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

secondary terms in counting functions for cubic fields we prove the existence of secondary terms of order x5 6 in the davenport heilbronn theorems on cubic fields and 3 torsion in class groups of quadratic fields for cubic fields this confirms a conjecture of datskovsky wright and roberts we also prove a variety of generalizations including to arithmetic progressions where we discover a curious bias in the secondary term roberts s conjecture has also been proved independently by bhargava shankar and tsimerman in contrast to their work our proof uses the analytic theory of zeta functions associated to the space of binary cubic forms developed by shintani and datskovsky wright

hermitian harmonic maps from complete hermitian manifolds to complete riemannian manifolds in this paper we study a nonlinear elliptic system of equations imposed on a map from a complete hermitian non kahler manifold to a riemannian manifold this system is more appropriate to hermitian geometry than the harmonic map system since it is compatible with the holomorphic structure of the domain manifold in the sense that holomorphic maps are hermitian harmonic maps it was first studied by jost and yau in j y and was applied to study the rigidity of compact hermitian manifolds we extend their existence and uniqueness results to the case where both domain and target manifolds are complete hopefully the results will be useful to study corresponding rigidity of complete hermitian manifolds let m be a complex manifold with hermitian metric h and let n be a riemannian manifold with metric gij and christoffel symbols i jk a hermitian harmonic map u m n satisfies the following elliptic system

energy release rate along any crack front in the thickness direction of shell elements abstract this paper deals with a numerical method for calculating the profile of the energy release rate g or j integral along any crack front in the thickness direction of shell elements for this purpose the virtual crack extension technique is firstly used for its average evaluation throughout the entire shell thickness secondly local values for points on the crack front are derived by assuming that the g parameter at any level in the shell s thickness direction is linearly proportional to the strain energy density of points at the same level in the region where hrr singular fields are dominant example computations show that the g parameter profile by the present method is in good agreement with that by alternative methods throughout a large range of elastic plastic deformation

nonvanishing of quadratic twists of modular l functions and applications to elliptic curves if d is square free or is a fundamental discriminant then let d d denote the kronecker character for the quadratic field q d whose fundamental discriminant is d throughout d shall denote a fundamental discriminant the d quadratic twist of f denoted f d is the newform corresponding to the twist of f by the character d in particular if gcd m d 1 then f d z n 1 d n a n q n and

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

rank one quadratic twists of an infinite family of elliptic curves a conjecture of goldfeld implies that a positive proportion of quadratic twists of an elliptic curve e q has analytic rank 1 this assertion has been confirmed by vatsal v1 and the first author by for only two elliptic curves here we confirm this assertion for infinitely many elliptic curves e q using the heegner divisors the 3 part of the class groups of quadratic fields and a variant of the binary goldbach problem for polynomials

rank one twists of a certain elliptic curve the purpose of this note is to give the first known example for a conjecture of goldfeld on the number of rank one curves appearing in a family of quadratic twists we show unconditionally that the curve x0 19 has the property that a positive proportion of its quadratic twists have analytic rank one this amounts to a strong nonvanishing statement for the derivatives of certain l functions when the sign of the functional equation is 1 we simultaneously obtain the fact that a positive proportion of twists have analytic rank zero when the sign of the functional equation is 1 let e be a modular elliptic curve over q with corresponding newform f z anq n s2 0 n let l s f l s e ann s be the usual l series then l s f satisfies a functional equation under s 2 s with sign 1 e let d be the fundamental discriminant of a quadratic field k k d and write d for the associated quadratic character in this setting we define the twisted l function l s fd d n ann if e is given by the equation y 2 p x then l s fd is the l function attached to the twisted elliptic curve ed given by dy 2 p x therefore l s fd also satisfies a functional equation and if d n 1 then the sign is given by d d n for r 0 1 and a positive real number x define m r f x d d x ords 1l s fd r the following conjecture of goldfeld gol79 is well known

selmer groups of quadratic twists of elliptic curves 1 1 e y a1xy a3y x 3 a2x 2 a4x a6 where a1 a2 a3 a4 a6 z let n e denote the conductor of e j e the j invariant of e and l e s n 1 a n n s the hasse weil l function of e if e is modular then let fe z n 1 ae n q n s2 n e 1 be the associated weight 2 cusp form here 1 denotes the trivial dirichlet character throughout d will denote a square free integer and d shall denote the kronecker character for the field q d let h d denote the order of cl d the ideal class group of q d if l is prime then define h d l by

