidence indicates that pontryagin duals of class groups behave better than the class groups themselves we also explore the behaviour of fitting ideals under projective limits and dualisation in a somewhat broader context

determination of algebraic relations among special zeta values in positive characteristic abstract as an analogue to special values at positive integers of the riemann zeta function we consider carlitz zeta values c n at positive integers n by constructing t motives after papanikolas we prove that the only algebraic relations among these characteristic p zeta values are those coming from the euler carlitz relations and the frobenius pth power relations

universal gauss thakur sums and l series we prove that the maximal abelian extension tamely ramified at infinity of the rational function field over mathbb f q is generated by the values at the points in the algebraic closure of mathbb f q of the higher derivatives of the so called anderson and thakur function omega we deduce a similar property for the special values of the higher derivatives of a new kind of l series introduced by the second author

on guided waves in 3 d photonic crystals the propagation of guided waves in 3 d photonic crystals is studied in this paper we suppose that the spectrum of reference medium a perfectly 3 d photonic crystal has band gaps if a line defect is introduced along one of the lattice vectors we prove that spectrum can be created inside the gaps of the reference spectrum if the line defect is sufficiently strong furthermore we also prove that the generalized eigenfunctions corresponding to the generalized eigenvalues created by line defect are exponentially decaying away from the line defect

gamma functions for function fields and drinfeld modules the purpose of this paper is to study gamma functions in the context of the theory of function fields of one variable over a finite field in particular we explore among other things their connection with the theory of drinfeld modules of rank one which is function field cyclotomic theory their interpolations functional equations the nature of their special values transcendence algebraicity connection with periods and various analogies most of the facts about the classical gamma function are recalled when necessary readers may refer to g k or artin s book on the classical gamma function now we describe more precisely the contents of this paper let fq be a finite field of characteristic p section 0 covers the background material on drinfeld modules it also describes the gamma function f for fq t with

linear independence of gamma values in positive characteristic we investigate the arithmetic nature of special values of thakur s func tion field gamma function at rational points our main result is that all linear dependence relations over the field of algebraic functions are consequences of the anderson deligne thakur bracket relations

congruences between derivatives of geometric l functions we prove a natural refinement of a theorem of lichtenbaum describing the leading terms of zeta functions of curves over finite fields in terms of weil etale cohomology we then use this result to prove the validity of chinburg s 3 conjecture for all abelian extensions of global function fields to prove natural refinements and generalisations of the refined stark conjectures formulated by amongst others gross tate rubin and popescu to prove a variety of explicit restrictions on the galois module structure of unit groups and divisor class groups and to describe explicitly the fitting ideals of certain weil etale cohomology groups in an appendix coauthored with k f lai and k s tan we also show that the main conjectures of geometric iwasawa theory can be proved without using either crystalline cohomology or drinfeld modules

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

values of certain l series in positive characteristic we introduce a class of deformations of the values of the goss zeta function we prove with the use of the theory of deformations of vectorial modular forms as well as with other techniques a formula for their value at 1 and some arithmetic properties of values at other positive integers our formulas involve anderson and thakur s function we discuss how our formulas may be used to investigate the existence of a kind of functional equation for the goss zeta function

the iwasawa main conjectures for gl2 we prove the one two and three variable iwasawa greenberg main conjectures for a large class of modular forms that are ordinary with respect to an odd prime p the method of proof involves an analysis of an eisenstein ideal for ordinary hida families for gu 2 2

shtukas and jacobi sums we show how the analogues of jacobi sums in the context of function fields introduced and studied in t1 t2 t3 can be obtained from shtukas introduced and studied in d2 d3 m we apply this to obtain some results on the prime factorization of analogues of gauss sums and to prove an analogue of the gross koblitz formula for general function field generalizing the results in t2 for this purpose we also introduce and interpolate a new analogue of gamma function

tannakian duality for anderson drinfeld motives and algebraic independence of carlitz logarithms we develop a theory of tannakian galois groups for t motives and relate this to the theory of frobenius semilinear difference equations we show that the transcendence degree of the period matrix associated to a given t motive is equal to the dimension of its galois group using this result we prove that carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent

