SCHEDULED TALKS AND THEIR ABSTRACTS FOR”DIOPHANTINE ANALYSIS AND RELATED FIELDS 2012”
Takao Komatsu (Hirosaki Univ.)
Title: Poly-Cauchy numbers and Riordan arrays(k)
Abstract: Poly-Cauchy numbers cn are deﬁned by
Z 1 Z 1(k)cn =· · · (x1 x2 · · · xk )(x1 x2 · · · xk − 1) · · · (x1 x2 · · · xk − n + 1)dx1 dx2 · · · dxk .
0| {z 0 }k(1)If k = 1, then cn = cn are the classical Cauchy numbers. Poly-Cauchy numbers havesome similar properties as poly-Bernoulli numbers which are introduced by M. Kaneko
and are some kinds of generalization of the classical Bernoulli numbers. In this talk wegive some characteristic properties and relations with Riordan arrays.
Takao Komatsu and Mari Yokohama (Hirosaki Univ.)
Title: A generalization of poly-Cauchy numbers
Abstract: In this talk we give a generalization of poly-Cauchy numbers. Namely, we(k)deﬁne the poly-Cauchy numbers with q parameter cn,q by
Z 1 Z 1(k)cn,q =· · · (x1 x2 · · · xk )(x1 x2 · · · xk − q) · · · (x1 x2 · · · xk − (n − 1)q)dx1 dx2 · · · dxk .
| 0 {z 0 }kPoly-Cauchy numbers with q parameter also have explicit formulae by using Stirlingnumbers of the ﬁrst kind, and have the corresponding generating functions.
Noriko Hirata-Kohno (Nihon Univ.)
Title: Arithmetic properties of p-adic elliptic polylogarithmic functions
Abstract: In the talk, we investigate arithmetic properties of p-adic elliptic polylogarithmic functions by using a formal group. Let t ∈ Cp with |t|p < 1. We deﬁne the p-adic
Z t
Li E,k−1 (u)k-th polylogarithmic function by Li E,1 (t) = logE (t) and Li E,k (t) =du (k =u02, 3, · · · ). We give bounds for the height of the Taylor coeﬃcients of our p-adic elliptic
polylogarithmic functions and show several new arithmetic properties.
Yuri V. Nesterenko (Moscow State Univ.)
Title: Some identities of Ramanujan type
Abstract: We will discuss identities of modular type that are proved for some functions gu,v (τ ) that looks like Eisenstein series. They generalize some classical identities ofS. Ramanujan and E. Grosswald. At the point i these identities give linear relations overQ connecting ζ-values at odd points and values gu,v (i). As a consequence we proved thatthe transcendence degree of the ﬁeld generated by the values of gu,v (τ ) is smaller that itis expected.
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DIOPHANTINE ANALYSIS AND RELATED FIELDS 2012

Hajime Kaneko (Nihon Univ.)
Title: Algebraic independence of the special values of power series
those derivatives
P and−n!
Abstract: Liouville proved the transcendence of the values of ∞b, where b is an
n=0integer greater than 1. In our talk we consider algebraic independence of the values ofpower series at certain rational points. In particular, we give new criteria for algebraic
independence of such numbers. Applying our criteria, we deduce algebraic independenceof the values of power series and those derivatives at certain rational points.
Koichiro Akiyama (Toshiba Corp.)
Title: Public key cryptosystem based on the diophantine problems on Fp [t]Abstract: We construct a new type of public-key cryptosystem whose security is basedon the Diophantine Problems on Fp [t]. Public-key cryptosystems requires two keys called
public key and private key. Public key is used for encrypt plaintext, and private key is used
for decrypt the ciphertext. Security of public key cryptosystem relies on the problem which
is diﬃcult to solve, such as integer factorization. We employ the Diophantine Problems
on Fp [t] called the section ﬁnding problem (SFP) to construct public key cryptosystem.
In this talk, I describe the encrypt/decrypt algorithms and key generation algorithm, anddiscuss about the diﬃculty of SFP.
Yu Yasufuku (Nihon Univ.)
Title: Finiteness integers in orbits of maps on Pn
Abstract: Silverman has proved ﬁniteness of integers in the orbit {P, φ(P ), φ(φ(P )), . . .}of a rational point P under a non-totally-ramiﬁed morphism φ on P1 . In this talk, wewill discuss several generalizations of this to maps on Pn . Some speciﬁc low-degree andlow-dimensional maps are treated, using ingredients such as Schmidt’s subspace theorem.
We also present a theoretical result treating more general morphisms, albeit assuming apowerful conjecture in Diophantine geometry by Vojta. The hypothesis required on themorphisms has been weakened from a prior paper, so we will highlight the diﬀerence.
Takehiro Hasegawa (Kogakuin Univ.)
Title: A polynomial with Franel numbers as coeﬃcients and its application to modularcurves¡ ¢3P
Abstract: In 1884 (19th century), Franel deﬁned the number an = nk=0 nk and heshowed that it satisﬁes the recurrence (n + 1)2 an+1 − (7n2 + 7n + 2)an − 8n2 an−1 = 0.
Ingave two kinds of functional equations for the generating function F (z) =P 2009, Maiernaz.
In
thistalk, we obtain two polynomial identities over ﬁnite ﬁelds from thesen≥0 nequations, and by using these identities, we explicitly describe behaviors of supersingularpoints in the towers {X0 (3 · 2n )} and {X0 (2 · 3n )}.
Takeshi Kurosawa (Tokyo Univ. Sci.)
Title: Algebraic dependence of inﬁnite products related with Fibonacci and Lucas numbers
Abstract: Let d be an integer greater than 1. In [1], we gave necessary and suﬃcientconditions for the inﬁnite products¶¶∞ µ∞ µYYaiai(i = 1, . . . , m)or(i = 1, . . . , m)
1+1+
Udk
Vdk
k=1
Udk 6=−ai

