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Seminar Yoshio Mimura, Mathematics, Kobe Pharmaceutical University

12 October 2011 H. Sasaki: Title: Representations of binary lattices over Q(\sqrt{5})
12 October 2011 H. Iwabuchi: Title: Universal non-classical forms over Q(\sqrt{5})
5 August 2011 K. Chinen: Title: On a distribution Property of the residual order of a mod p with a condition
5 August 2011 H. Sasaki: Title: Representations of binary lattices ocer Q(\sqrt{5})
22 June 2011 H. Sasaki: Title: On 2-Universal Lattices over Q(\sqr{5})
10 March 2011 H. Sasaki: Title: On Representations of binary lattices over Q(\sqr{5})
21 October 2010 H. Iwabuchi: Title: Universal non-classical integral forms over Q(\sqr{5})
15 September 2010 H. Sasaki: Title: On Class Number Three Problem (3)
6 August 2010 H. Sasaki: Title: On Class Number Three Problem (2)
30 June 2010 H. Iwabuchi: Title: Universal non-classical integral forms on Q(\sqr{5})
12 May 2010 H. Sasaki: Title: On Class Number Three Problem
5 March 2010 H. Sasaki: Title: Remarks on quarternary unimodular lattices over Q(\sqr{13})
27 January 2010 H. Iwabuchi
25 December 2009 H. Sasaki & K. Chinen
25 November 2009 H. Sasaki: Title: Remarks on 2-universal lattices over real quadratic fields(3)
Abstract: We discuss about whether there exist 2-universal O-lattices with rank 7 over several real quadratic fields F.
28 October 2009 H. Iwabuchi: Title: Non-classical integral universal forms over Q(\sqrt{5})
30 September 2009 H. Sasaki: Title: Remarks on 2-universal lattices over real quadratic fields(2)
Abstract: We discuss about whether there exist 2-universal O-lattices over several real quadratic fi elds F with rank 7.
7 August 2009 H. Iwabuchi: Title: Non-classical integral universal forms over Q(\sqrt{5})
15 July 2009 H. Sasaki: Title: On 2-universal lattices over Q(\sqrt{13})
17 June 2009 H. Iwabuchi: Title: Non-classical integral universal forms over Q(\sqrt{5})
22 April 2009 H. Sasaki: Title: A remark on an even 3-universal lattice over Q(\sqrt{5})
27 March 2009 H. Iwabuchi: Title: Universal non-classic integral quadratic forms over Q(\sqrt{5})
18 February 2009 H. Sasaki: Title: Even 2-universal lattices over real quadratic fields -3
26 December 2008 H. Iwabuchi & H. Sasaki
26 November 2008 H. Sasaki: Title: 3-universal even lattice over real quadratic fields
Abstract: In this talk we show that there exists a 3-universal even lattice over Q(\sqrt{5}) with rank 6 and we also consider whether there exist such lattices over another real quadratic fields.
22 Octoberr 2008 H. Iwabuchi
17 September 2008 H. Sasaki: Title: Remarks on 2-universal O-lattices over real quadratic fields(2)
Abstract: We continue the discussion on whether there exist 2-universal O-lattices over real quadratic fields F with rank 7 except F=Q(\sqrt{2}), Q(\sqrt{5}).
5 August 2008 K. Chinen & T. Ishikawa
16 July 2008 H. Sasaki: Title: Remarks on 2-universal O-lattices over real quadratic fields
Abstract: We discuss about whether there exist 2-universal O-lattices over real quadratic fields F w ith rank 7 except F=Q(\sqrt{2}), Q(\sqrt{5}).
18 June 2008 H. Iwabuchi
28 May 2008 H. Sasaki: Title: Representations of binary O-lattices by an O-lattice with rank 6 over Q(\sqrt{5}) (7)
Abstract: We continue the discussion of the representations of some totally positive definite binary quadratic O-lattices by an O-lattice with rank 6 which is one of the candidates for 2-universal lattices over Q(\sqrt{5}).
27 February 2008 H. Iwabuchi
26 December 2007 K. Chinen
21 November 2007 H. Sasaki
24 October 2007 H. Iwabuchi
21 September 2007 H. Sasaki: Title: 2-universal lattices over Q(\sqrt{2}) with rank greater than 6
Abstract: In this talk we show that there exists no 2-universal lattice over Q(\sqrt{2}) with rank 7 which contains no 2-universal sublattice with rank 6, and we investigate on 2-universal lattices (over Q(\sqrt{2}) with rank 8 which contains no 2-universal sublattice with rank 7.
