a parametric family of quintic thue equations for an integral parameter t z we investigate the family of thue equations f x y x 5 t 1 2 x 4 y 2t 3 4t 4 x 3 y 2 t 4 t 3 2t 2 4t 3 x 2 y 3 t 3 t 2 5t 3 xy 4 y 5 1 originating from emma lehmer s family of quintic fields and show that for t 3 28 10 15 the only solutions are the trivial ones with x 0 or y 0 our arguments contain some new ideas in comparison with the standard methods for thue families which gives this family a special interest

families or irreducible polynomials of gaussian periods and matrices of cyclotomic numbers given an odd prime p we show a way to construct large families of polynomials p q x e q x q c where c is a set of primes of the form q 1 mod p and p q x is the irreducible polynomial of the gaussian periods of degree p in q q examples of these families when p 7 are worked in detail we also show given an integer n 2 and a prime q 1 mod 2n how to represent by matrices the gaussian periods 0 n of degree n in q q and how to calculate in a simple way with the help of a computer irreducible polynomials for elements of q 0

note on a polynomial of emma lehmer recently emma lehmer constructed a parametric family of units in real quintic fields of prime conductor p t 5t i5t 25 25 as translates of gaussian periods later schoof and washington showed that these units were fundamental units in this note we observe that lehmer s family comes from the covering of modular curves x 25 y0 25 this gives a conceptual explanation for the existence of lehmer s units they are modular units which have been studied extensively by relating lehmer s construction with ours one finds expressions for certain gauss sums as values of modular units on y 25 1 lehmer s polynomial throughout the discussion we fix a choice of primitive th roots of

p metrics on gl n c un and their busemann compactifications abstract let gl n c u n be the group of n n complex valued invertible matrices and the subgroup of unitary matrices respectively in this paper we study finsler p metrics on the homogeneous space x n gl n c u n for p 1 which are induced by schatten p norms on the tangent bundle of x n and are invariant under the action of gl n c we show that for p 1 the busemann p compactification is the visual compactification for p 1 the busemann p compactification is not the visual compactification we give a complete description of busemann 1 compactification

class numbers of real cyclotomic fields of prime conductor the class numbers hl of the real cyclotomic fields q l l 1 are notoriously hard to compute indeed the number hl is not known for a single prime l 71 in this paper we present a table of the orders of certain subgroups of the class groups of the real cyclotomic fields q l l 1 for the primes l 10 000 it is quite likely that these subgroups are in fact equal to the class groups themselves but there is at present no hope of proving this rigorously in the last section of the paper we argue on the basis of the cohen lenstra heuristics that the probability that our table is actually a table of class numbers hl is at least 98

computation of class numbers of quadratic number fields we explain how one can dispense with the numerical computation of approximations to the transcendental integral functions involved when computing class numbers of quadratic number fields we therefore end up with a simpler and faster method for computing class numbers of quadratic number fields we also explain how to end up with a simpler and faster method for computing relative class numbers of imaginary abelian number fields

connection between gaussian periods and cyclic units this paper finds that all known parametric families of units in real quadratic cubic quartic and sextic fields with prime conductor are linear combinations of gaussian periods and exhibits these combinations this approach is used to find new units in the real quintic field for prime conductors p n4 5n3 15n2 25n 25

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we

a relaxation of the intrinsic biharmonic energy the tension field u of a map u from a domain omega subset mathbb r m into a manifold n is the negative l2 gradient of the dirichlet energy in this paper we study the intrinsic biharmonic energy functional t u int omega tau u 2 in order to overcome the lack of coercivity of t we extend it to a larger space we construct minimizers of the extended functional via the direct method and we study the relation between these minimizers and critical points of t our results are restricted to dimensions m 4

jacobi sums and new families of irreducible polynomials of gaussian periods let m 2 m an m th primitive root of 1 q 1 mod 2m a prime number s sq a primitive root modulo q and f fq q 1 m we study the jacobi sums j a b q 1 k 2 a inds k b inds 1 k m 0 a b m 1 where ind s k is the least nonnegative integer such that s ind s k k mod q we exhibit a set of properties that characterize these sums some congruences they satisfy and a maple program to calculate them then we use those results to show how one can construct families p q x q e p of irreducible polynomials of gaussian periods i f 1 j 0 si mj q of degree m where p is a suitable set of primes 1 mod 2m we exhibit examples of such families for several small values of m and give a maple program to construct more of them