k=1
Vdk 6=−aiTITLES AND ABSTRACTS FOR DARF 2012

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to be algebraically dependent, where a1 , . . . , am are nonzero integers and {Un }n≥0 and{Vn }n≥0 are generalized Fibonacci numbers and Lucas numbers, respectively. In thistalk, we relax the condition a1 , . . . , am nonzero integers to real algebraic numbers. By
generalizing the condition we show new algebraically dependent cases.
References[1] T. Kurosawa, Y. Tachiya and T. Tanaka, Algebraic independence of inﬁnite products generated by
Fibonacci numbers, Tsukuba Journal of Mathematics 34 (2010), no. 2, 255–264.

Yohei Tachiya (Hirosaki Univ.)
Title: Algebraic independence of inﬁnite products generated by distinct binary recurrences
Abstract: Let α and β be real algebraic numbers with |α| > 1 and αβ = −1. Deﬁne
Un =αn − β nα−βand Vn = αn + β n(n ≥ 0).

√If a = (1 + 5)/2, we have Un = Fn and Vn = Ln (n ≥ 0), where the sequences {Fn }n≥0and {Ln }n≥0 are Fibonacci and Lucas numbers, respectively.
Let d1 , d2 ≥ 2 be integers and γ1 , γ2 be real algebraic numbers. The purpose of thistalk is to discuss the algebraic independence over Q of the inﬁnite products
Ã!
Ã!
∞∞YY
γ1
γ21+and1+.
Udk1
Vdk2
k=1
Udk 6=−γ11

k=1
Vdk 6=−γ2As an application, we show that the inﬁnite products
are algebraically independent over Q.

2Q∞

k=1 (1+1/F2k ) andQ∞

k=1 (1+1/L2k )Florian Luca (Inst. de Mat., UNAM)
Title: Balancing with powers of Fibonacci numbers
Abstract: Let (Fn )n≥0 be the Fibonacci sequence. In my talk, I will show that the only
solution of the Diophantine equationk``
F1k + F2k + · · · + Fn−1
= Fn+1+ · · · + Fn+r

in positive integers (k, `, n, r) is (8, 2, 4, 3). This conﬁrms a conjecture of Behera, Liptai,Panda and Szalay. This is joint work with A. Dujella (Zagreb) and S. Dı́az Alvarado(Toluca).
Attila Pethö and Michael E. Pohst (Univ. Debrecen)
Title: On the indices of multiquadratic number ﬁelds
Abstract: We present explicit formulae for the index forms of multiquadratic number
ﬁelds. For octic ﬁelds we calculate all potential ﬁeld indices and characterize the corresponding ﬁelds. We also show that any prime power pk divides the ﬁeld index if the degree
2r is suﬃciently large. Our method diﬀers from all others which have been previously used.

Isao Wakabayashi (Tokyo)

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DIOPHANTINE ANALYSIS AND RELATED FIELDS 2012

Title: Number of solutions for quartic simple Thue equations
Abstract: Put F (X, Y ) = bX 4 −aX 3 Y −6bX 2 Y 2 +aXY 3 +bY 4 with integer coeﬃcients.
We show that the number of solutions for the Thue equation F (x, y) = 1, −1 is 0 or 4,
except for a few already known cases. To obtain an upper bound for the size of solutions,
we use Padé approximation method. To obtain a lower bound for the size of solutions, weconstruct a continued fraction with positive or negative rational partial quotients. This
construction is carried out carefully by using special properties of the form F . Combining
these lower and upper bounds, we obtain the result.
Yasutsugu Fujita (Nihon Univ.) and Nobuhiro Terai (Ashikaga Inst. Tech.)
Title: Generators and integer points on the elliptic curve y 2 = x3 − nx
Abstract: Let E be an elliptic curve over the rationals Q given by y 2 = x3 − nx witha positive integer n. We are interested in integer points on E and generators for theMoedell-Weil group E(Q).
Consider ﬁrst the case where n is a perfect square. Assume that n = N 2 for a squarefree integer N . Then, we show that if E(Q) has rank one, there exist at most 17 integer
points on E. Moreover, we show that for some parameterized N a point P can be in asystem of generators for E(Q) and completely determine the integer points in the group
generated by the torsion points and the point P .
Secondly, consider the case where n is non-square, more precisely, where n = s4 + t4 forcoprime positive integers s and t. It is known that the rank of E(Q) is greater than orequal to two. We here show that if n is fourth-power-free, then the points P1 = (−t2 , s2 t)
and P2 = (−s2 , st2 ) can be in a system of generators for E(Q). Furthermore, we prove that
if n is square-free, then there exist at most nine integer points in the group Γ generated
by the points P1 , P2 and (0, 0). In particular, in case n = s4 + 1 the integer points in Γare completely determined.
Taka-aki Tanaka (Keio Univ.)
Title: Algebraic independence of the values of Mahler functions with ‘negative’ transformations
Abstract: Mahler’s method gives algebraic independence results for the values of functions of several variables satisfying certain functional equations, called Mahler type functional equations, under the transformations of the variables represented as the multiplicative action of matrices with integer entries. In the Mahler’s method, the matrices
representing the transformations of the variables are restricted to nonnegative matrices;
however, in a special case, one can admit having a negative entry. In this talk I will show
the algebraic independence of the values of certain functions satisfying Mahler type functional equations under the transformation represented by a matrix including a negative
entry by expressing those values as linear combinations of the values of ordinary Mahler
functions.