dwork s conjecture on unit root zeta functions annals of mathematics 150 1999 867 927 arxiv math 9911270v1 math nt 1 nov 1999 dwork s conjecture on unit root zeta functions by daqing wan 1 introduction in this article we introduce a systematic new method to investigate the conjectural p adic meromorphic continuation of professor bernard dwork s unit root zeta function attached to an ordinary family of algebraic varieties defined over a finite field of characteristic p after his pioneer p adic investigation of the weil conjectures on the zeta function of an algebraic variety over a finite field dwork went on to study the p adic analytic variation of a family of such zeta functions when the variety moves through an algebraic family in the course of doing so he was led to a new zeta function called the unit root zeta function which goes beyond the reach of the existing theory he conjectured 8 that such a unit root zeta function is p adic meromorphic everywhere these unit root zeta functions contain important arithmetic information about a family of algebraic varieties they are truly p adic in nature and are transcendental functions sometimes seeming quite mysterious in fact no single nontrivial example has been proved to be true about this conjecture other than the trivial overconvergent or log convergent case for which dwork s classical p adic theory already applies see 6 11 26 for various attempts in this article we introduce a systematic new method to study such unit root zeta functions our method can be used to prove the conjecture in the case when the involved unit root f crystal has rank one in particular this settles the first nontrivial case the rank one unit root f crystal coming from the family of higher dimensional kloosterman sums our method further allows us to understand reasonably well about analytic variation of an arithmetic family of such rank one unit root zeta functions motivated by the gouv ea mazur conjecture about dimension variation of classical and p adic modular forms we shall introduce another this work was partially supported by nsf the author wishes to thank p deligne and n katz for valuable discussions at one stage of this work during 1993 and 1994 the author would also like to thank s sperber for his careful reading of the manuscript and for his many detailed comments which led to a significant improvement of the exposition

on the density of modular points in universal deformation spaces based on comparison theorems for hecke algebras and universal deformation rings with strong restrictions at the critical prime l as provided by the results of wiles taylor diamond et al we prove under rather general conditions that the corresponding universal deformation spaces with no restrictions at l can be identified with certain hecke algebras of l adic modular forms as conjectured by gouvea thus generalizing previous work of gouvea and mazur along the way we show that the universal deformation spaces we consider are complete intersections flat over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i l of relative dimension three in which the modular points form a zariski dense subset furthermore the fibers above inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i l of these spaces are generically smooth

geometric invariants for liaison of space curves let k be an algebraically closed field and let s k x x x2 x3 for a curve cc pi the hartshorne rao module m c bneie h p3 xc n is a graded s module of finite length and up to duals and shifts is a complete invariant of liaison cf 12 1 given a graded s module m z m of finite length the action of s p p 6 1 between two consecutive components i e ql s hom m m gives rise to a degeneracy locus which can be thought of as lying in psi p namely let i p les irk4 l r and let v v s where s max r i v 5 p3 or else v 0 these loci are isomo hism invariants and are preserved under duals and shifts for m n c these loci are related to the geometry of c itself the main philosophy that emerges section 2 is that the degeneracy locus generally corresponds to those planes h in p3 which meet c nongenerically either containing a component of c or having cn n impose an unusually small number of conditions on some plane curves on h we conclude section 2 by applying these ideas to derive some necessary conditions for m c to have components in negative degrees the remaining sections give various apphcations of the techniques introduced in section 2 first in section 3 we classify sets of skew lines up to liaison that is if c and c consist of t 2 3 and t skew lines respectively then we ask when c can be linked to c the answer in general is never but the situation changes signi cantly if c lies on a quadric surface cf theorem 3 1 a large part of the answer can be read from the degeneracy locus v

dimension variation of classical and p adic modular forms a quadratic bound is obtained for a conjecture of gouvea mazur on arithmetic variation of dimensions of classical and p adic modular forms

