Tin Flux, Crystal Structure, Magnetic Properties, Structural Relationships U Cr6P4 was prepared from a tin flux in two forms at low (a) and high (ß) temperatures (880 °C and 1000 °C), respectively. The crystal structures o f both m odifications were deter mined from single-crystal data. a-U C r6P4: P6m 2 (N o. 187), a = 698.5(3) pm, c = 350.8(1) pm, Z = 1, R = 0.052 for 18 variable parameters and 410 structure factors; /?-UCr6P4: Pmmn (N o. 59), a = 698.6(1) pm, b = 350.85(4) pm, c = 1196.1(2) pm, Z = 2, R = 0.047 for 21 variables and 656 structure factors. Although the lattice constants o f both m odifications are closely re lated, the two forms can be transformed into each other only by a very sluggish, reconstructive phase transformation. Nevertheless, both structures have very similar coordination polyhedra. The U atom s have 6 P neighbours in trigonal prismatic arrangement. H a lf o f the Cr atoms have tetrahedral, the other half square pyramidal P coordinations. As is typical for phosphides with high metal content, all metal atoms additionally have many metal neighbours. The P at oms are located in trigonal prisms o f metal atom s with two or three additional metal atoms outside the rectangular faces o f the prisms. Both m odifications o f UCr6P4 show relatively high, almost temperature independent paramagnetism, as is frequently observerd for intermetallic phases o f uranium.

We compute the arithmetic L-invariants (of Greenberg-Benois) of twists of symmetric powers of p-adic Galois representations attached to Iwahori level Hilbert modular forms (under some technical conditions). Our method uses the automorphy of symmetric powers and the study of analytic Galois representations on p-adic families of automorphic forms over symplectic and unitary groups. Combining these families with some explicit plethysm in the representation theory of GL(2), we construct global Galois cohomology classes with coefficients in the symmetric powers and provide formulae for the L-invariants in terms of logarithmic derivatives of Hecke eigenvalues.

We determine the shapes of pure cubic fields and show that they fall into two families based on whether the field is wildly or tamely ramified (of Type I or Type II in the sense of Dedekind). We show that the shapes of Type I fields are rectangular and that they are equidistributed, in a regularized sense, when ordered by discriminant, in the one-dimensional space of all rectangular lattices. We do the same for Type II fields, which are however no longer rectangular. We obtain as a corollary of the determination of these shapes that the shape of a pure cubic field is a complete invariant determining the field within the family of all cubic fields. 1 2 Robert Harron which we denote L n−1 and call the space of rank (n − 1) lattices, carries a natural GL n−1 (R)-invariant measure.Little is known about the shapes of number fields. Their study was first taken up in the PhD thesis of David Terr [Ter97], a student of Hendrik Lenstra's. In it, Terr shows that the shapes of both real and complex cubic fields are equidistributed in L 2 when the fields are ordered by the absolute value of their discriminants. This result has been generalized to S 4 -quartic fields and to quintic fields by Manjul Bhargava and Piper Harron in [BH13,Har16] (where the cubic case is treated, as well). Aside from this, Bhargava and Ari Shnidman [BS14] have studied which real cubic fields have a given shape, and in particular which shapes are possible for fields of given quadratic resolvent. Finally, Guillermo Mantilla-Soler and Marina Monsurrò [MSM16] have determined the shapes of cyclic Galois extensions of Q of prime degree. Note however that [MSM16] uses a slightly different notion of shape: they instead restrict the quadratic form in (1) to the space of elements of trace zero in O K . It is possible to carry out their work with our definition and our work with theirs; the answers are slightly different as described below. 1 This article grew out of the author's desire to explore an observation he made that, given a fixed quadratic resolvent in which 3 ramifies, the shapes of real S 3 -cubic fields sort themselves into two sets depending on whether 3 is tamely or wildly ramified; a tame versus wild dichotomy, if you will. Considering complex cubics, the pure cubics-that is, those of the form Q(m 1/3 )-are in many ways the simplest. These were partitioned into two sets by Dedekind, who, in the typical flourish of those days, called them Type I and Type II. In our context, pure cubics are exactly those whose quadratic resolvent is Q(ω), ω being a primitive cube root of unity, and Type I (resp. Type II) corresponds to 3 being wildly (resp. tamely) ramified. The first theorem we prove (Theorem A) computes the shape of a given pure cubic field and shows that, just as in the real case, pure cubic shapes exhibit a tame versus wild dichotomy. 2 In the real cubic case, [BS14] shows that for fields of a fixed quadratic resolvent, there are only finitely many options for the shape. However, there are infinitely many possibilities for the shape of a pur...

We prove the exceptional zero conjecture for the symmetric powers of CM cuspidal eigenforms at ordinary primes. In other words, we determine the trivial zeroes of the associated p-adic Lfunctions, compute the L-invariants, and show that they agree with Greenberg's L-invariants.In an appendix, we prove a functional equation for some of the p-adic L-functions we construct.

In [PS11], efficient algorithms are given to compute with overconvergent modular symbols. These algorithms then allow for the fast computation of p-adic L-functions and have further been applied to compute rational points on elliptic curves (e.g. [DP06, Tri06]). In this paper, we generalize these algorithms to the case of families of overconvergent modular symbols. As a consequence, we can compute p-adic families of Hecke-eigenvalues, two-variable p-adic L-functions, L-invariants, as well as the shape and structure of ordinary Hida-Hecke algebras.