variation of iwasawa invariants in hida families let gq gl2 k be an absolutely irreducible modular galois representation over a finite field k of characteristic p assume further that is p ordinary and p distinguished in the sense that the restriction of to a decomposition group at p is reducible and non scalar the hida family h of is the set of all p ordinary p stabilized newforms f with mod p galois representation isomorphic to if is unramified at p then one must also fix an unramified line in and require that the ordinary line of f reduces to this fixed line these newforms are a dense set of points in a certain p adic analytic space of overconvergent eigenforms consisting of an intersecting system of branches i e irreducible components t a indexed by the minimal primes a of a certain hecke algebra to each modular form f h one may associate the iwasawa invariants an f an f alg f and alg f the analytic resp algebraic invariants are the number of zeroes of the p adic l function resp of the characteristic power series of the dual of the selmer group of f while the invariants are the exponents of the powers of p dividing the same objects in this paper we prove the following results on the behavior of these iwasawa invariants as f varies over h

optimal quotients of modular jacobians let n be a positive integer and f a normalized newform of weight two on 0 n attached to f is an optimal quotient a f of the jacobian j 0 n we prove two theorems concerning such optimal quotients a let denote the algebra of endomorphisms of j 0 n generated by the hecke operators and let f denote the ideal of fusion of f in if f denotes the dual abelian variety to a f then the canonical polarization of j 0 n induces a polarization f f a f we show that there is an embedding ker f j 0 n f whose cokernel is supported at maximal ideals of for which j 0 n is not two dimensional b if n is prime let c denote the subgroup of j 0 n generated by the divisor 0 on x 0 n mazur has shown that c is equal to the full torsion subgroup of j 0 n and that specialization modulo n induces an isomorphism of c with the group of connected components j 0 n of the characteristic n fibre of the neron model of j 0 n we prove that analogous results hold for every optimal quotient of prime conductor thereby generalizing results of mestre and oesterle who treated the case of strong weil curves and confirming william stein s refined eisenstein conjecture the key idea in the proof of these two theorems is encapsulated in corollary 2 5 below which allows us to apply multiplicity one results in a novel way to the study of optimal quotients

the cuspidal group and special values of l functions the structure of the cuspidal divisor class group is investigated by relating this structure to arithmetic properties of special values of l functions of weight two eisenstein series a new proof of a theorem of kubert proposition 3 1 concerning the group of modular units is derived as a consequence of the method the key lemma is a nonvanishing result theorem 2 1 for values of the l function attached to a one dimensional cohomology class over the relevant congruence subgroup proposition 4 7 provides data regarding eisenstein series and associated subgroups of the cuspidal divisor class group which the author hopes will simplify future calculations in the cuspidal group let xr q be shimura s canonical model over q of the complete modular curve associated to the congruence group r c sl2 z where r is one of the groups r n r1 n or ro n the cuspidal divisor class group cr c pic xr plays an important role in the theory of the arithmetic of the modular curve xr for example in mazur s proof of ogg s conjecture 13 and in the proof by mazur and wiles of the main conjecture of iwasawa theory 14 subgroups of the cuspidal group lead to the isogenies used in the eisenstein descent theory these subgroups can also be used to give congruence formulas for the universal special values of the l function of xr which are compatible with the descent theoretic results and the conjecture of birch and swinnerton dyer 3 5 12 19 the structure of the cuspidal group has been investigated extensively by kubert and lang 6 7 their approach is based on a study of the group of modular units in the present paper we propose another approach based on a study of integrality and divisibility properties of special values of l functions of weight two eisenstein series typically such an l function is a product of two dirichlet l functions whose special values are closely related to the arithmetic of cyclotomic fields using a theorem of l tllashington 20 on the non p part of the class number in cyclotomic zp extensions we describe a method of computing subgroups of cr which should be useful for the descent theory the corresponding congruence formulas for universal special values of l functions are an immediate consequence of the method since eisenstein series occur in many settings our point of view offers the prospect of generalization such a generalization to hilbert blumenthal varieties may be an received by the editors september 4 1984 1980 mathematics subject classification primary llf67 llg16 secondary llfll llf33 l partially supported by nsf grant mcs 82 01762 t1985 american mathematical society 0002 9947 8s 1 00 25 per page