20 June 2007 H. Sasaki: Title: A proper 2-universal O-lattice over Q(\sqrt{2})
Abstract: We call an n-universal O-lattice L {\it proper} if L contains no n-universal O-lattice as a proper O-submodule of L. In this talk we investigate 2-universality of an O-lattice over Q(\sqrt{2}) which is the candidate for a proper 2-universal lattice.
14 March 2007 H. Sasaki: Title: Representations of binary O-lattices by an O-lattice with rank 6 over Q(\sqrt{5}) (7)
Abstract: We investigate the representations of some totally positive definite binary quadratic O-lattices by an O-lattice with rank 6 which is one of the candidates for 2-universal lattices over Q(\sqrt{5}).
27 December 2006 K. Chinen: Title: On the Riemann hypothesis for invariant polynomials from the general Hamming codes
Abstract: We constructed invariant polynomials and their zeta functions from the general Hamming codes Ham(r,q) (r >= 3, q >= 2, see Seminar 12/5/2006). In this talk we prove the Riemann hypothesis for the cases r >= 3 and q >= 4.
27 December 2006 K. Wada
13 December 2006 H. Sasaki: Title: Representations of binary O-lattices by an O-lattice with rank 6 over Q(\sqrt{5}) (6)
Abstract: We continue the discussion of the representations of some totally positive definite binary quadratic O-lattices by an O-lattice with rank 6 which is one of the candidates for 2-universal lat tices over Q(\sqrt{5}).
17 November 2006 K. Chinen: Title: Distribution of the zeros of certain self-reciprocal polynomials
Abstract: We give a simple sufficient condition for all the zeros of a self-reciprocal polynomial to lie on the unit circle.
15 November 2006 H. Iwabuchi
13 October 2006 K. Wada
4 October 2006 H. Sasaki
3 August 2006 K. Chinen & H. Sasaki: Title: Self-reciprocal polynomials and orthogonal polynomials by K. Chinen
Abstract: We study a relation between self-reciprocal polynomials and (ordinary) polynomials. We construct a linear mapping from the linear space of polynomials of degree n to that of self-reciprocal polynomials of degree 2n. Using it, we discuss the distribution of the roots of self-reciprocal polynomials and orthogonal polynomials.; Title: A universal forms over Q(\sqrt{17}), II by H. Sasaki
19 July 2006 H. Sasaki: Title: Representations of binary O-lattices by an O-lattice with rank 6 over Q(\sqrt{5}) (5)
Abstract: We continue the discussion of the representations of some totally positive definite binary quadratic O-lattices by an O-lattice with rank 6 which is one of the candidates for 2-universal lattices over Q(\sqrt{5}).
30 June 2006 K. Wada
14 June 2006 H. Iwabuchi
12 May 2006 K. Chinen: Title: Zeta functions for non-self-dual linear codes (2)
Abstract: We deduce explicit expressions of the zeta functions for the invariant polynomials associated to the general Hamming codes. We also discuss non-self-dual Golay codes.
10 May 2006 H. Sasaki: Title: Representations of binary O-lattices by an O-lattice with rank 6 over Q(\sqrt{5}) (4)
Abstract: We continue the discussion of the representations of some totally positive definite binary quadratic O-lattices by an O-lattice which is one of the candidates for 2-universal lattices over Q(\sqrt{5}) with rank 6.
17 March 2006 K. Chinen & H. Sasaki: Title: Zeta functions for non-self-dual linear codes by K. Chinen
Abstract: We consider zeta functions for linear codes which are not self-dual. We propose a new functional equation for them. As an application, we show that there are abundant invariant polynomials which satisfy the Riemann hypothesis.; Title: Representations of binary O-lattices by an O-lattice with rank 6 over Q(\sqrt{5}) (3) by H. Sasaki
Abstract: We continue the discussion of the representations of some totally positive definite binary quadratic O-lattices by an O-lattice which is one of the candidates for 2-universal lattices over Q(\sqrt{5}) with rank 6.
8 February 2006 K. Wada
28 December 2005 K. Chinen: Title: Zeta functions for invariant polynomials without x^n (2)
Abstract:We show the Riemann hypothesis for a certain infinite sequence of invariant homogeneous polynomials which do not have the term x^n. Moreover, we show that there are polynomials of this type which have the same zeta polynomials as the extremal weight enumerators up to some simple factors.
