Diophantine Analysis and Related Fields 2020
Scheduled Talks and Their Abstracts
Thursday 5th March
13:40–14:30 Yasutsugu Fujita (Nihon University)
Title: Generators and integral points on quartic curves of the forms u2 ± v 4 = m
Abstract: In the preceding work, we investigated generators and integral points on twists ofFermat’s cubic of the form x3 + y 3 = m for a non-zero integer m. In this talk, using a similar
strategy we study generators and integral points on quartic curves of the forms u2 ± v 4 = mfor a non-zero integer m. The theorems assert that certain integral points on the curves canbe extended to bases for the Mordell-Weil groups of the elliptic curves birationally equivalent
to the quartic curves in the cases where the Mordell-Weil ranks are at most two. Moreover,the integral points on the quartic curves are explicitly described in the rank one or two case.
14:40–15:30 Alan Filipin (University of Zagreb)
Title: The extension of parametric Diophantine triples in the ring of Gaussian
integer
Abstract: A Diophantine m-tuple is a set of m distinct integers such that the product ofany two of its distinct elements increased by 1 is a perfect square. In this talk we consider
the extendibility of a parametric Diophantine triple {k − 1, k + 1, 16k 3 − 4k} in the ring ofGaussian integers to a Diophantine quadruple. Similar parametric family {k − 1, k + 1, 4k}
was studied by Franušić and it was proven that the extension to a Diophantine quadruple isunique.
The family of the form {k − 1, k + 1, 16k 3 − 4k} was studied in rational integers by
Bugeaud, Dujella and Mignotte and it was a special case while solving the extendibility
problem of Diophantine pairs of the form {k − 1, k + 1}, in which it was not possible touse the same method as in the other cases. As authors pointed out, the diﬃculty appears
because the gap between elements k + 1 and 16k 3 − 4k is not suﬃciently large. We find thesame diﬃculty here while trying to use Diophantine approximations. However, we partiallysolve this problem by using linear forms in logarithms.
During the proof we have also proven some results on the extendibility of general triple{a, b, c} in Gaussian integers which can be useful in improving the bound for the size ofDiophantine m-tuples in Gaussian integers.
This is joint work with Nikola Adžaga and Zrinka Franušić.
15:50–16:40 Shinichi Yasutomi (Toho University)(Joint work with Rao Hui (Central China Normal University) and Jiang Zhu (Huazhong
University of Science and Technology))
Title: Ergodicity for p-adic multidimensional continued fraction algorithms
Abstract: Schweiger provided a generalization of multidimensional continued fraction algorithms as piecewise fractional linear maps. In this talk following Schweiger’s generalizationof multidimensional continued fraction algorithms we consider a family of p-adic multidimensional continued fraction algorithms, which include Schneider’s algorithm and Ruban’s
1algorithms. We give the ergodicity of transformations associated with algorithms which isincluded in the family. We also give some numerical experiments related to the topic. Thisis a joint work with Rao Hui and Jiang Zhu.
16:50–17:40 Kota Saito (Nagoya University)
Title: Distribution of arithmetic progressions of Piatetski-Shapiro sequences
Abstract: For every r ≥ 1 and 1 < α < 2, we get the asymptotic formula of the distributionof positive integers n such the set of integer parts of nα , (n + r)α , · · · , (n + r(k − 1))α is anarithmetic progression of length k. The asymptotic density of this set is equal to 1/(k − 1)that is independent of r ≥ 1 and 1 < α < 2.

Friday 6th March
10:00–10:50 Hajime Kaneko (Tsukuba University)
Title: Application of Hensel’s lemma for the base-b expansions of integersAbstract: Let b ≥ 2 be a fixed integer. The base-b expansions of special integers have been
investigated by many mathematicians. For instance, Dupuy and Weirich conjectured theuniformity of the digits in the ternary expansions of 2n (n = 0, 1, 2, . . .). For investigating
the digits of base-b expansions of integers, we generalize Hensel’s lemma, which gives ananswer for an open problem by Axelsson and Khrennikov. This is a joint work with Thomas
Stoll.
11:10–12:00 István Pink (University of Debrecen)
Title: Some applications of Baker’s method to Diophantine equations
Abstract: In this talk we will concentrate on some applications of the famous method ofBaker to Diophantine equations. After a concise introduction to the theory of linear forms
in logarithms of algebraic numbers we give a brief survey on some classical results in Diophantine number theory involving Baker’s method. Then we point out several refinementsof the original work of Baker and we indicate how the combination of these refinements withsome other eﬀective methods (e.g. LLL algorithm, local method, hypergeometric method,etc.) may lead to eﬀective resolution of Diophantine equations of certain type. We end thetalk with explicit resolution of some concrete Diophantine equations illustrating the powerof the method.
14:00–14:50 Seiji Nishioka (Yamagata University)
Title: Applications of diﬀerence algebra to Mahler functions
Abstract: Proofs of the transcendence or algebraic independence of values of Mahler functions require that of the functions themselves. The algebraic relation of Mahler function
2f (x) and its transforms f (xd ), f (xd ), etc. can be regarded as an diﬀerence equation bythe operator transforming y(x) to y(xd ) in diﬀerence algebra which is a field considering
algebraic properties of solutions. Hence results in diﬀerence algebra would be applicable. Inthis talk, some examples will be introduced.

