Showing papers on "Ring of integers published in 2006"

CM-values of Hilbert modular functions

Abstract: This is a joint work with Jan Bruinier, and is a generalization of the well-known work of Gross and Zagier on singular moduli [GZ]. Here is the main result. For detail, please see [BY]. Let p ≡ 1 (mod 4) be a prime number and F = Q(√p). We write OF for the ring of integers of F , and x 7→ x′ for the conjugation in F . Let Γ = SL2(OF ) be the Hilbert modular group associated to F . The corresponding Hilbert modular surface X = Γ\H2 is a normal quasi-projective algebraic variety defined over Q. Let K = F ( √ ∆) be a non-biquadratic quartic CM number field (containing F ) with discriminant dK = p q for some prime q ≡ 1 mod 4 (technical condition). Let σ and σ′ be the complex embeddings of K given by σ( √ ∆) = σ( √ ∆′), and σ( √ ∆) = −√∆′. Then Φ = {1, σ} and Φ′ = {1, σ′} are two CM types. Let CM(K, Φ) be the (formal) sum of CM points in X of CM type (K, Φ) by OK . Then CM(K) = CM(K, Φ)+CM(K, Φ′) is an 0-cycle on X defined over Q. If Ψ is a rational modular function on X, then Ψ(CM(K)) is a rational number. An interesting and in general very hard question is to find a factorization formula for this number. We did it successfully when Ψ is a Borcherds product or equivalently has its divisor supported on the Hirzebruch-Zagier divisors, which were constructed in their seminar work in 1970’s [HZ]. Let K be the reflex field of (K, Φ) with real quadratic subfield F . For a nonzero element t ∈ d−1 K/F (relative discriminant) and a prime ideal l of F , we define

Lattices that Admit Logarithmic Worst-Case to Average-Case Connection Factors.

Abstract: We exhibit an average-case problem that is as hard as finding γ(n)-approximate shortest nonzero vectors in certain n-dimensional lattices in the worst case, for γ(n) = O(√log n). The previously best known factor for any non-trivial class of lattices was γ(n) = O(n).Our results apply to families of lattices having special algebraic structure. Specifically, we consider lattices that correspond to ideals in the ring of integers of an algebraic number field. The worst-case problem we rely on is to find approximate shortest vectors in these lattices, under an appropriate form of preprocessing of the number field.For the connection factors γ(n) we achieve, the corresponding decision problems on ideal lattices are not known to be NP-hard; in fact, they are in P. However, the search approximation problems still appear to be very hard. Indeed, ideal lattices are well-studied objects in computational number theory, and the best known algorithms for them seem to perform no better than the best known algorithms for general lattices.To obtain the best possible connection factor, we instantiate our constructions with infinite families of number fields having constant root discriminant. Such families are known to exist and are computable, though no efficient construction is yet known. Our work motivates the search for such constructions. Even constructions of number fields having root discriminant up to O(n2/3-e) would yield connection factors better than O(n).As an additional contribution, we give reductions between various worst-case problems on ideal lattices, showing for example that the shortest vector problem is no harder than the closest vector problem. These results are analogous to previously-known reductions for general lattices.

Non-unique factorizations: a survey

Abstract: It is well known that the ring of integers of an algebraic number field may fail to have unique factorization. In the development of algebraic number theory in the 19th century, this failure led to Dedekind's ideal theory and to Kronecker's divisor theory. Only in the late 20th century, starting with L. Carlitz' result concerning class number 2, W. Narkiewicz began a systematic combinatorial and analytic investigation of phenomena of non-unique factorizations in rings of integers of algebraic number fields (see Chapter 9 of [24] for a survey of the early history of the subject). In the sequel several authors started to investigate factorization properties of more general integral domains in the spirit of R. Gilmer's book [18] (see for example the series of papers [2], [3], [4] and the survey article [19] by R. Gilmer). It soon turned out that the investigation of factorization problems can successfully be carried out in the setting of commutative cancellative monoids, and this point of view opened the door to further applications of the theory. Among them the most prominent ones are the arithmetic of congruence monoids, the theory of zero-sum sequences over abelian groups and the investigation of Krull monoids describing the deviation from the Krull-Remak-Azumaya-Schmidt Theorem in certain categories of modules.