18 November 2005 K. Chinen: Title: Zeta functions for invariant polynomials without x^n
Abstract: We extend the zeta functions for linear codes (after Duursma) to the case of invariant homogeneous polynomials which do not have the term x^n. They have various properties different from the zeta functions of the original Duursma theory, but there is an infinite sequence of polynomials which satisfies the Riemann hypothesis.
19 October 2005 H. Iwabuchi
28 September 2005 H. Sasaki: Title: Property of universal lattices over Q(\sqrt{5}) (2)
Abstract: We continue the discussion on the proof of universality of several quaternary lattices over Q(\sqrt{5}).
26 July 2005 K. Chinen, H. Iwabuchi: Title: An analogue of the Mallows-Sloane bound for the formal weight enumerators by K. Chinen
Abstract: The Mallows-Sloane bound is an upper bound for the minimum distances of certain sequences of self-dual codes. It gives the best possible bounds for the extremal self-dual codes. In this talk, we give a similar bound for the formal weight enumerators, which gives the best possible bound for the extremal formal weight enumerators.; Title: Quaternary universal lattices over Z[\sqrt{6}] by H. Iwabuchi
15 June 2005 H. Sasaki: Title: Property of universal lattices over Q(\sqrt{5})
Abstract: We discuss on the proof of the next conjecture: "Let L be a universal lattice over Q(\sqrt{5}). Then L contains some ternary or quaternary universal lattice over Q(\sqrt{5})."
10 June 2005 K. Chinen: Title: On the Mallows-Sloane bound for self-dual codes.
Abstract: In 2003, I Duursma found a new, purely algebraic proof of the Mallows-Sloane bound for self-dual codes. In this seminar, his proof for the type II codes is reported. We apply his method to the formal weight enumerators and some partial results are mentioned.
25 May 2005 H. Iwabuchi
13 May 2005 K. Wada
20 April 2005 H. Sasaki: Title: Relatively non-trivial quaternary universal lattices over Q(\sqrt{5})
Abstract: Let F be a real quadratic field and O the ring of integers in F. We call an n-ary O-lattice L is relatively non-trivial (n-ary) universal if L is universal and L does not contain any (n-1)-ary universal lattice as an O-submodule. We discuss on non-trivial quaternary universal lattices over Q(\sqrt{5}).
25 March 2005 H. Iwabuchi
4 March 2005 K. Chinen: Title: On zeta functions for the formal weight enumerators (2)
Abstract: We defined the zeta functions for "the formal weight enumerators" in the last seminar (22/12/2004). We look into them more carefully. It turned out that there seemed to be a similar structure to that of the zeta functions for ordinary weight enumerators.
9 January 2005 H. Sasaki: Title: Remarks on quaternary universal forms over Q(\sqrt{13}) (2)
Abstract: In this talk we give the detail of proof for universality of a quaternary universal form over Q(\sqrt{13}), and conclude all proofs for such forms.
22 December 2004 H. Sasaki & K. Chinen: Title: On zeta functions for the formal weight enumerators by K. Chinen
Abstract: Iwan Duursma defined the zeta functions for linear codes as generating functions of weight enumerators of linear codes. In this talk, we try to extend his result for so called "the formal weight enumerators".
Title: Remarks on quaternary universal forms over Q(\sqrt{13}) by H. Sasaki
Abstract: We have already proved that there are two quaternary universal forms over Q(\sqrt{13}), up to isometry. However, we investigate more simple proofs of universality for those lattices.
10 December 2004 H. Wada
24 November 2004 H. Iwabuchi
5 November 2004 K. Chinen: Title: Zeta functions for linear codes and a Riemann hypothesis analog
Abstract: I. Duursma defined the zeta function for the geometric Goppa code in 1999 and later he extended the definition to any linear codes. For self-dual codes, the zeta functions contain deep structures similar to those of algebraic curves and we can formulate an analogue of the Riemann hypothesis. This talk is a survey of Duursma's work, with a very short introduction to the coding theory.
8 October 2004 K. Wada
6 October 2004 H. Sasaki: Title: On quaternary universal forms over Q(\sqrt{17}) (2)
Abstract: In this talk we give the proof of universality for two quaternary unimodular lattices over Q(\sqrt{17}).
30 July 2004 H. Sasaki: Title: On quaternary universal forms over Q(\sqrt{17})
Abstract: In this talk we give candidates for quaternary universal forms over Q(\sqrt{17}) and consider universality for those forms. There are at most three candidates for such form.