families of modular eigenforms this article is an expansion of the notes to a one hour lecture for an msri workshop on computational number theory the editors of mathematics of computation kindly asked us to submit these notes for publication and we are enormously pleased to do so our original audience did not consist of experts in the field of modular forms and we have tried to keep this article accessible to nonexperts we have made an experimental investigation of certain arithmetic conjec tures using macsyma this investigation requires a search for certain mod ular eigenforms of high weight these computations pose problems which we feel may be interesting on their own we are novices here and we seek advice from people more experienced in making computations of an analogous sort we are in fact deeply grateful to j f mestre who came to our aid and who vastly extended our computations using pari 1 mestre has graciously allowed us to present his computations in this article the kind of families we have in mind as in the title of this lecture has as its prototypical example the standard family of classical eisenstein series of level 1 and weight k for k 4 6 8 whose fourier expansions are given by oo

on the cohen macaulay and buchsbaum property for unions of planes in affine space the coordinate ring a of a union of planes in affine space is studied and it is asked when a is a cohen macaulay or buchsbaum ring these properties are related to the position of the planes via the notion of seminormality it is shown that a is cohen macaulay iff a is connected in codimension 2 and seminormal in an appropriate sense consideration of the cohen macaulification then yields a simple criterion for a to be buchsbaum methods with several examples are illustrated

a cubic counterpart of jacobi s identity and the agm we produce exact cubic analogues of jacobi s celebrated theta function identity and of the arithmetic geometric mean iteration of gauss and legendre the iteration in question is a n 1 a n 2b n 3 and b n 1 formula cannot be replicated the limit of this iteration is identified in terms of the hypergeometric function f 1 3 2 3 1 which supports a particularly simple cubic transformation

ramanujan s theories of elliptic functions to alternative bases in his famous paper 3 10 pp 23 39 ramanujan offers several beautiful series representations for 1 pi he first states three formulas one of which is frac 4 pi sum limits mathbf n 0 infty frac left 6 mathbf n 1 right left frac 1 2 right 3 n left mathbf n right 3 4 n where a o 1 and for each positive integer n left mathbf a right n mathbf a left mathbf a 1 right left mathbf a 2 right left mathbf a mathbf n 1 right

slopes of eigencurves over boundary disks let p be a prime number we study the slopes of u p eigenvalues on the subspace of modular forms that can be transferred to a definite quaternion algebra we give a sharp lower bound of the corresponding newton polygon the computation happens over a definite quaternion algebra by jacquet langlands correspondence it generalizes a prior work of jacobs slopes of compact hecke operators thesis university of london imperial college 2004 who treated the case of p 3 with a particular level in case when the modular forms have a finite character of conductor highly divisible by p we improve the lower bound to show that the slopes of u p eigenvalues grow roughly like arithmetic progressions as the weight k increases this is the first very positive evidence for buzzard kilford s conjecture on the behavior of the eigencurve near the boundary of the weight space that is proved for arbitrary p and general level we give the exact formula of a fraction of the slope sequence

flasque sheaves on posets and cohen macaulay unions of regular varieties the goal of this paper is to find new examples of cohen macaulay and non cohen macaulay rings among the coordinate rings of the unions of regular varieties the cohen macaulayness as well as other properties seminormality buchsbaumness etc was studied extensively for twodimensional varieties z 5 111 however for larger dimensions not much was known besides the reisner theorem lo which gives an equivalent combinatorial condition for the coordinate ring of the unions of coordinate linear subspaces stanley reisner or face rings 123 to be cohen macaulay the method used in this paper is to represent a ring as the section ring of a sheaf on a poset and then to apply theorem 6 4 of 131 more precisely for a finite set q of alline regular varieties we consider the poset x a of all intersections of elements of 0 and the sheaf a on x o of the coordinate rings of these intersections the restriction of regular functions defines a monomorphism p from the coordinate ring r of u u to the section ring s of d c the above mentioned result form 13 can be applied to r only if the two following restrictive conditions hold

on the mean iteration a b a 3b 4 ab b 2 ce processus iteratif est etudie en detail sa vitesse de convergence est quadratique une limite est explicitement donnee