Distributions of discriminants of cubic algebras

Abstract: Let $k$ be a number field and $O$ the ring of integers. In the previous paper [T06] we study the Dirichlet series counting discriminants of cubic algebras of $O$ and derive some density theorems on distributions of the discriminants by using the theory of zeta functions of prehomogeneous vector spaces. In this paper we consider these objects under imposing finite number of splitting conditions at non-archimedean places. Especially the explicit formulae of residues at $s=1$ and 5/6 under the conditions are given.

Multiple planar coincidences with N-fold symmetry

Abstract: Planar coincidence site lattices and modules with N-fold symmetry are well understood in a formulation based on cyclotomic fields, in particular for the class number one case, where they appear as certain principal ideals in the corresponding ring of integers. We extend this approach to multiple coincidences, which apply to triple or multiple junctions. In particular, we give explicit results for spectral, combinatorial and asymptotic properties in terms of Dirichlet series generating functions.

Additive number theory and the ring of quantum integers

Abstract: Let m and n be positive integers. For the quantum integer [n]q = 1+q+q2+⋯+ qn−−1 there is a natural polynomial addition such that [m]q ⊕q [n]q = [m+n]q and a natural polynomial multiplication such that [m]q⊗q [n]q = [mn]q. These definitions are motivated by elementary decompositions of intervals of integers in combinatorics and additive number theory. This leads to the construction of the ring of quantum integers and the field of quantum rational numbers.

Some examples of Hecke algebras for two-dimensional local fields

Abstract: Let $\mathbf{K}$ be a local non-archimedian field, $\mathbf{F} = \mathbf{K}((t))$ and let $\mathbf{G}$ be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical and Iwahori Hecke algebras for representations of the group $\mathbb{G} = G(\mathbf{F})$ and its central extension $\hat{\mathbb{G}}$. For instance our spherical Hecke algebra corresponds to the subgroup $G(\mathcal{A}) \subset G(\mathbf{F})$ where $\mathcal{A} \subset \mathbf{F}$ is the subring $\mathcal{O}_{\mathbf{K}}((t))$ where $\mathcal{O}_{\mathbf{K}} \subset \mathbf{K}$ is the ring of integers. It turns out that for generic level (cf. [4]) the spherical Hecke algebra is trivial; however, on the critical level it is quite large. On the other hand we expect that the size of the corresponding Iwahori-Hecke algebra does not depend on a choice of a level (details will be considered in another publication).

Squares and Cubes Modulo n

Abstract: We study the asymptotics of the average number of squares (or quadratic residues) in Zn and Z ∗ Similar analyses are performed for cubes, square roots of 0 and 1, and cube roots of 0 and 1 Let Zn denote the ring of integers modulo n, and let Z ∗ denote the group (under multiplication) of integers relatively prime to n The number of elements in Z ∗ is ϕ(n), where ϕ is Euler's totient function What is the average number of elements in Z ∗ , given an arbitrary n? One way to answer this question is to apply the Selberg- Delange method (1, 2, 3) to the Dirichlet series ∞ X n=1 ϕ(n) n s+1 = Y p 1 + ∞ X r=1 ϕ(p r ) p r(s+1) !

The abc–conjecture for Algebraic Numbers

Abstract: The abc–conjecture for the ring of integers states that, for every e > 0 and every triple of relatively prime nonzero integers (a, b, c) satisfying a + b = c, we have max(|a|, |b|, |c|) ≤ rad(abc)1 + e with a finite number of exceptions. Here the radical rad(m) is the product of all distinct prime factors of m. In the present paper we propose an abc–conjecture for the field of all algebraic numbers. It is based on the definition of the radical (in Section 1) and of the height (in Section 2) of an algebraic number. From this abc–conjecture we deduce some versions of Fermat's last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat's equation in fields of small degrees over ℚ.

On the restriction of representations of $\GL_2(F)$ to a Borel subgroup

Abstract: Let $F$ be a non-Archimedean local field and let $p$ be the residual characteristic of $F$. Let $G=GL_2(F)$ and let $P$ be a Borel subgroup of $G$. In this paper we study the restriction of irreducible representations of $G$ on $E$-vector spaces to $P$, where $E$ is an algebraically closed field of characteristic $p$. We show that in a certain sense $P$ controls the representation theory of $G$. We then extend our results to smooth $\oK[G]$- modules of finite length and unitary $K$-Banach space representations of $G$, where $\oK$ is the ring of integers of a complete discretely valued field $K$, with residue field $E$.

