This chapter is of an auxiliary nature, being mainly concerned with the relationship between bounds for linear forms in the logarithms of algebraic numbers in different (archimedean and non-archimedean) metrics. This material will later be used in the analysis of Thue and Thue-Mahler equations. Elliptic and hyperelliptic equations, and equations of hyperelliptic type, will be analysed using direct bounds for linear forms in the logarithms of algebraic numbers, and the necessary results are stated without proof at the end of the chapter. *** DIRECT SUPPORT *** A00I6B17 00002

Linear forms in the logarithms of algebraic numbers

LECTURES ON p‐ADIC L‐FUNCTIONS

On p-adic Hurwitz-type Euler zeta functions

Let F be a totally real number field, and let p be a finite prime of F such that p splits completely in the finite abelian extension H of F. Tate has proposed a conjecture [22, Conjecture 5.4] stating the existence of a p-unit u in H with absolute values at the places above p specified in terms of the values at zero of the partial zeta functions associated to H/F. This conjecture is an analogue of Stark's conjecture, which Tate called the Brumer-Stark conjecture. Gross [12, Conjecture 7.6] proposed a refinement of the Brumer-Stark conjecture that gives a conjectural formula for the image of u in Fp×/E∧, where Fp denotes the completion of F at p and E∧ denotes the topological closure of the group of totally positive units E of F. We present a further refinement of Gross's conjecture by proposing a conjectural formula for the exact value of u in Fp×

Shintani zeta functions and Gross-Stark units for totally real fields

We define an integral version of Sczech’s Eisenstein cocycle on GLn by smoothing at a prime l. As a result we obtain a new proof of the integrality of the values at nonpositive integers of the smoothed partial zeta functions associated to ray class extensions of totally real fields. We also obtain a new construction of the p-adic Lfunctions associated to these extensions. Our cohomological construction allows for a study of the leading term of these p-adic L-functions at s = 0. We apply Spiess’s formalism to prove that the order of vanishing at s = 0 is at least equal to the expected one, as conjectured by Gross. This result was already known from Wiles’ proof of the Iwasawa Main Conjecture.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cjm/2014/0002/0001/CJM-2014-0002-0001-a002.pdf

Integral Eisenstein cocycles on $\mathbf{GL}_n$, I: Sczech’s cocycle and $p$-adic $L$-functions of totally real fields

Shintani expressed the first derivative at $s = 0$ of a partial $\zeta$-function of an algebraic number field in terms of the multiple gamma function. Cassou-Nogues constructed a $p$-adic analogue of the partial $\zeta$-function and calculated the derivative at $s = 0$. In this paper, we will define a $p$-adic analogue of the multiple gamma function and give a $p$-adic analogue of Shintani’s formula. This formula has a strong resemblance to the original Shintani’s formula. Using this formula, we get a partial result toward Gross’ conjecture concerning the order at $s = 0$ of the $p$-adic $L$-function.

/pdf/on-a-p-adic-analogue-of-shintani-s-formula-49fktk3kyx.pdf

On a p-adic analogue of Shintani's formula

The second author defined the absolute CM-period symbol using the multiple gamma function. This symbol is conjectured to give the same value as Shimura's period symbol up to multiplication by algebraic numbers. In this paper, we define a p-adic analogue of the absolute CM-period symbol, using the p-adic multiple gamma function studied by the first author. In the completely split case, we present a conjecture which predicts the exact value of this symbol as well as supporting evidences.

https://www.kurims.kyoto-u.ac.jp/~prims/pdf/45-1/45-1-6.pdf

On p-adic absolute CM-periods I

In part I of this paper, we studied the p-adic absolute CM-period symbol in the completely split case. We presented a conjecture which predicts the exact value of this symbol with supporting evidences. In this part II, we study the properties of this symbol in the general case and clarify its relation to p-adic periods.

https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=45&iss=1&rank=7

On p-Adic Absolute CM-Periods II

\n We define a “period-ring-valued beta function” and give a reciprocity law on its special values.\nThe proof is based on some results of Rohrlich and Coleman concerning Fermat curves.\nWe also have the following application.\nStark’s conjecture implies that the exponentials of the derivatives at \n \n \n \n s=0\n \n of partial zeta functions are algebraic numbers which satisfy a reciprocity law\nunder certain conditions.\nIt follows from Euler’s formulas and properties of cyclotomic units when the base field is the rational number field.\nIn this paper, we provide an alternative proof of a weaker result by using the reciprocity law on the period-ring-valued beta function.\nIn other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units.

Fermat curves and a refinement of the reciprocity law on cyclotomic units

We provide an equivalent condition for the monogenity of the ring of integers of any cyclic cubic field. We show that if a cyclic cubic field is monogenic then it is a simplest cubic field $K_t$ which is the splitting field of a Shanks cubic polynomial $f_t(x):=x^3-tx^2-(t + 3)x-1$ with $t \in \mathbb Z$. Moreover we give an equivalent condition for when $K_t$ is monogenic, which is explicitly written in terms of $t$.

Tomokazu Kashio

Papers

On a p-adic analogue of Shintani's formula

On p-adic absolute CM-periods I

On p-Adic Absolute CM-Periods II

Fermat curves and a refinement of the reciprocity law on cyclotomic units

The characterization of cyclic cubic fields with power integral bases