9 June 2004 H. Sasaki: Title: Universal quaternary forms over Q(\sqrt{13}) (2)
Abstract: We continue the discussion on the proof of the universality of quaternary forms over Q(\sqrt{13}) and determine all such forms.
4 June 2004 K. Chinen: Title: Two isospectral non-isomorphic graphs
Abstract: Let G_1 and G_2 be finite graphs. We call G_1 and G_2 isospectral if their adjacency operators have the same spectrum. It is known that there are pairs of non-isomorphic finite graphs which are isospectral. This is a finite analog of the famous problem of M. Kac "Can you hear the shape of a drum ?" which asks the existence or nonexistence of two different bounded domains in R2 whose Dirichlet spectra are the same. In this talk, we review a result which gives a pair of non-isomorphic directed graphs with the same spectrum. They are constructed as the Schreier graphs using some group theory, the proof is based on so-called "pre-trace formula" for certain induced representations of the group.
12 May 2004 H. Iwabuchi
23 April & 7 May 2004 K. Wada
14 April 2004 H. Sasaki: Title: Universal quaternary forms over Q(\sqrt{13})
Abstract: We discuss on universal quaternary forms over Q(\sqrt{13}). There are at most two candidates for such forms, and we investigate universality for those forms.
15 March 2004 K. Chinen: Title: On normal numbers
Abstract: Take a real number x, a positive integer r (> 1) and we consider the expansion of x into a decimal fraction with the scale r. We call x simply normal in the scale r if each number b (0 <= b < r) appears in the expansion with the same frequency. This seminar is a survey talk of the classical theorem of E. Borel: "Almost all numbers are simply normal in any scale" ("almost all" = "except for a set of Lebesgue measure 0"). In addition, a new problem is proposed.
27 February 2004 H. Sasaki: Title: Some topics on universal forms over real quadratic fields
Abstract: We will introduce two works of B. M. Kim on universal forms over real quadratic fields:
(i) If D is large enough, there are no positive integral diagonal septanary universal quadratic forms over Q(\sqrt{D}).
(ii) There are infinitely many integers n such that n^{2} - 1 is square-free and Q(\sqrt{n^2 - 1}) admits universal octonary diagonal quadratic forms.
26 December 2003 K. Chinen: Title: On a distribution property of the residual order of a (mod p) (10). -- A joint work with L. Murata
Abstract: Take a positive integer a(>1), and let D_a(p) be the order of a in the group (Z/pZ)*. Moreover we denote by Q_a(x; k,l), the set of primes p< x such that D_a(p) is congruent to l mod k.
We consider the most general case, i.e. k and l are arbitrary integers. In this talk, we show that we can get under GRH, an algorithm for effective computation of the density of Q_a(k,l) and we can obtain the exact value of it for any k and l.
16 December 2003 H. Sasaki: Title: Integral spinor norm groups over dyadic local fields (by C. N. Beli)
Abstract: Review of a paper of C. N. Beli (JNT 2003). The spinor norms of integral rotations of an arbitrary quadratic lattice over an arbitrary dyadic local field are determined. The results are given in terms of BONGs, short for "bases of norm generators", which provides a new way to describe lattices over dyadic local fields.
04 November 2003 H. Iwabuchi
24 October 2003 K. Wada: Title: Zeros of 2-adic L-functions and congruences for class numbers and functional units ?
Abstract: The aim of this talk is to studied the imaginary quadratic fields such that the Iwasawa \lanbada_2-invariant equals 1, obtaining infomation on zeros of 2-adic L-functions.
7 October 2003 H. Sasaki: Title: Representations of binary O-lattices by an O-lattice with rank 6 over Q(\sqrt{5})
Abstract: We consider the representations of some totally positive definite binary quadratic O-lattices by an O-lattice which is one of the candidates for 2-universal lattices over Q(\sqrt{5}) with rank 6.
12 September 2003 H. Iwabuchi
30 July 2003 H. Sasaki & K. Chinen: Title: Totally positive definite non-classical ternary universal forms over real quadratic fields(4) by H. Sasaki
Abstract: We will talk about the non-existence of totally positive definite non-classical ternary universal forms over Q(\sqrt{m}) for some m \equiv 1 mod 4 in which the norm N(\epsilon) of the fundamental unit \epsilon is one.
Title: Ramanujan Graphs and Networks by K. Chinen
Abstract: Ramanujan graphs have various good properties from the viewpoint of the network theory. In this seminar, we show the estimation of several parameters of the graph by its eigenvalues, and see why the notion of the Ramanujan graph is important in applied mathematics.