Global applications of relative (ϕ,) -modules I

Abstract: In this paper, given a smooth proper scheme X over a p-adic dvr and a p-power torsionlocal system L on it, we study a family of sheaves associated to the cohomology of local, relative (',)-modules of L and their cohomology. As applications we derive descriptions of thecohomology groups on the geometric generic fiber of X with values in L, as well as of their classical (',)-modules, in terms of cohomology of the above mentioned sheaves. Let p be a prime integer, K a finite extension of Qp and V its ring of integers. In (Fo), J.-M. Fontaine introduced the notion of (ϕ,)-modules designed to classify p-adic representations of the absolute Galois group GV of K in terms of semi-linear data. More precisely, if T is a p-adic representation of GV , i.e. T is a finitely generated Zp-module (respectively a Qp-vector space of finite dimension) with a continuous action of GV , one associates to it a (ϕ,)-module, denoted D V (T). This is a finitely generated module over a local ring of dimension two AV (respectively a finitely generated free module over BV := AV ⊗Zp Qp) endowed with a semi-linear Frobenius endomorphism ϕ and a commuting, continuous, semi-linear action of the group V := Gal(K(� p∞)/K) such that (DV (T), ϕ) isetale. This construction makes the group whose representations we wish to study simpler with the drawback of making the coefficients more complicated. It could be seen as a weak arithmetic analogue of the Riemann-Hilbert correspondence between representations of the fundamental group of a complex manifold and vector bundles with integrable connections. The main point of this construction is that one may recover T with its GV -action directly from DV (T) and, therefore, all the invariants which can be constructed from T can be described, more or less explicitly, in terms of DV (T). For example (∗) one can express in terms of DV (T) the Galois cohomology groups H i (K, T) = H i (GV , T) of T.

Endomorphisms of superelliptic jacobians

Abstract: Let K be a field of characteristic zero, n>4 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group is doubly transitive simple non-abelian group. Let p be an odd prime, Z[\zeta_p] the ring of integers in the p-th cyclotomic field, C_{f,p}:y^p=f(x) the corresponding superelliptic curve and J(C_{f,p}) its jacobian. Assuming that either n=p+1 or p does not divide n(n-1), we prove that the ring of all endomorphisms of J(C_{f,p}) coincides with Z[\zeta_p].

Phase transitions with spontaneous symmetry breaking on Hecke C*-algebras from number fields

Abstract: When one attempts to generalize the results of Bost and Connes [BC] to algebraic number fields, one has to face sooner or later the fact that in a number field there is no unique factorization in terms of primes. As is well known, this failure is twofold: the ring of integers has nontrivial units, and even if one considers integers modulo units, (equivalently the principal integral ideals), it turns out that factorization in terms of these can fail too, essentially because ‘irreducible’ does not mean ‘prime’. The first difficulty, with the units, already arises in the situation of [BC], Remark 33.b], but is easily dealt with by considering elements fixed by a symmetry corresponding to complex conjugation. In the existing generalizations the lack of unique factorization has been dealt with in various ways. It has been eliminated, through replacing the integers by a principal ring that generates the same field [HLe], it has been sidestepped, by basing the construction of the dynamical system on the additive integral adeles with multiplication by (a section of) the integral ideles [Coh], and it has been ignored, by considering an almost normal subgroup that makes no reference to multiplication [ALR]. These simplifications make the construction and analysis of interesting dynamical systems possible, but they come at a price. Indeed, the noncanonical choices introduced in [HLe] and [Coh] lead to phase transitions with groups of symmetries that are not obviously isomorphic to actual Galois groups of maximal abelian extensions, and have slightly perturbed zeta functions in the case of [HLe], while the units not included in the almost normal subgroup in [ALR], reappear as a (possibly infinite) group of symmetries under which KMS states have to be invariant, which causes severe difficulties in their computation.

Relative Galois module structure of rings of integers of absolutely Abelian number fields

Abstract: Let L/K be an extension of number fields where L/\Q is abelian. We define such an extension to be Leopoldt if the ring of integers O_L of L is free over the associated order A_L/K. Furthermore we define an abelian number field K to be Leopoldt if every finite extension L/K with L/Q abelian is Leopoldt in the sense above. Previous results of Leopoldt, Chan & Lim, Bley, and Byott & Lettl culminate in the proof that the n-th cyclotomic field Q^(n) is Leopoldt for every n. In this paper, we generalize this result by giving more examples of Leopoldt extensions and fields, along with explicit generators.