a family of cyclic quartic fields arising from modular curves we study a family of cyclic quartic fields arising from the covering of modular curves x1 16 xo 16 an integral basis and a fundamental system of units are found it is shown that a root of the quartic polynomial we construct is a translate of a cyclotomic period by an integer of the quadratic subfield of the quartic field recently 0 lecacheux 9 10 and h darmon 4 showed how to use coverings of modular curves to obtain cyclic extensions of q in particular they were able to give a geometric construction of a family of cyclic quintic fields discovered by e lehmer 11 the covering x1 n x0 n for n 2 has degree q n 2 and group z nz x i 1 for the quintic case they took n 25 which gave a cyclic covering of degree 10 then took the subcovering of degree 5 an important ingredient in the construction was the fact that xo 25 has genus 0 this also occurs for n 1 10 12 13 16 18 these all give trivial or quadratic coverings except for n 7 9 13 16 18 the values n 7 9 18 yield cubic extensions and can be shown to yield the family of polynomials x3 ax2 a 3 x 1 namely the simplest cubic fields 17 however it should be remarked that every cyclic cubic extension of q comes from a polynomial of this form if a is allowed to be rational similarly we are guaranteed that the quadratic extensions obtained from the coverings mentioned above correspond to polynomials of the form x2 ax 1 with a rational the case n 13 is treated by lecacheux 9 it might be suspected that the sextic fields she obtains are the same as the simplest sextics constructed by m n gras 6 however these latter fields were found by taking the fixed field of an element of order 6 in pgl2 q aut q x therefore they come from a covering of curves of genus 0 but x1 13 has genus 2 alternatively these sextic fields must be different because the discriminants of the quadratic and cubic subfields are different in the present paper we study the case n 16 as above it might be hoped that this case would give a geometric construction of the quartic fields studied received march 14 1990 revised october 22 1990 1980 mathematics subject classification 1985 revision primary 1 i r 16 research supported in part by n s f 1991 american mathematical society 0025 5718 91 1 00 25 per page

the spin of prime ideals fixing a nontrivial automorphism of a number field k we associate to ideals in k an invariant with values in 0 1 which we call the spin and for which the associated l function does not possess euler products we are nevertheless able using the techniques of bilinear forms to handle spin value distribution over primes obtaining stronger results than the analogous ones which follow from the technology of l functions in its current state the initial application of our theorem is to the arithmetic statistics of selmer groups of elliptic curves

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

families of cyclic polynomials obtained from geometric generalization of gaussian period relations a general method of constructing families of cyclic polynomials over q with more than one parameter will be discussed which may be called a geometric generalization of the gaussian period relations using this we obtain explicit multi parametric families of cyclic polynomials over q of degree 3 e 7 we also give a simple family of cyclic polynomials with one parameter in each case by specializing our parameters

special units in real cyclic sextic fields we study the real cyclic sextic fields generated by a root w of x l 6 r2 108 x2 a 2 t e z 0 6 26 we show that when t1 108 is square free except for powers of 2 and 3 and t t 0 10 54 then w is a generator of the module of relative units the details of the proofs are given in 3

efficient computation of root numbers and class numbers of parametrized families of real abelian number fields let k m be a parametrized family of simplest real cyclic cubic quartic quintic or sextic number fields of known regulators e g the so called simplest cubic and quartic fields associated with the polynomials p m x x3 mx2 m 3 x 1 and p m x x 4 mx 3 6x 2 mx 1 we give explicit formulas for powers of the gaussian sums attached to the characters associated with these simplest number fields we deduce a method for computing the exact values of these gaussian sums these values are then used to efficiently compute class numbers of simplest fields finally such class number computations yield many examples of real cyclotomic fields q p of prime conductors p 3 and class numbers h p greater than or equal to p however in accordance with vandiver s conjecture we found no example of p for which p divides h p

stratification for the singular set of approximate harmonic maps the aim of this note is to extend the results in naber and valtorta ann math 2 185 131 227 https doi org 10 4007 annals 2017 185 1 3 2017 to the case of approximate harmonic maps more precisely we will proved that the singular strata mathcal s k u of an approximate harmonic map are k rectifiable and we will show effect bounds on the quantitative strata in the process we will simplify many of the arguments from naber and valtorta 2017 and in particular we produce a new main covering lemmas which vastly simplifies the older argument

nonstationary weak limit of a stationary harmonic map sequence let m and n be two compact riemannian manifolds let uk be a sequence of stationary harmonic maps from m to n with bounded energies we may assume that it converges weakly to a weakly harmonic map u in h1 2 m n as k in this paper we construct an example to show that the limit map u may not be stationary 2002 wiley periodicals inc

partial regularity for stationary harmonic maps into spheres in an interesting recent paper 12 f hi lein has shown that any weakly harmonic mapping from a two dimensional surface into a sphere is smooth i present here a kind of generalization to higher dimensions asserting in effect that a stationary harmonic mapping from an open subset of nn n 3 into a sphere is smooth except possibly for a closed singular set of n 2 dimensional hausdorff measure zero my proof crucially depends upon several of h l in s observations as streamlined by p l lions to state the result precisely let us suppose that m n 2 u is a smooth open subset of nn and s m 1 denotes the unit sphere in nm a function u in the sobolev space hi u rm u u 1 urn belongs to h i u s ml provided t u 1 a e in u