on special l values of t modules we show that taelman s conjecture on special l values of ander son t modules holds for a large class of t modules this class contains all mixed a finite and uniformizable t modules whose hodge pink weights are at least 1 as a consequence we deduce various log algebraicity identities for tensor powers of the carlitz module generalizing the work of anderson thakur

on equations defining arithmetically cohen macaulay schemes i in this paper we continue the study of the simplest arithmetically cohen macauly cm for short schemes cf 111 and 12 i we describe the hilbert function of an arbitrary cm scheme in p of given degree e lying on a hypersurface of given degree k and such that its general curvilinear section has minimal genus proposition 1 3 we present a geometric construction of all reduced irreducible codimension two cm varieties whose general curvilinear sections have minimal genus with respect to degree theorem 1 7 we also prove the following assertion essentially due to severi there exist smooth nonspecial cm curves of degree n and genus g in p iff r y1 g r d t y 1 2 proposition 1 6 any such curve has minimal genus with respect to degree in the last section we consider d dimensional cm schemes of degree e mvd j jed in p whose general curvilinear sections have minimal genus these are the most accessible cm schemes cf 12 i we prove that the corresponding part of the hilbert scheme is a connected smooth variety naturally birationally equivalent to nd symemvd p provided v d 2 theorem 2 6 this was known for m v d 2 3 1 cf 9 our proof relies on an explicit description of the infinitesimal deformations of a thick subspace of arbitrary codimension see 2 4 we work over an algebraically closed field of characteristic 0 by the genus of a cm curve we mean its arithmetic genus

transcendence and drinfeld modules it has been ten years since v g drinfeld introduced in 3 the concept of elliptic a module which we now call drinfeld a module these modules are analogues of elliptic curves for global function field k and we have witnessed a beautiful algebraic theory gradually taking shape in 13 we have asked a few natural t ranscendence questions conjectures concerning these modules having in mind the analogue of the classical work of siegel schneider about elliptic integrals and based on evidences provided by the old papers of carlitz and wade cf 1 11 here in this article we will prove those conjectures we will show that indeed there is an interesting transcendence theory for each algebraic curve over a finite field fol lowing drinfeld we let m be a lattice in i e a finitely generated discrete a module in k and form

on finite drinfeld modules in 5 deuring determined the possible isomorphism types of endomorphism rings of elliptic curves notably for those curves that are defined over a finite field his results were later generalized to abelian varieties of higher rank by tate 17 and honda 13 now in the fundamental paper 6 drinfeld transports the modular theory of elliptic curves to the function field case he found the kind of diophantine objects called by him elliptic modules that over global function fields play the role of elliptic curves in number theory by his theory he was able to prove analogues of the theorem of kroneckerweber the main theorem of complex multiplication and parts of the langlands conjectures for gl 2 over function fields actually in the course of the last few years the theory of drinfeld modules has shown to be the key tool in the arithmetic of function fields over finite fields this comes from the fact that drinfeld modules lead to moduli problems that are related to gl r r arbitrary and to galois representations in local fields of positive characteristic which one needs in order to describe the absolute galois group of a global function field in this paper we treat deuring s problem of endomorphism rings in the drinfeld module setting i e we study drinfeld modules that are defined over a finite field and their endomorphism rings let k co be a pair consisting of a function field k in one variable over a finite field and a place co of k let further a be the ring of elements of k with poles at most at co and p a prime of a with finite residue field if a p as for elliptic curves the classification up to isogeny of drinfeld modules 4 over extensions of if is given by the isomorphism type of end d k theorem 3 5 this ring turns out to be a certain division algebra central over the subfield e generated over k by the frobenius endomorphism f of 4 theorem 2 9 these two results are stated in 7 prop 2 11 in a somewhat disguised