Lattices that Admit Logarithmic Worst-Case to Average-Case Connection Factors.

Abstract: We demonstrate an average-case problem which is as hard as finding γ(n)-approximate shortest vectors in certain n-dimensional lattices in the worst case, where γ(n) = O( √ log n). The previously best known factor for any class of lattices was γ(n) = O(n). To obtain our results, we focus on families of lattices having special algebraic structure. Specifically, we consider lattices that correspond to ideals in the ring of integers of an algebraic number field. The worst-case assumption we rely on is that in some `p length, it is hard to find approximate shortest vectors in these lattices, under an appropriate form of preprocessing of the number field. Our results build upon prior works by Micciancio (FOCS 2002), Peikert and Rosen (TCC 2006), and Lyubashevsky and Micciancio (ICALP 2006). For the connection factors γ(n) we achieve, the corresponding decisional promise problems on ideal lattices are not known to be NP-hard; in fact, they are in P. However, the search approximation problems still appear to be very hard. Indeed, ideal lattices are well-studied objects in computational number theory, and the best known algorithms for them seem to perform no better than the best known algorithms for general lattices. To obtain the best possible connection factor, we instantiate our constructions with infinite families of number fields having constant root discriminant. Such families are known to exist and are computable, though no efficient construction is yet known. Our work motivates the search for such constructions. Even constructions of number fields having root discriminant up to O(n2/3− ) would yield connection factors better than the current best of O(n). ∗SRI International, cpeikert@alum.mit.edu †Harvard CRCS, DEAS, alon@eecs.harvard.edu

On the (2, 3, 7)-Generation of Some Special Linear Groups

Abstract: In Lucchini et al. (2000), proved, in particular, that the groups SL n (q) and SL n (Z) are (2, 3, 7)-generated for each prime power q and each integer n ≥ 287, where Z is the ring of integers. Moreover, the method used in their article also applies for 93 smaller ranks less than 287, and our interest here is in finding other small ranks for which (2, 3, 7)-generation can be established. In this article, we find altogether 50 new ranks n for which the groups SL n (q) and SL n (Z) are (2, 3, 7)-generated. Communicated by P. Higgins.

On the trace map between absolutely abelian number fields of equal conductor

Abstract: After first determining criteria for wild ramification of L/K (which can only happen at primes above 2), the above result is obtained for n = 2 (e ≥ 3) by computing TL/K(OL) explicitly, and is then extended to the general case. This approach does not rely on Leopoldt’s Theorem, in contrast to the techniques used in [6]. The explicit nature of the calculations used to compute I(L/K) leads to the definition of an “adjusted trace map” TQ(n)/K with the property that TQ(n)/K(O) = OK (here Q denotes the n cyclotomic field and O(n) its ring of integers). Using this map, we restate Leopoldt’s Theorem and show that its proof can be reduced to the (easier) cyclotomic case.

Realizability of the adams-novikov spectral sequence for formal a-modules

Abstract: We show that the formal A-module Adams-Novikov spectral sequence of Ravenel does not naturally arise from a filtration on a map of spectra by examining the case A = Z[i]. We also prove that when A is the ring of integers in a nontrivial extension of Qp, the map (L, W) → (L A , W A ) of Hopf algebroids, classifying formal groups and formal A-modules respectively, does not arise from compatible maps of E ∞ -ring spectra (MU, MU^MU)→ (R, S).

The resolution of the universal ring for finite length modules of projective dimension two

Abstract: Hochster established the existence of a commutative noetherian ring $\Cal R$ and a universal resolution $\Bbb U$ of the form $0\to \Cal R^{e}\to \Cal R^{f}\to \Cal R^{g}\to 0$ such that for any commutative noetherian ring $S$ and any resolution $\Bbb V$ equal to $0\to S^{e}\to S^{f}\to S^{g}\to 0$, there exists a unique ring homomorphism $\Cal R\to S$ with $\Bbb V=\Bbb U\otimes_{\Cal R} S$. In the present paper we assume that $f=e+g$ and we find a resolution $\Bbb F$ of $\Cal R$ by free $\Cal P$-modules, where $\Cal P$ is a polynomial ring over the ring of integers. The resolution $\Bbb F$ is not minimal; but it is straightforward, coordinate free, and independent of characteristic. Furthermore, one can use $\Bbb F$ to calculate $\operatorname{Tor}^{\Cal P}_{\bullet}(\Cal R, \Bbb Z)$. If $e$ and $g$ both at least 5, then $\operatorname{Tor}^{\Cal P}_{\bullet}(\Cal R, \Bbb Z)$ is not a free abelian group; and therefore, the graded betti numbers in the minimal resolution of $\pmb K\otimes_{\Bbb Z} \Cal R$ by free $\pmb K\otimes_{\Bbb Z} \Cal P$-modules depend on the characteristic of the field $\pmb K$. We record the modules in the minimal $\pmb K\otimes_{\Bbb Z} \Cal P$ resolution of $\pmb K\otimes_{\Bbb Z} \Cal R$ in terms of the modules which appear when one resolves divisors over the determinantal ring defined by the $2\times 2$ minors of an $e\times g$ matrix.