2

15:00–15:50 Yu Yasufuku (Nihon University)
Title: Towards Vojta’s conjecture on blowups of P2
Abstract: Vojta’s conjecture is a deep conjecture in Diophantine geometry, quantitatively
describing how the Diophantine approximation on a variety is controlled by the geometry.
In the case of blowups of P2 , this conjecture becomes a GCD inequality, and this inequality
has been proved for some subsets (Corvaja–Zannier, Luca). In this talk, we discuss a weakerversion of the conjectured inequality for the full set.
16:10–17:00 Makoto Kawashima (Osaka University)
Title: Rodligues’ formula and irrationality
Abstract: The Padé approximants of logarithm function are given by A. M. Legendre in1782. Nowadays they are called as Legendre polynomials. After the work of Legendre, in1816, O. Rodligues found a simple representation of Legendre polynomials which is known
as Rodligues’ formula. In this talk, we will show a generalization of Rodligues’ formulaand give explicit construction of Padé approximants of certain class of formal Laurent series
including Lerch functions. We also give some applications of the Padé approximants, forexample, irrationality and linear independence of values of Lerch functions and some otherrelated functions.
17:10–18:00 Tomohiro Yamada (Osaka University)
Title: Irrationality measure of (arctan 1/2)/π
Abstract: We shall show that for all positive integers p, q with q suﬃciently large,|pπ/4 − q arctan 1/2| > q −9.04722

Saturday 7th March
9:30–10:20 Masanori Katsurada (Keio University)
Title: Complete asymptotic expansions for the product average of higher derivatives of Lerch zeta-functions
Abstract: Let s = σ +it and sj = σj +itj (σ, σj , t, tj ∈ R) for j = 1, 2 be complex variables, xand λ real parameters with x > 0,∑and write e(s) = e2πis . The Lerch zeta-function ϕ(s, x, λ)−s(σ > 1), and its meromorphic continuis defined by the Dirichlet series ∞l=0 e(λl)(x + l)(m)ation over the whole s-plane. We write ϕ (s, x, λ) = (∂/∂s)m ϕ(s, x, λ) (m = 0, 1, . . .), andintroduce the product average of ϕ(m) (s, x, λ), in the form
∫ 1
Im1 ,m2 (s1 , s2 ; a, λ) =ϕ(m1 ) (s1 , a + x, λ)ϕ(m2 ) (s2 , a + x, −λ)dx0for any nonnegative integers m1 and m2 , and any real a and λ with a > 0. It is then shownin the talk that complete asymptotic expansions exist for Im1 ,m2 (σ1 + it, σ2 − it; a, λ) in thedescending order of t as t → ±∞, being valid in a satisfactorily wide region of (σ1 , σ2 ) andfor any nonnegative integers m1 and m2 . The case (σ1 , σ2 ) = (σ, σ) and (m1 , m2 ) = (m, m)of our main
result in particular reduces to complete asymptotic expansions for the mean
∫ 1 (m)square 0 |ϕ (σ + it, a + x, λ)|2 dx (m = 0, 1, . . .) as t → ±∞, except the cases σ = n/2(n = 2, 1, 0, −1, . . .), where the important situations on the critical line σ = 1/2 and on thevertical line σ = 1 are treated as their limiting forms when σ → 1/2 and σ → 1, respectively.
3

Crucial rôles in the proofs are played by the complete asymptotic expansions (together withexact reminders) for a certain quotient of gamma functions.
10:40–11:30 Takuya Aoki (Waseda University)
Title: On class number√ calculations of the intermediate fields of the cyclotomic
Z2 -extension over Q( 5)Abstract: Let h2,n be the class number of Q(2 cos (2π/2n+2 )). It is known as Weber’s classnumber problem whether h2,n = 1 holds for all positive integer n or not. This problem has
been studied deeply by K. Horie, K. Horie-M. Horie, Fukuda-Komatsu, Okazaki, MorisawaOkazaki and J. Miller.
√In this talk, we attempt to generalize this problem for the case of Q( 5) and introduce theresults we obtained as analogies of the known results for the original Weber’s class numberproblem.
11:40–12:30 Ryotaro Okazaki
Title: On total capitulation in CM-fields and Scholz reflection theorem
Abstract: We use A.Scholz’s construction of class field to prove:
Total Capitulation Thereom in CM-fields:
Let K be a given CM-field. Then, there exists some extension field L of K, so that
• The extension field L is also a CM-field.
• All ideal classes of the subfield K capitulate in the extension field L.
This result is surprising since Horie’s Divisibility Theorem asserts the relative ideal classnumber of K divides that of L; and Fujii’s Stability Theorem (2019) asserts the relative idealclass group of K almost survives in L if K and L are cyclotomic fields.

4