15 July 2003 H. Iwabuchi
20 June 2003 K. Chinen: Title: Explicit Construction of Ramanujan Graphs
Abstract: Several ways are known so far to construct Ramanujan graphs. Among them, Winnie Li constructed an infinite family of Ramanujan graphs using finite fields (J. Number Theory, 1992). The aim of this talk is to review some basic facts about finite graphs and to introduce Li's result.
10 June 2003 H. Sasaki: Title: Totally positive definite non-classical ternary universal forms over real quadratic fields(3)
Abstract: We will talk about the non-existence of totally positive definite non-classical ternary universal forms over Q(\sqrt{m}) for some m \equiv 1 mod 4.
23 May 2003 K. Wada
13 May 2003 H. Iwabuchi
25 April 2003 K. Chinen: Title: On a distribution property of the residual order of a (mod p) (9). -- A joint work with L. Murata.
Abstract: Take a positive integer a(> 1), and let D_a(p) be the order of a in the group (Z/pZ)*. Moreover we denote by Q_a(x; k,l), the set of primes p< x such that D_a(p) is congruent to l mod k. Under GRH, we determine the natural density of Q_a(x; q^i, j) where i>0 and j is an arbitrary natural number. When q|(the square free part of a), we have to deal with the condition of Legendre symbol to find the density of Q_a(x; q, j). It follows that we complete the determination of the densities of Q_a(x; q^i, j) for an arbitrary prime power q^i (including q=2) and a non-negative integer j.
15 April 2003 H. Sasaki: Title: Totally positive definite non-classical ternary universal forms over real quadratic fields(2)
Abstract: Let F=Q(\sqrt{m}) be a real quadratic field with m a square free positive integer. We will discuss whether there exist totally positive definite non-classical ternary universal forms over Q(\sqrt{m}) if m > 40 (m \equiv 2,3 mod 4), m > 145 (m \equiv 1 mod 4).
28 March 2003 H. Iwabuchi
14 March 2003 K. Wada
21 February 2003 K. Chinen: Title: On a distribution property of the residual order of a (mod p) (8). -- A joint work with L. Murata.
Abstract: Take a positive integer a(>1), and let D_a(p) be the order of a in the group (Z/pZ)*. Moreover we denote by Q_a(x; k,l), the set of primes p<x such that D_a(p) is congruent to l mod k. In this seminar, we think the natural density of Q_a(x; 2^i, j) where i>1 and j is an arbitrary natural number. In most cases, calculation of the natural density of Q_a(x; 2^i, j) can be reduced to that of Q_a(x; 4, j), which is already done.
7 February 2003 H. Sasaki: Title: 2-universal O-lattices over real quadratic fields with rank 7
Abstract: We discuss about whether there exist 2-universal O-lattices over real quadratic fields F with rank 7 except F=Q(\sqrt{2}), Q(\sqrt{5}).
10 January 2003 H. Iwabuchi
27 December 2002 K. Wada
13 December 2002 K. Chinen: Title: On a distribution property of the residual order of a (mod p) (6). -- A joint work with L. Murata.
Abstract: Take a positive integer a greater than 1, and we consider the set Q_a(x; q, j) for an odd prime q (see the abstract of 20/4/2001 for the definition of Q_a(x; k,l)). Under GRH, we give an explicit formula for the natural densities d(q, j) of the above sets for nonzero j and for a with slight restrictions. Moreover, we take q=5 as an example. Then it turns out that d(5, j) is represented by a combination of several absolute constants containing imaginary numbers. This phenomenon seems a little surprising, but it can be explained in terms of the Dirichlet characters.
22 November 2002 H. Sasaki
1 November 2002 H. Iwabuchi
4 & 18 October 2002 K. Wada: Title: Computing class fields via Artin map by C. Fieker
Abstract: review
13 September 2002 K. Chinen: Title: On a distribution of the order of g (mod p) over residue classes (1). -- A joint work with P. Moree and L. Murata.
Abstract: This is the first talk of the above mentioned problem, which is a revised version of the talks with the title "On a distribution property of the residual order of a (mod p)" (cf. 20/4/2001, 14/9/2001, 2/11/2001 for example). In this talk, we discuss a new method of determining the natural density of N_g(j,4), the set of primes p s.t. the order of g (mod p) in (Z/pZ)* is congruent to j mod 4.
19 July 2002 H. Sasaki: Title: A 2-universal lattice over Q(\sqrt{5}) with rank 6 (2)
Abstract: In this talk we discuss the 2-universality for a quadratic lattice which is one of (three) candidates for 2-universal lattices over Q(\sqrt{5}) with rank 6.