The Growth of Selmer Ranks of an Abelian Variety with Complex Multiplication

Abstract: Let K be a CM field and O be its ring of integers. Let p be an odd prime integer and p be a prime in K lying above p. Let F be a Galois extension of K unramified over p. For an Abelian variety A defined over F with complex multiplication by O, we study the variation of the p-ranks of the Selmer groups in pro-p algebraic extensions. We first study the Zpextension case. When K is quadratic imaginary and E is an elliptic curve, we also study the p-ranks of the Selmer groups in an unramified p-class field tower.

Reduction in the case of imperfect residue fields

Abstract: Let K be a complete field with a discrete valuation v, ring of integers OK , and maximal ideal (πK) Let k := OK/(πK) be the residue field, assumed to be separably closed of characteristic p ≥ 0 LetX/K be a smooth proper geometrically irreducible curve of genus g ≥ 1 Let X /OK denote a regular model of X/K Let Xk = ∑v i=1 riCi be its special fiber, where Ci/k is an irreducible component of Xk of multiplicity ri Let e(Ci) denote the geometric multiplicity of Ci (see [BLR], 91/3) In particular, e(Ci) = 1 if and only if Ci/k is geometrically reduced Any reduced curve C/k is geometrically reduced when k is perfect Associate to X /OK the field extension kX/k generated by the following three types of subfields: by the fields H(C,OC), where C is any irreducible component of Xk; by the fields of rationality of all points P such that P is the intersection point of two components of X red k ; and by the fields of rationality of all points Q that belong to geometrically reduced components and such that Q is not smooth

On the cardinality of the set of solutions to congruence equation associated with quintic form

Abstract: The exponential sum associated with f is defined as where the sum is taken over a complete set of residues modulo q and let x = (x1, x2, ... , xn) be a vector in the space Zn with Z ring of integers and q be a positive integer, f a polynomial in x with coefficients in Z. The value of S(f; q) has been shown to depend on the estimate of the cardinality |jV|, the number of elements contained in the set where fx is the partial derivative of f with respect to x = (x1, x2, ..., xn). This paper will give an explicit estimate of |V| for polynomial f(x; y) in Zp[x; y] of degree five. Earlier authors have investigated similar polynomials of lower degrees. The polynomial that we consider in this paper is as follows: The approach is by using p-adic Newton Polyhedron technique associated with this polynomial.

Operator algebras and conjugacy problem for the pseudo-Anosov automorphisms of a surface

Abstract: The conjugacy problem for the pseudo-Anosov automorphisms of a compact surface is studied. To each pseudo-Anosov automorphism f, we assign an AF-algebra A(f) (an operator algebra). It is proved that the assignment is functorial, i.e. every f', conjugate to f, maps to an AF-algebra A(f'), which is stably isomorphic to A(f). The new invariants of the conjugacy of the pseudo-Anosov automorphisms are obtained from the known invariants of the stable isomorphisms of the AF-algebras. Namely, the main invariant is a triple (L, [I], K), where L is an order in the ring of integers in a real algebraic number field K and [I] an equivalence class of the ideals in L. The numerical invariants include the determinant D and the signature S, which we compute for the case of the Anosov automorphisms. A question concerning the p-adic invariants of the pseudo-Anosov automorphism is formulated.

