P-adic Logarithms Research Articles

Lower bounds for linear forms in two p-adic logarithms

We prove explicit lower bounds for linear forms in two p-adic logarithms. More specifically, we establish explicit lower bounds for the p-adic distance between two integral powers of algebraic numbers, that is, |Λ|p=|α1b1−α2b2|p (and corresponding explicit upper bounds for vp(Λ)), where α1,α2 are numbers that are algebraic over Q and b1,b2 are positive rational integers.This work is a p-adic analogue of Gouillon's explicit lower bounds in the complex case. Our upper bound for vp(Λ) has an explicit constant of reasonable size and the dependence of the bound on B (a quantity depending on b1 and b2) is log⁡B, instead of (log⁡B)2 as in the work of Bugeaud and Laurent in 1996.

Restriction of Eisenstein series and Stark–Heegner points

In a recent work of Darmon, Pozzi and Vonk, the authors consider a particular p-adic family of Hilbert–Eisenstein series E k (1,ϕ) associated with an odd character ϕ of the narrow ideal class group of a real quadratic field F and compute the first derivative of a certain one-variable twisted triple product p-adic L-series attached to E k (1,ϕ) and an elliptic newform f of weight 2 on Γ 0 (p). In this paper, we generalize their construction to include the cyclotomic variable and thus obtain a two-variable twisted triple product p-adic L-series. Moreover, when f is associated with an elliptic curve E over ℚ, we prove that the first derivative of this p-adic L-series along the weight direction is a product of the p-adic logarithm of a Stark–Heegner point of E over F introduced by Darmon and the cyclotomic p-adic L-function for E.

From category [formula omitted] to locally analytic representations

Let G be a p-adic reductive group and g its Lie algebra. We construct a functor from the extension closure of the Bernstein-Gelfand-Gelfand category O associated to g into the category of locally analytic representations of G, thereby expanding on an earlier construction of Orlik-Strauch. A key role in this new construction is played by p-adic logarithms on tori. This functor is shown to be exact with image in the subcategory of admissible representations in the sense of Schneider and Teitelbaum. En route, we establish some basic results in the theory of modules over distribution algebras and related subalgebras, such as a tensor-hom adjunction formula. We also relate our constructions to certain representations constructed by Breuil and Schraen in the context of the p-adic Langlands program.

Stark points on elliptic curves via Perrin-Riou’s philosophy

In the early 90’s, Perrin-Riou (Ann Inst Fourier 43(4):945–995, 1993) introduced an important refinement of the Mazur–Swinnerton-Dyer p-adic L-function of an elliptic curve E over $$\mathbb {Q}$$ , taking values in its p-adic de Rham cohomology. She then formulated a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for this p-adic L-function, in which the formal group logarithms of global points on E make an intriguing appearance. The present work extends Perrin-Riou’s construction to the setting of a Garret–Rankin triple product (f, g, h), where f is a cusp form of weight two attached to E and g and h are classical weight one cusp forms with inverse nebentype characters, corresponding to odd two-dimensional Artin representations $$\varrho _g$$ and $$\varrho _h$$ respectively. The resulting p-adic Birch and Swinnerton-Dyer conjecture involves the p-adic logarithms of global points on E defined over the field cut out by $$\varrho _g\otimes \varrho _h$$ , in the style of the regulators that arise in Darmon et al. (Forum Math 3(e8):95, 2015), and recovers Perrin-Riou’s original conjecture when g and h are Eisenstein series.