28 June 2002 H. Iwabuchi: Title: Binary Quadratic Forms over Q(\sqrt(6))
Abstract:
7 June 2002 K. Wada: Title: On Iwasawa \lambda_3-Invariants of Relative Real Cyclic Extensions of Degree p
Abstract: A criterion of the vanishing of for relative cyclic extensions of totally real number fields with degree p using Iwasawa's result ofRiemann-Hurwit type which is an analogue to Kida's formula. (Review of T.Fukuda, K.Komatsu, M.Ozaki and H.Taya, Tokyo J.Math. vol.20, No.2, 1997)
24 May 2002 K. Chinen: Title: On generalizing Artin's conjecture on primitive roots to composite moduli
Abstract: This is a survey talk of a preprint by Shuguang Li and Carl Pomerance (2001?). Let a be an integer and p be a prime. Artin's conjecture asks whether a generates the multiplicative group (Z/pZ)^* for infinitely many p. In this preprint, the authors replace (Z/pZ)^* by (Z/nZ)^* where n is a positive integer. In this case, (Z/nZ)^* is not always cyclic, so they call a "a primitive root for n" when a generates the maximal cyclic subgroup of (Z/nZ)^* (this definition is due to Carmichael). Let N_a(x) be the number of n less than x such that (a,n)=1 and a is a primitive root for n. The authors found a very interesting phenomenon: the natural density of N_a(x) oscillates and does not exist.
10 May 2002 H. Sasaki: Title: Sums of squares of integral linear forms over Q(\sqrt(5)).
26 April 2002 H. Iwabuchi: Title: a representation of universal quaternary quadratic form over Q(\sqrt(6)).
Abstract: We think representational properties of the universal form.
5 April 2002 K. Wada: Title: Review : On p-adic zeta functions and Z_p-extensions of certain totally real number fields by H. Taya
22 March 2002 K. Chinen: Title: On a distribution property of the residual order of a (mod p) (6). -- A joint work with L. Murata.
Abstract: Take a positive integer a greater than 1, and we consider the set Q_a(x; q, j) for an odd prime q (see the abstract of 20/4/2001) for the definition of Q_a(x; k,l)). Some new (but partial) results about the natural densities of Q_a(x; q, j) for nonzero j's will be reported.
15 March 2002 H. Sasaki: Title: Non-classical 3-universal integral lattices over real quadratic fields
Abstract: Let F=Q(\sqrt{m}) be a real quadratic field with m a square free positive integer. In this talk we will show that there exists a (totally positive definite) non-classical 3-universal integral lattice with rank 6 over F=Q(\sqrt{5}). We also try to determine other m's such that there exist non-classical 3-universal lattices with rank 6 over F.
1 March 2002 H. Iwabuchi: Title: a certain boundedness of slight totally positive integer --- concerned with universal quaternary quadratic form.
Abstract: We say a totally positive integer a is farther totally positive than b, if a-b is totally positive. In this talk, we show integers which is not farther positive than a fixed positive rational integer are bounded except for units. Then we complete a proof of universality of the quaternary quadratic form in the previous talk.
15 February 2002 K. Wada: Title: Norm Residue Symbol and the First Case of Fermat's Equation
Abstract: We show how a solution of the first case of Fermat's Equation (of exponent an odd number p) in a number field F leads to the construction of element in Q(\zeta)* which is orthogonal to the real cyclotomic units fo the norm residue symbol. Bruno Angles,JNT 91,297-311(2001)
1 February 2002 H. Sasaki: Title: Representations by unit lattices over real quadratic fields
Abstract: Let F=Q(\sqrt{m}) be a quadratic field with m a square-free positive rational integer. O be the ring of integers in F. Let V be a totally positive definite quadratic space over F and L be a lattice on V, that is L be a finitely generated O-module in V. We define g_m(n) to be the smallest positive rational integer such that every totally positive definite lattice with rank n that can be represented by the unit lattice I_N is represented by the unit lattice I_{g_m(n)} with rank g_m(n). In this talk we show that g_m(2) is greater than 5 if m>5 and g_m(2)=5 if m=2, 5.
11 January 2002 K. Chinen: Title: Ramanujan graphs and finite upper half planes
Abstract: The notion of Ramanujan graphs was introduced by Lubotzky, Phillips and Sarnak in 1988, and the finite upper half planes are important examples of them. In this talk, we will review some basic graph theory, study some properties of Ramanujan graphs, and introduce the finite upper half planes.