localization of classical waves ii electromagnetic waves we consider electromagnetic waves in a medium described by a position dependent dielectric constant we assume that is a random perturbation of a periodic function and that the periodic maxwell operator has a gap in the spectrum where we prove the existence of localized waves i e finite energy solutions of maxwell s equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times localization of electromagnetic waves is a consequence of anderson localization for the self adjoint operators we prove that in the random medium described by the random operator exhibits anderson localization inside the gap in the spectrum of this is shown even in situations when the gap is totally filled by the spectrum of the random operator we can prescribe random environments that ensure localization in almost the whole gap

iwasawa main conjecture for the carlitz cyclotomic extension and applications we prove an iwasawa main conjecture for the class group of the mathfrak p cyclotomic extension mathcal f of the function field mathbb f q theta mathfrak p is a prime of mathbb f q theta showing that its fitting ideal is generated by a stickelberger element we use this and a link between the stickelberger element and a mathfrak p adic l function to prove a close analog of the ferrero washington theorem for mathcal f and to provide information on the mathfrak p adic valuations of the bernoulli goss numbers beta j i e on the values of the carlitz goss zeta function at negative integers

explicit formulae for l values in positive characteristic we focus on the generating series for the rational special values of pellarin s l series in 1 le s le 2 q 1 indeterminates and using interpolation polynomials we prove a closed form formula relating this generating series to the carlitz exponential the anderson thakur function and the anderson generating functions for the carlitz module we draw several corollaries including explicit formulae and recursive relations for pellarin s l series in the same range of s and divisibility results on the numerators of the bernoulli carlitz numbers by monic irreducibles of degrees one and two

uniformizable families of t motives abelian t modules and the dual notion of t motives were introduced by anderson as a generalization of drinfeld modules for such anderson defined and studied the important concept of uniformizability it is an interesting question and the main objective of the present article to see how uniformizability behaves in families since uniformizability is an analytic notion we have to work with families over a rigid analytic base we provide many basic results and in fact a large part of this article concentrates on laying foundations for studying the above question building on these we obtain a generalization of a uniformizability criterion of anderson and among other things we establish that the locus of uniformizability is berkovich open

transcendence in positive characteristic and special values of hypergeometric functions we prove a simple transcendence criterion suitable for function field arithmetic we apply it to show the transcendence of special values at non zero rational arguments or more generally at algebraic arguments which generate extension of the rational function field with less than q places at infinity of the entire hypergeometric functions in the function field over fq context and to obtain a new proof of the transcendence of special values at non natural p adic integers of the carlitz goss gamma function we also characterize in the balanced case the algebraicity of hypergeometric functions giving an analog of the result of f r villegas based on beukers heckman results in the classical hypergeometric case 0 introduction number theorists study the number fields and the function fields over finite fields together as global fields in view of the strong analogies that exist between them transcendental number theory originated with liouville s result of 1844 see 17 that an algebraic irrational cannot be well approximated by rationals k mahler 18 showed in 1949 that liouville s theorem holds for function fields over any field and also that the liouville s inequality is best possible if the ground field k is of positive characteristic p by considering t p i k 1 t which is algebraic of degree p as well as of the approximation exponent p in particular naive analog of roth s theorem fails see 30 cha 9 for a survey of general situation which is not even conjecturally fully understood yet so we need stronger approximations in function fields than in number fields counterparts to conclude transcendence by diophantine approximation on the other hand for many naturally occurring quantities in the function field arithmetic related to drinfeld modules and anderson s t motives such as periods special values of exponential logarithm gamma and zeta functions etc different techniques that exist algebraic group techniques more developed motivic machinery automata method strong diophantine approximation criterion of special kind etc in function fields have been used to prove very powerful results see 30 cha 10 11 for a survey of older results and 4 21 22 7 for some recent results many quantities of interest that occur naturally in the function field arithmetic have expansions that are fq linear or closely related to such so that they are date may 12 2010 1991 mathematics subject classification primary 11j81 11j93 secondary 11t55