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

kummer s criterion for the special values of hecke l functions of imaginary quadratic fields and congruences among cusp forms here arrow i refers to our previous papers 9 12 and 13 in which we have shown that a fixed primitive cusp form f of sk f i n has congruences with other cusp forms of sk f n modulo the special value at s k of a certain zeta function off in this paper we will construct the second arrow for the cusp form f associated with a hecke character of an imaginary quadratic field k and consequently obtain the third namely in w167 6 we will show as in the works of coates wiles 3 and robert 23 that the split primes of k which divide this special value are irregular in an appropriate sense in this case the zeta function o f f mentioned above is a hecke l function of k such a criterion of irregularity for hurwitz numbers has already been obtained in 3 and 233 however our result together with those of 3 and 23 does not cover all the l values of hecke l functions of k whose algebraicity was given by damerell 5 for details see below our method is analogous to that of ribet 21 see also wiles 35 who used congruences between cusp forms and eisenstein series to prove the non vanishing of certain eigenspaces relative to the action of gal q q of the p primary part of the class group of the field of p th roots of unity by adopting this approach we will obtain such an information of eigenspaces again in our case see theorem 0 1 below to be more specific let 2 be a hecke character of k satisfying

class numbers of quadratic fields and shimura s correspondence it is well known see fueter 8 and aigner 1 that fermat s equation x y 3 1 has no solutions in a quadratic field k q d for d 1 mod 3 provided the class number of k is not divisible by 3 this subject was further investigated by frey 6 who gave estimates for the 3 rank of the class group of k in terms of 3 descent on a curve y 2 x d in this paper we consider curves ed dy 2 4x 27 which are quadratic twists of the fermat s curve x y 3 1 we give a precise formula for the rank of the selmer group corresponding to the complex multiplication 3 ed e 3d in terms of the 3 rank of the class group of q d resp q 3d this result may be considered as a quantitative version of 1 and 8 we discuss also birch and swinnerton dyer s conjecture for curves ed according to a theorem of waldspurger 22 23 natural rational factor of l ed q 1 may be expressed in terms of coefficients of certain modular forms of weight 3 2 we identify these forms explicitely and verify mod 3 part of birch and swinnertondyer s conjecture for ed in most cases for d of density 5 6

non vanishing of quadratic twists of modular l functions k have been the subject of much study both because of their intrinsicinterest and because of the prominent role they have played inkolyvagin s work on the birch and swinnerton dyer conjecture see b f h i j ko ma m m m1 o s p p waldspurger proved a fundamental theorem the ore me 1 w1 relating these central critical values to the fourier coe cients of half integral weight cusp forms for notational convenience if d is afundamental discriminant of a quadratic number eld then de ne d

parametrizations of elliptic curves by shimura curves and by classical modular curves abstract fix an isogeny class of semistable elliptic curves over q the elements of have a common conductor n which is a square free positive integer let d be a divisor of n which is the product of an even number of primes i e the discriminant of an indefinite quaternion algebra over q to d we associate a certain shimura curve x0d n d whose jacobian is isogenous to an abelian subvariety of j0 n there is a unique a for which one has a nonconstant map d x0d n d a whose pullback a pic0 x0d n d is injective the degree of d is an integer d which depends only on d and the fixed isogeny class we investigate the behavior of d as d varies

visible evidence for the birch and swinnerton dyer conjecture for modular abelian varieties of analytic rank zero this paper provides evidence for the birch and swinnerton dyer conjecture for analytic rank 0 abelian varieties a f that are optimal quotients of j 0 n attached to newforms we prove theorems about the ratio l a f 1 af develop tools for computing with af and gather data about certain arithmetic invariants of the nearly 20 000 abelian varieties a f of level 2333 over half of these a f have analytic rank 0 and for these we compute upper and lower bounds on the conjectural order of iii a f we find that there are at least 168 such af for which the birch and swinnerton dyer conjecture implies that iii a f is divisible by an odd prime and we prove for 37 of these that the odd part of the conjectural order of iii a f really divides iii a f by constructing nontrivial elements of iii a f using visibility theory we also give other evidence for the conjecture the appendix by cremona and mazur fills in some gaps in the theoretical discussion in their paper on visibility of shafarevich tate groups of elliptic curves