21 December 2001 H. Iwabuchi: Title: Universal quaternary quadratic forms over some real quadratic fields.
Abstract: In this talk, we consider a proof of the universality of some forms over the integer rings of some real quadratic fields.
7 December 2001 K. Wada: Title: On Iwasawa \lambda_3-Invariants of Cyclic Cubic Fields of Prime Conductor
Abstract: For certain cyclic cubic fields k,it was verified by T.Fukuda and K.Komatu that Iwsawa invariants Lambda_3(k) vanished by calculating units of abelian number field of degree 27.
16 November 2001 H. Sasaki: Title: 2-universal quadratic lattices over Q(\sqrt{5}) with rank 6
Abstract: We have seen that there is one 2-universal quadratic lattice over Q(\sqrt{5}) with rank 6, and there are three candidates for such lattices besides. In this talk we discuss the 2-universality for those lattices.
2 November 2001 K. Chinen: Title: On a distribution property of the residual order of a (mod p) (5). -- A joint work with L. Murata.
Abstract: Take a positive integer a greater than 1, and we consider the set Q_a(x; q^i,j) for an odd prime q (see the abstract of 20/4/2001 for the definition of Q_a(x; k,l)). Under GRH, natural densities d(q^i,j) of the above sets are expressed by certain infinite sums. So far, it is hard to find the explicit values of them for all j, but we can determine d(q^i,j) when q|j. The main result of this talk is
"For any a and any j s.t. q|j, d(q^i,j)=1/q^{i-2}(q^2-1)."
26 October 2001 H. Iwabuchi: Title: Real quadratic fields with a universal quaternary form
Abstract: We give an upper bound for the discriminant of real quadratic fields which have a universal quaternary totally positive (classic) integral quadratic lattice.
12 October 2001 M. Kuroda: Title: Gauss sums on matrices I
Abstract: A generalization of Gauss sums over a finite field to ones over a matrix ring with coefficients in the field is given. Quadratic Gauss sums on matirices are evaluated. Numbers of solutions of a diagonal equation over a finite field can be expressed using character sums like Jacobi sums and Gauss sums, and formulas using Gauss sums are also generalized to those of diagonal equations over matrix rings.
28 September 2001 K. Wada: Title: Ankeny-Artin-Chowla Conjecture and Continued Fraction Expansion
Abstract: Let p be a prime congurent to 1 modulo 4, and let t,u be rational integers such that (t+u\sqrt{p})/2 is the fundamental unit of the real quadratic field Q(\sqrt{p}). The Ankeny-Artin-Chowla Conjecture asserts that p will not divide u. We investigate a certain relation between the conjecture and the continued fraction expansion of (t+u\sqrt{p})/2.
14 September 2001 K. Chinen: Title: On a distribution property of the residual order of a (mod p) (4). -- A joint work with L. Murata.
Abstract: Take a positive integer greater than 1, and we consider the set Q_a(x; 4,l) for l=1 and 3 (see the abstract of 20/4/2001 for the definition of Q_a(x; k,l)). Under GRH and a weak condition that a is not a h-th power for h >= 2, we can determine the natural densities of Q_a(x; 4,l) for all a. In this talk, this result will be reported.
24 July 2001 H. Sasaki: Title: Non-classical 2-universal integral lattices over real quadratic fields
Abstract: Let F=Q(\sqrt{m}) be a real quadratic field with m a square free positive integer. In this talk we will try to determine m's such that there exist totally positive definite non-classical 2-universal integral lattices with rank 4 over F.
13 July 2001 H. Iwabuchi: Title: Universal Quaternary Quadratic Forms over Z
Abstract: We investigate universality of some forms by the method that was used in the paper "Universal quaternary quadratic form with the maximal discriminant (to appear in Mathematika)".
29 June 2001 K. Wada
15 June 2001 K. Chinen: Title: Uniform distribution of primes having a prescribed primitive root
Abstract: Review of a paper by Pieter Moree. Let P_g be the set of primes p such that a fixed integer g is a primitive root mod p. Take an integer d. We consider classifying P_g mod d. This paper determines the condition of g and d for P_g to be uniformly distributed over the irreducible residue classes mod d.
1 June 2001 H. Sasaki: Title: Totally positive definite non-classical ternary universal forms over real quadratic fields
Abstract: Let F=Q(\sqrt{m}) be a real quadratic field with m a square free positive integer. In this talk we will try to determine m's such that there exist totally positive definite non-classical ternary universal forms over Q(\sqrt{m}).