energy concentration for almost harmonic maps and the willmore functional let be an open domain in 3 or 4 and n a smooth compact riemannian manifold we consider the dirichlet energy e u for maps u n and its negative l2 gradient the tension field u we study sequences of maps ui n with open image in new window if the maps are sufficiently regular we find strong h1 subconvergence away from a generalized submanifold in if the limit map is regular too we can estimate a willmore type energy of this generalized submanifold

cyclotomy and delta units in this paper we examine cyclic cubic quartic and quintic number fields of prime conductor p containing units that bear a special relationship to the classical gaussian periods qj qj 1 c is a unit for periods q and c e z

a family of real 2ⁿ tic fields we study the family of polynomials pn x a 9 x i 2n a 3 x i 2n and determine when p x a a e z is irreducible the roots are all real and are permuted cyclically by a linear fractional transformation defined over the real subfield of the 2nth cyclotomic field the families of fields we obtain are natural extensions of those studied by m n gras and y y shen but in general the present fields are non galois for n 4 from the roots we obtain a set of independent units for the galois closure that generate an almost fundamental piece of the full group of units finally we discuss the two examples where our fields are galois namely a 2n and a 24 239

quintic polynomials and real cyclotomic fields with large class numbers we study a family of quintic polynomials discoverd by emma lehmer we show that the roots are fundamental units for the corresponding quintic fields these fields have large class numbers and several examples are calculated as a consequence we show that for the prime p 641491 the class number of the maximal real subfield of the pth cyclotomic field is divisible by the prime 1566401 in an appendix we give a characterization of the simplest quadratic cubic and quartic fields

revisiting the siegel upper half plane i in this paper we study the automorphisms of siegel upper half plane of complex dimension 3 we give the normal forms and classify the set of fixed points of such transformations

irregular primes to one million using fast algorithms for power series inversion based on the fast fourier transform and multisectioning of power series we have calculated all irregular primes up to one million including their indices of irregularity and associated irregular pairs using this data we verified that fermat s last theorem and vandiver s conjecture are true for these primes two primes with index of irregularity five were already known we find that there are nine other primes less than one million with index five and that the prime 527377 is the unique prime less than one million with index six a pair of integers p k is said to be an irregular pair if p is a prime k is an even integer satisfying 2 p are known see 8 our calculations show that vandiver s conjecture is true for all p 106 the table of irregular pairs could also be used to calculate iwasawa invariants for the corresponding primes we did not do this as part of our calculations see 9 or 4 for a discussion of this problem previous computations of irregular pairs have used algorithms that take 0 p2 arithmetic operations for each prime p the bernoulli numbers are defined in received by the editor april 30 1991 and in revised form september 10 1991 1991 mathematics subject classification primary 11b68 secondary 1 1d41 65t20 1 1y99 the first author was partially supported by national science foundation grant dms 9012989 d 1992 american mathematical society 0025 5718 92 1 00 25 per page

bubble tree convergence for harmonic maps let be a compact riemann surface any sequence fn m of harmonic maps with bounded energy has a bubble tree limit consisting of a harmonic map o m and a tree of bubbles fk s2 m we give a precise construction of this bubble tree and show that the limit preserves energy and homotopy class and that the images of the fn converge pointwise we then give explicit counterexamples showing that bubble tree convergence fails i for harmonic maps fn when the conformal structure of varies with n and ii when the conformal structure is fixed and n is a palais smale sequence for the harmonic map energy

a lower bound for the class number of certain cubic number fields let k be a cyclic number field with generating polynomial x3 a 3 32 a 3 x 1 2 2 and conductor m we will derive a lower bound for the class number of these fields and list all such fields with prime conductor m a2 27 4 or m 1 27b2 4 and small class number 1 introduction let h denote the class number of the cyclotomic field q m and h the class number of its maximal real subfield q cos 2t m it is a well known conjecture of vandiver that p h holds for all primes p e p this is a customary assumption for proving the second case of fermat s last theorem for more details see washington 16 since h grows slowly h 1 for p 1 the exact value of h is known without using grh masley suggested that p p perhaps h 1 using the quadratic subfield ankeny chowla and hasse 1 showed that h 1 if p belongs to certain quadratic sequences in n and s d lang 9 and takeuchi 15 found more such sequences similar results were

on the computation of class numbers of real abelian fields in this paper we give a procedure to search for prime divisors of class numbers of real abelian fields and present a table of odd primes 10000 not dividing the degree that divide the class numbers of fields of conductor 2000 cohen lenstra heuristics allow us to conjecture that no larger prime divisors should exist previous computations have been largely limited to prime power conductors

power integral bases in a parametric family of totally real cyclic quintics we consider the totally real cyclic quintic fields k n q v n generated by a root v n of the polynomial f n x x 5 n 2 x 4 2n 3 6n 2 10n 10 x 3 n 4 5n 3 11n 2 15n 5 x 2 n 3 4n 2 10n 10 x 1 assuming that m n 4 5n 3 15n 2 25n 25 is square free we compute explicitly an integral basis and a set of fundamental units of k n and prove that k n has a power integral basis only for n 1 2 for n 1 2 both values presenting the same field all generators of power integral bases are computed