l series with nonzero central critical value suppose that f zn 1 an f qtn is a cusp form of weight 2k k e n we denote by l f s the l function of f for re s sufficiently large the value of l f s is given by l f s en 1 an and one can show that l f s has analytic continuation to the entire complex plane the value of l f s at s k will be of particular interest to us and we will refer to this value as the central critical value of l f s let xd denote the dirichlet character associated to the extension q d q that is xd n va where ad denotes the discriminant of q d q define the dth quadratic twist of f to be fxd en 1 an f xd n qn for any integer d the l function of fx is the dth quadratic twist of l f s that is l fxd s zn aan f xd n we will be interested in determining how often l fxd s has nonzero central critical value as d varies over all integers since xd 2 xd we will restrict our attention to the square free integers d we expect that as we let d vary over all of the square free integers a positive proportion of the l functions l fxd s will have nonzero central critical value indeed goldfeld 7 conjectures that for newforms f of weight 2 l fxd 1 78 0 for 1 of the square free integers given an elliptic curve e y2 x3 ax2 bx c a b c e z with conductor ne and an integer d we define the dth quadratic twist of e to be the curve ed y2 x3 adx2 bd2x cd3 let l ed s denote the l function associated to ed for square free d coprime to 2ne l ed s is simply the dth quadratic twist of l ei s if f e s2 n is a newform with integer coefecients we know via the theory of eichler and shimura that there is an elliptic curve e over q having conductor n so that l e s l f s thus if d is coprime to 2n then l ed s l fxd s also one knows from the work of kolyvagin 13 as supplemented by the work of murty and murty 17 or that of bump friedberg and hoffstein 3 see also 10 for a shorter proof that if e is a modular elliptic curve and if l e 1 7 0 then the rank of e is 0 thus if f has the property that a positive proportion of the twists of l f s have nonzero central critical value then this implies that a positive density of the quadratic twists ed have rank 0 there have been many papers which have proved results in this direction for example in 2 3 6 10 16 17 19 28 one can find general theorems on

arithmetic properties of the shimura shintani waldspurger correspondence we prove that the theta correspondence for the dual pair widetilde textup sl 2 textit pb times for b an indefinite quaternion algebra over acting on modular forms of odd square free level preserves rationality and p integrality in both directions as a consequence we deduce the rationality of certain period ratios of modular forms and even p integrality of these ratios under the assumption that p does not divide a certain l value the rationality is applied to give a direct construction of isogenies between new quotients of jacobians of shimura curves completely independent of faltings isogeny theorem

canonical periods and congruence formulae the purpose of this article is to show how congruences between the fourier coefficients of hecke eigenforms give rise to corresponding congruences between the algebraic parts of the critical values of the associated l functions this study was initiated by b mazur in his fundamental work on the eisenstein ideal see maz77 and maz79 where it was made clear that congruences for analytic l values were closely related to the integral structure of certain hecke rings and cohomology groups the results of maz79 also showed that congruences were useful in the study of nonvanishing of l functions this idea was then further developed by stevens ste82 and rubin wiles rw82 the work of rubin and wiles in particular used congruences to study the behavior of elliptic curves in towers of cyclotomic fields a key ingredient here was a theorem of washington which states roughly that almost l values in certain families are nonzero modulo p this theme has recently been taken up again in the work of ono skinner osa osb james jam and kohnen koh97 while the earlier history was primarily concerned with cyclotomic twists the current emphasis is on families of twists by quadratic characters here one wants quantitative estimates for the number of quadratic twists of a given modular form which have nonvanishing l function at s 1 we continue this trend in the present work by using our general results to obtain a strong nonvanishing theorem for the quadratic twists of modular elliptic curves with rational points of order three this generalizes a beautiful example due to kevin james and provides new evidence for a conjecture of goldfeld gol79 it should however be pointed out that even the study of quadratic twists may be traced back to mazur the reader is urged to look at pages 212 213 of maz79 and especially at the footnote at the bottom of page 213 the theorems of davenport heilbronn dh71 and washington was78 which are crucial in this paper are both mentioned in mazur s article we want to begin by discussing the congruences that lie at the heart of this article thus let f anq n be an elliptic modular cuspform of level m and weight k 2 assume that f is a simultaneous eigenform for all the hecke operators and that a1 f 1 the l function associated to f is defined by the dirichlet series l s f ann s which converges for the real part of s sufficiently large and has analytic continuation to s c a fundamental theorem of shimura shi76 states that l s f enjoys the following algebraicity property

on the bloch kato conjecture for elliptic modular forms of square free level let kappa ge 6 be an even integer m an odd square free integer and f in s 2 kappa 2 gamma 0 m a newform we prove that under some reasonable assumptions that half of the lambda part of the bloch kato conjecture for the near central critical value l kappa f is true we do this by bounding the ell valuation of the order of the appropriate bloch kato selmer group below by the ell valuation of algebraic part of l kappa f we prove this by constructing a congruence between the saito kurokawa lift of f and a cuspidal siegel modular form

explicit expressions for energy release rates using virtual crack extensions a new finite element technique for calculating energy release rates is presented an explicit expression for energy changes due to virtual crack extensions is formulated based on a variation of isoparametric element mappings energy release rates are calculated directly from integral expressions evaluated over singular quarter point isoparametric elements surrounding the crack tip since the energy release rates are expressed in variational form there is no need for the analyst to select a small finite crack extension to simulate a virtual crack extension the method is shown to produce very accurate solutions even with fairly coarse element meshes a similar technique for mixed mode fracture based on mutual potential energy release rates is described