Diagonal restrictions of p-adic Eisenstein families

We compute the diagonal restriction of the first derivative with respect to the weight of a p-adic family of Hilbert modular Eisenstein series attached to a general (odd) character of the narrow class group of a real quadratic field, and express the Fourier coefficients of its ordinary projection in terms of the values of a distinguished rigid analytic cocycle in the sense of Darmon and Vonk (Duke Math J, to appear, 2020) at appropriate real quadratic points of Drinfeld’s p-adic upper half-plane. This can be viewed as the p-adic counterpart of a seminal calculation of Gross and Zagier (J Reine Angew Math 355:191–220, 1985, §7) which arose in their “analytic proof” of the factorisation of differences of singular moduli, and whose inspiration can be traced to Siegel’s proof of the rationality of the values at negative integers of the Dedekind zeta function of a totally real field. Our main identity enriches the dictionary between the classical theory of complex multiplication and its extension to real quadratic fields based on RM values of rigid meromorphic cocycles, and leads to an expression for the p-adic logarithms of Gross–Stark units and Stark–Heegner points in terms of the first derivatives of certain twisted Rankin triple product p-adic L-functions.

Tubular approaches to Baker’s method for curves and varieties

Baker’s method, relying on estimates on linear forms in logarithms of algebraic numbers, allows one to prove in several situations the effective finiteness of integral points on varieties. In this article, we generalize results of Levin regarding Baker’s method for varieties, and explain how, quite surprisingly, it mixes (under additional hypotheses) with Runge’s method to improve some known estimates in the case of curves by bypassing (or more generally reducing) the need for linear forms in p-adic logarithms. We then use these ideas to improve known estimates on solutions of S-unit equations. Finally, we explain how a finer analysis and formalism can improve upon the conditions given, and give some applications to the Siegel modular variety A2(2).

P-Adic measures associated with zeta values and p-adic log multiple gamma functions

We study a relation between two refinements of the rank one abelian Gross–Stark conjecture. For a suitable abelian extension [Formula: see text] of number fields, a Gross–Stark unit is defined as a [Formula: see text]-unit of [Formula: see text] satisfying certain properties. Let [Formula: see text]. Yoshida and the author constructed the symbol [Formula: see text] by using [Formula: see text]-adic [Formula: see text] multiple gamma functions, and conjectured that the [Formula: see text] of a Gross–Stark unit can be expressed by [Formula: see text]. Dasgupta constructed the symbol [Formula: see text] by using the [Formula: see text]-adic multiplicative integration, and conjectured that a Gross–Stark unit can be expressed by [Formula: see text]. In this paper, we give an explicit relation between [Formula: see text] and [Formula: see text] and prove that two refinements are consistent.

Les variétés de Hecke–Hilbert aux points classiques de poids parallèle 1

We show that the Eigenvariety attached to Hilbert modular forms over a totally real field F is smooth at the points corresponding to certain classical weight one theta series and we give a precise criterion for etaleness over the weight space at those points. In the case where the theta series has real multiplication, we construct a non-classical overconvergent generalised eigenform and compute its Fourier coefficients in terms of p-adic logarithms of algebraic numbers. Our approach uses deformations of Galois representations.

Plus and minus logarithms and Amice transform

We give a new description of Pollack's plus and minus p-adic logarithms logp± in terms of distributions. In particular, if μ± denote the pre-images of logp± under the Amice transform, we give explicit formulae for the values μ±(a+pnZp) for all a∈Zp and all integers n≥1. Our formulae imply that the distribution μ− agrees with a distribution studied by Koblitz in 1977. Furthermore, we show that a similar description exists for Loeffler's two-variable analogues of these plus and minus logarithms.

Q-Analogue of p-Adic log Γ Type Functions Associated with Modified q-Extension of Genocchi Numbers with Weight α and β

The p-adic log gamma functions associated with q-extensions of Genocchi and Euler polynomials with weight α were recently studied . By the same motivation, we aim in this paper to describe q-analogue of p-adic log gamma functions with weight alpha and beta. Moreover, we give relationship between p-adic q-log gamma functions with weight (α,β) and q-extension of Genocchi numbers with weight alpha and beta and modified q-Euler numbers with weight α.