Integral bases for an infinite family of cyclic quintic fields

Abstract: see [5, p. 5391. Schoof and Washington [6, p. 5481 have shown that f n ( x ) is irrreducible for all n E Z . Let 0 E @ be a root of f,(x) = 0. Set K = Q ( 0 ) so that [ K : Q ] = 5. It is known that K is a cyclic field [6, p. 5481. We denote the ring of integers of K by OK. The discriminant d ( K ) of K has been determined by Jeannin [4, p. 761, see also Spearman and Williams [7, p. 2151, namely d ( K ) = f ( K ) 4 , where the conductor f (K) of K is given by

2-universal -lattices over real quadratic fields

Abstract: Let Open image in new window be a real quadratic field with m a square-free positive rational integer, and Open image in new window be the ring of integers in F. An Open image in new window-lattice L on a totally positive definite quadratic space V over F is called r-universal if L represents all totally positive definite Open image in new window-lattices l with rank r over Open image in new window. We prove that there exists no 2-universal Open image in new window-lattice over F with rank less than 6, and there exists a 2-universal Open image in new window-lattice over F with rank 6 if and only if m=2, 5. Moreover there exists only one 2-universal Open image in new window-lattice with rank 6, up to isometry, over Open image in new window.

Bounded generation and lattices that cannot act on the line

Abstract: Let D be an irreducible lattice in a connected, semisimple Lie group G with finite center. Assume that the real rank of G is at least two, that G/D is not compact, and that G has more than one noncompact simple factor. We show that D has no orientation-preserving actions on the real line. (In algebraic terms, this means that D is not right orderable.) Under the additional assumption that no simple factor of G is isogenous to SL(2,R), applying a theorem of E.Ghys yields the conclusion that any orientation-preserving action of D on the circle must factor through a finite, abelian quotient of D. The proof relies on the fact, proved by D.Carter, G.Keller, and E.Paige, that SL(2,A) is boundedly generated by unipotents whenever A is a ring of integers with infinitely many units. The assumption that G has more than one noncompact simple factor can be eliminated if all noncocompact lattices in SL(3,R) and SL(3,C) are virtually boundedly generated by unipotents.

On pi-adic expansion of singular integers of the p-cyclotomic field

Abstract: Let p be an odd prime. Let F_p^* be the no-null part of the finite field of p elements. Let K = Q(zeta) be the p-cyclotomic field and let O_K be the ring of integers of K. Let pi be the prime ideal of K lying over p. An integer B \in O_K is said singular if B^{1/p} not \in K and if B O_K = b^p where b is an ideal of O_K. An integer B \in O_K is said semi-primary if B = beta mod pi^2 where the natural beta is coprime with p. Let sigma be a Q-isomorphism of the field K generating the Galois group Gal(K/Q). When p is irregular, there exists at least one subgroup Gamma of order p of the class group of K annihilated by a polynomial sigma - mu with mu \in F_p^*. We prove the existence, for each Gamma, of singular semi-primary integers B where B O_K= b^p with class Cl(b) \in Gamma and B^{sigma-mu} \in K^p and we describe their pi-adic expansion. This paper is at a strictly elementary level.

Integral Loop Rings of Loops Whose Derived Subloop Is Central

Abstract: It is proven that finite loops whose derived subloop is central are determined by Z, the ring of integers, that is, if L and M are two such loops and Z L ≅ Z M, then L ≅ M.

Singular Integers and Kummer-Stickelberger relation

Abstract: Let p be an odd prime. Let F_p^* be the no-null part of the finite field of p elements. Let K=\Q(zeta) be a p-cyclotomic field and O_K be its ring of integers. Let pi be the prime ideal of K lying over p. Let sigma : zeta --> zeta^v be the Q-isomorphism of K for a primitive root v mod p. The subgroup of exponent p of the class group of K can be seen as a direct sum oplus_{i=1}^r Gamma_i of groups of order p where the group Gamma_i is annihilated by a polynomial sigma-mu_i with mu_i \in F_p^*. Let Gamma be one of the Gamma_i. From Kummer, there exist prime non-principal prime ideals Q of inertial degree 1 with Cl(Q) \in Gamma. We show that there exists singular semi-primary integers A such that A O_K= Q^p with A^{sigma-mu} = alpha^p where alpha in K. Let E = \frac{alpha^p}{\bar{\alpha}^p}-1 and nu the positive integer defined by nu = v_pi(E) -(p-1), where v_pi(.) is the pi-adic valuation. The aims of this article are to describe the pi-adic expansions of the Gauss Sum g(Q) and of E and to derive an upper bound of nu from the Jacobi resolvents and Kummer-Stickelberger relation of the cyclotomic field K.