rank one case of dwork s conjecture journal of the american mathematical society volume 00 number 0 xxxx xxxx pages 000 000 s 0894 0347 xx 0000 0 arxiv math 0005308v1 math nt 9 may 2000 rank one case of dwork s conjecture daqing wan 1 introduction in the higher rank paper 17 we reduced dwork s conjecture from higher rank case over any smooth affine variety x to the rank one case over the simplest affine space a n in the present paper we finish our proof by proving the rank one case of dwork s conjecture over the affine space a n which is called the key lemma in 17 the key lemma had already been proved in 16 in the special case when the frobenius lifting is the simplest q th power map x x q thus the aim of the present paper is to treat the general frobenius lifting case our method here is an improvement of the limiting method in 16 it allows us to move one step further and obtain some explicit information about the zeros and poles of the unit root l function as in 16 to handle the rank one case we are forced to work in the more difficult infinite rank setting see section 2 for precise definitions of the various basic infinite rank notions let f q denote the finite field of characteristic p 0 our main result of this paper is the following theorem theorem 1 1 let be a nuclear overconvergent module over the affine n space a n f q ordinary at the slope zero side let unit be the unit root slope zero part of assume that unit has rank one let be another nuclear overconvergent module over a n f q then for each integer k the l function l k unit t is p adic meromorphic furthermore the family l k t of l functions unit parametrized by integers k in each residue class modulo q 1 is a strong family of meromorphic functions with respect to the p adic topology of k a finite rank module is automatically nuclear thus theorem 1 1 includes the key lemma of 17 over a n as a special case the basic ideas of the present paper are the same as the limiting approach in 16 the details are however quite different in the simplest q th power frobenius lifting case one has the fundamental dwork trace formula available which is completely explicit for uniform estimates this makes it easy to extend the dwork trace formula to infinite rank setting it also makes it possible to see the various analytic subtleties involved in a concrete case as a result we were able to prove analytically optimal results in 16 for a general frobenius lifting even over the simplest affine n space a n as we shall work in this paper one has to use the much more difficult monsky trace formula which is a generalization of dwork s trace formula thus the first task of this paper is to extend the monsky trace formula to infinite rank setting and to make it sufficiently 1991 mathematics subject classification primary 11g40 11s40 secondary 11m41 14g15 key words and phrases l functions fredholm determinants p adic meromorphic continua tion nuclear modules and banach modules this work was partially supported by nsf c american mathematical society

quadratic iterations to associated with elliptic functions to the cubic and septic base in this paper properties of the functions a d q b d q and c d q are derived specializing at d 1 and 2 we construct two new quadratic iterations to these are analogues of previous iterations discovered by the borweins 1987 j m borwein and f g garvan 1997 and h h chan 2002 two new transformations of the hypergeometric series 2 f 1 1 3 1 6 1 z are also derived

p adic banach spaces and families of modular forms let p be a prime cp the completion of an algebraic closure of the p adic numbers qp and k a nite extension of qp contained in cp let v be the valuation on cp such that v p 1 and let be the absolute value on cp such that x p v x for x cp suppose n is a positive integer prime to p let x1 np denote the modular curve over k which represents elliptic curves with 1 np structure and let up be the hecke operator on modular forms on x1 np which takes a form with q expansion n anq n to the modular form with q expansion n anpq n a modular form f is said to have slope q if there is a polynomial r t over cp such that r up f 0 and such that the newton polygon of r t has only one side and its slope is

seminormality and cohomology of projective varieties let x be a reduced closed subscheme of p proj r y k a field and r k x a r 2 2 let a be the homogeneous co ordinate ring of x in 2 3 51 the various authors have attempted to determine whether a is cohenmacaulay or seminormal or more generally to study the seminormalization or cohen macaulification of a the papers mentioned above work primarily with unions of straight lines in s corollary 5 91 it is proved for example that if x is a connected union of lines which have linearly independent directions at each intersection point then a is seminormal if and only if a is cohen macaulay the aim of this paper is to give a better understanding of the seminormalization of a and the relationship between seminormality and cohenmacaulayness hinted at by the above mentioned corollary 5 9 this we do in section 2 by interpreting the seminormalization of a in terms of sheaf cohomology we are then able to give a reasonable generalization our corollary 3 5 of s corollary 5 91 and also to resolve several computational problems left open in 2 31 and 4 the hilbert function of a graded ring a will be denoted by ha that is ha i dim ai if x proj a the hilbert polynomial of a is denoted hp x in some computations of section 6