galois theory for iterative connections and nonreduced galois groups this article presents a theory of modules with iterative connection this theory is a generalisation of the theory of modules with connection in characteristic zero as given by n katz to modules over rings of arbitrary characteristic we show that these modules with iterative connection and also the modules with integrable iterative connection form a tannakian category assuming some nice properties for the underlying ring and we show how this generalises to modules over schemes stratifications on modules as introduced by a grothendieck generalise the notion of integrable ordinary connections in another way however over smooth rings we obtain an equivalence of stratifications and integrable iterative connections furthermore over a regular ring in positive characteristic we show that the category of modules with integrable iterative connection is also equivalent to the category of flat bundles as defined by d gieseker in the second part of this article we set up a picard vessiot theory for fields of solutions for such a picard vessiot extension we obtain a galois correspondence which takes into account even nonreduced closed subgroup schemes of the galois group scheme on one hand and inseparable intermediate extensions of the picard vessiot extension on the other hand finally we compare our galois theory with the galois theory for purely inseparable field extensions given by s chase

integrable systems and number theory in finite characteristic the purpose of this paper is to give an overview of applications of the concepts and techniques of the theory of integrable systems to number theory in finite characteristic the applications include explicit class field theory and langlands conjectures for function fields effect of the geometry of the theta divisor on factorization of analogs of gauss sums special values of function field gamma zeta and l functions analogs of theorems of weil and stickelberger control of the intersection of the jacobian torsion with the theta divisor the techniques are the krichever drinfeld dictionaries and the theory of solitons akhiezer baker and tau functions developed in this context of arithmetic geometry by anderson 2001 elsevier science b v all rights reserved

on periods of the third kind for rank 2 drinfeld module in analogy with the periods of abelian integrals of differentials of the third kind for an elliptic curve defined over a number field we introduce a notion of periods of the third kind for a rank 2 drinfeld mathbb f q t module defined over an algebraic function field in this paper we establish explicit formulae for these periods of the third kind for combining with the main result in chang and papanikolas j am math soc 25 123 150 2012 we show the algebraic independence of the periods of first second and third kinds for

gauss sums for fq t the purpose of this paper is to define gauss sums taking values in function fields of one variable over a finite field and to prove analogues of various classical and recent results these results include stickelberger s theorem the hasse davenport theorem weil s theorem on jacobi sums as hecke characters and the gross koblitz theorem for comparison the reader may consult g k and references given there

alternating multizeta values in positive characteristic we introduce alternating multizeta values in positive characteristic which are generalizations of thakur multizeta values we establish their fundamental properties including non vanishing sum shuffle relations period interpretation and linear independence which is a direct sum result for these values

estimating the order of vanishing at infinity of drinfeld quasi modular forms we introduce and study certain deformations of drinfeld quasi modular forms by using rigid analytic trivialisations of corresponding anderson s t motives we show that a sub algebra of these deformations has a natural graduation by the group z 2 x z q 1 z and an homogeneous automorphism and we deduce from this and other properties multiplicity estimates

taylor coefficients of anderson thakur series and explicit formulae for each positive characteristic multiple zeta value defined by thakur in function field arithmetic world scientific publishing river edge 2004 the first and third authors in chang and mishiba invent math 2020 https doi org 10 1007 s00222 020 00988 1 constructed a t module together with an algebraic point such that a specified coordinate of the logarithmic vector of the algebraic point is a rational multiple of that multiple zeta value the objective of this paper is to use the taylor coefficients of anderson thakur series and t motivic carlitz multiple star polylogarithms to give explicit formulae for all of the coordinates of this logarithmic vector

gaps in the spectrum of the maxwell operator with periodic coefficients the periodic maxwell operator is considered piecewise constant coefficients are constructed in such a way that the spectrum of operator has the gaps

special values of drinfeld modular forms and algebraic independence let a be a polynomial ring in one variable over a finite field and k be its fraction field let f be a drinfeld modular form of nonzero weight for a congruence subgroup of gl2 a so that the coefficients of the q expansion of f are algebraic over k we consider n cm points 1 n on the drinfeld upper half plane for which the quadratic fields k 1 k n are pairwise distinct suppose that f is non vanishing at these n points then we prove that f 1 f n are algebraically independent over k