18 May 2001 H. Iwabuchi: Title: Universal quaternary quadratic lattices over real quadratic integer rings ---the third report.
Abstract: To give an upper bound of the discriminants of real quadratic fields for which there exist universal forms, and an application of negative fraction expansions.
27 April 2001 K. Wada: Title: Families of irreducible polynomials of Gaussian Periods and Matrices of Cyclotomic Numbers
Abstract: For a given odd prime p, put C = { q : prime | q \equiv 1 \mod p }. We give a method which construct big families of irreducible polynomials P_q(x) \in Q[x] (q \in C) of Gaussian periods belonging to Q(\zeta_q) of degree p.
20 April 2001 K. Chinen: Title: On a distribution property of the residual order of a (mod p) (3). -- A joint work with L. Murata
Abstract: Take a positive square free integer a(> 2), and let D_a(p) be the order of a in the group (Z/pZ)*. Moreover we define the following prime set: Q_a(x; k,l)={p < x ; D_a(p) \equiv l (mod k)}. In the previous two talks of the same title (29/9/2000 and 17/11/2000), we considered the natural densities of Q_a(x; k,l)'s for (i) k: prime and l=0, (ii) k=4 and l=0,1,2,3. In this talk, we consider the case where k=2^i (i > 1), and the natural density of Q_a(x; 2^i,j) (j: odd) is determined for certain types of a's.
6 April 2001 H. Sasaki: Title: 2-universal O-lattices with rank 6 over real quadratic fields
Abstract: Let F=Q(\sqrt{m}) (m a square-free positive integer) be a real quadratic field and O be the ring of integers in F. Let V be a quadratic space over F and L be an O-lattice on V. We suppose that O-lattices are integral, i.e., the scale ideal of L is in O, in the following. A totally positive definite O-lattice L is called 2-universal if L represents all totally positive definite binary O-lattices. We show that there exist 2-universal (integral) O-lattices with rank 6 over real quadratic fields F=Q(\sqrt{m}) (m>0 square-free) if and only if F=Q(\sqrt{2}) and Q(\sqrt{5}).
30 March 2001 H. Iwabuchi: Title: Universal quaternary quadratic lattices over real quadratic integer rings --- the second report.
Abstract: In this talk, we show the strong method to decide whether universal integral quaternary quadratic lattices exist or not over the integer ring of a real quadratic field.
16 March 2001 K. Wada: Title: Quinitic Polinomials and Real Cyclotomic Fields with Large Class Numbers
Abstarct: Review of R. Schoof and L.C. Washington's paper. For a family of quinitic polinomials discovered by E. Lehmer, it is shown that their roots are units of the corresponding fields and the fields have large class numbers. Some examples are shown.
2 March 2001 K. Chinen: Title: On the density of primes dividing the Lucas numbers.
Abstract: Let L_n (n=1,2,..) be the Lucas sequence, and let S be the set of primes p such that p|L_n for some n. Then S has a natural density 2/3. This was proved by J. C. Lagarias in 1985. The proof was made under the inspiration of H. Hasse's result (1965) to answer the problem proposed by W. Sierpinski (1958).
In this talk, the proof by Lagarias, as well as some brief history of the problem is surveyed.
16 February 2001 H. Iwabuchi: Title: Universal quaternary quadratic lattices over real quadratic integer rings---the first report.
Abstract: We will (would?) decide conditions that an integral quaternary quadratic lattice over the integer ring of some real quadratic fields represent all totally positive integers.
16 February 2001 H. Sasaki: Title: 2-universal quadratic lattices of rank 6 over real quadratic integer rings
02 February 2001 K. Wada: Title: Connection Between Gaussian Periods and Cyclic Units
Abstract: Survey of works of E.Lehmer, R.Schoof & L.Washington and F.Thaine. Cf. F.Thaine: Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers (Math. Comp. vol.69, 2000, 1653-1666)
19 January 2001 K. Chinen: Title: On Artin's conjecture for primitive roots
Abstract: Let a be an integer other than 0, ±1, or a perfect square. In 1927, E. Artin conjectured that there exist infinitely many primes p for which a is a primitive root mod p. This is now known as "Artin's conjecture for primitive roots". It is still an open problem, but in 1967, C. Hooley proved it under the assumption of the Riemann hypothesis for the Dedekind zeta functions of certain types of the Kummer fields. In this talk, Hooley's result will be surveyed.

Last updated: 21 October, 2011