on modular representations of bar q q arising from modular forms where g is the galois group gai i q and f is a finite field of characteristic i 3 suppose that p is modular of level n i e that it arises from a weight 2 newform of level dividing n and trivial nebentypus character then p is an odd representation the matrix p c where c is a complex conjugation in g has eigenvalues 1 1 since 1 and 1 are distinct in f p is absolutely irreducible and has a model over every subfield of f containing the set trace p we assume that f has been chosen so that it is generated by this set assume that p is a prime which exactly divides n we write p i n and restrict p to a decomposition group of g for the prime p view the restriction as a representation p p of gal t p qp we say that p is finite at p 34 or 35 w if there is a finite flat f vector space scheme h over zp for which the action of gal t p qp on the f vector space h 1 p gives pp i f p 9 is finite at p if and only if pp is unramified it is clear that p is finite at p whenever p is already modular of level n p in w of 34 cf 35 serre conjectured the converse i e that if p is finite then p is modular of level n p soon thereafter mazur 1 20 proved this conjecture in the case

congruence formula for certain dihedral twists in this article we prove a congruence formula for the special values of certain dihedral twists of two primitive modular forms of weight two with isomorphic residual galois representation at a prime p

special values of adjoint l functions and congruences for automorphic forms on gl n over a number field we prove an integrality result for the value at s 1 of the adjoint l function associated to a cohomological cuspidal automorphic representation on rm gl n over any number field we then show that primes outside an exceptional set dividing this special value give rise to congruences between automorphic forms we also prove a non vanishing property at infinity for the relevant rankin selberg l functions on rm gl n times rm gl n

calculation of fracture mechanics parameters for an arbitrary three dimensional crack by the equivalent domain integral method summary in this paper an equivalent domain integral edi method and the attendant numerical algorithms arc presented for the computation of a near crack tip field parameter the vector je integral and its variation along the front of an arbitrary three dimensional crack in a structural component account is taken of possible non elastic strains present in the structure in this case the near tip je values may be significantly different from the far field values jf especially under non proportional loading

supersingular elliptic curves theta series and weight two modular forms this paper deals with two subjects and their interaction the first is the problem of spanning spaces of modular forms by theta series the second is the commutative algebraic properties of hecke modules arising in the arithmetic theory of modular forms let p be a prime and let b denote the quaternion algebra over q that is ramified at p and and at no other places if l is a left ideal in a maximal order of b then l is a rank four z module equipped in a natural way with a positive definite quadratic form 6 1 we shall say that l is a rank four quadratic space and remark that the isomorphism class of l as a quadratic space depends only on the left ideal class of l in its maximal order eichler 5 proved that the theta series of l is a weight two modular form on 0 p and that as l ranges over a collection of left ideal class representatives of all left ideals in all maximal orders of b these theta series span the vector space of weight two modular forms on 0 p over q in this paper we strengthen this result as follows if l is as above then the q expansion of its theta series l has constant term equal to one and all other coefficients equal to even integers suppose that f is a modular form whose qexpansion coefficients are even integers except perhaps for its constant term which we require merely to be an integer it follows from eichler s theorem that f may be written as a linear combination of l with l ranging over a collection of left ideals of maximal orders of b with rational coefficients we show that in fact these coefficients can be taken to be integers let t denote the z algebra of hecke operators acting on the space of weight two modular forms on 0 p the proof that we give of our result hinges on analyzing the structure of a certain t module x we can say what x is it is the free zmodule of divisors supported on the set of singular points of the reducible nodal curve x0 p in characteristic p the key properties of x which imply the above result on theta series are that the natural map t endt x is an isomorphism and that furthermore x is locally free of rank one in a zariski neighbourhood of the eisenstein ideal of t we remark that it is comparatively easy to prove the analogous statements after tensoring with q for they then follow from the fact that x is a faithful t module indeed combining this with the semi simplicity of the q algebra t z q one deduces that x z q is a free t z q module of rank one and in particular that the map t z q endt zq x z q is an isomorphism

liouville theorems for harmonic maps we prove several liouville theorems for harmonic maps between certain classes of riemannian manifolds in particular the results can be applied to harmonic maps from the euclidean space r m g 0 to a large class of riemannian manifolds our assumptions on the harmonic maps concern the asymptotic behavior of the maps at