Gross–Stark units and p-adic iterated integrals attached to modular forms of weight one

This article can be read as a companion and sequel to the authors’ earlier article on Stark points and p-adic iterated integrals attached to modular forms of weight one, which proposes a conjectural expression for the so-called p-adic iterated integrals attached to a triple (f, g, h) of classical eigenforms of weights (2, 1, 1). When f is a cusp form, this expression involves the p-adic logarithms of so-called Stark points: distinguished points on the modular abelian variety attached to f, defined over the number field cut out by the Artin representations attached to g and h. The goal of this paper is to formulate an analogous conjecture when f is a weight two Eisenstein series rather than a cusp form. The resulting formula involves the p-adic logarithms of units and p-units in suitable number fields, and can be seen as a new variant of Gross’s p-adic analogue of Stark’s conjecture on Artin L-series at \(s=0\).

A distribution formula for Kashio's p-adic log-gamma function

We study a special case of Kashio's p-adic log⁡Γ-function, that we call LogΓp, which combines these of Morita and Diamond. It agrees with each of these on large parts of its domain and has the advantage of being a locally analytic function. We prove a distribution formula for LogΓp which generalizes and links the known distribution formulas for Diamond's and Morita's functions.

Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields

This article examines the Fourier expansions of certain non-classical p-adic modular forms of weight one: the eponymous generalised eigenforms of the title, so called because they lie in a generalised eigenspace for the Hecke operators. When this generalised eigenspace contains the theta series attached to a character of a real quadratic field K in which the prime p splits, the coefficients of the attendant generalised eigenform are expressed as p-adic logarithms of algebraic numbers belonging to an idoneous ring class field of K. This suggests an approach to “explicit class field theory” for real quadratic fields which is simpler than the one based on Stark's conjecture or its p-adic variants, and is perhaps closer in spirit to the classical theory of singular moduli.

On the norm of Krasner analytic functions with applications to transcendence results

Given a prime number p let Cp be the Tate field i.e. the topological completion of the algebraic closure of the field of p-adic numbers with respect to the p-adic absolute value. We give an estimate of the norm of Krasner analytic functions defined on the complement of a fundamental set X of Cp, which are Cauchy transforms obtained by integrating against strongly Lipschitz distributions on X. The functions from a specific class of the above representations are transcendental over Cp(Z) and, as a consequence, we obtain transcendence results related to the twisted p-adic log gamma function and the trace functions. The twisted p-adic log gamma function and its derivatives are linearly independent over Cp(Z) and, moreover, all their zeros are algebraic.

Explicit Chabauty-Kim theory for the thrice punctured line in depth 2

Let X = P 1 ∖ { 0 , 1 , ∞ } , and let S denote a finite set of prime numbers. In an article of 2005, Kim gave a new proof of Siegel's theorem for X: the set X ( Z [ S − 1 ] ) of S-integral points of X is finite. The proof relies on a ‘nonabelian’ version of the classical Chabauty method. At its heart is a modular interpretation of unipotent p-adic Hodge theory, given by a tower of morphisms h n between certain Q p -varieties. We set out to obtain a better understanding of h 2 . Its mysterious piece is a polynomial in 2 | S | variables. Our main theorem states that this polynomial is quadratic, and gives a procedure for writing its coefficients in terms of p-adic logarithms and dilogarithms.

On p-adic Diamond–Euler Log Gamma functions

Abstract Text In this paper, using the fermionic p-adic integral on Z p , we define the corresponding p-adic Log Gamma functions, so-called p-adic Diamond–Euler Log Gamma functions. We then prove several fundamental results for these p-adic Log Gamma functions, including the Laurent series expansion, the distribution formula, the functional equation and the reflection formula. We express the derivative of p-adic Euler L-functions at s = 0 and the special values of p-adic Euler L-functions at positive integers as linear combinations of p-adic Diamond–Euler Log Gamma functions. Finally, using the p-adic Diamond–Euler Log Gamma functions, we obtain the formula for the derivative of the p-adic Hurwitz-type Euler zeta function at s = 0 , then we show that the p-adic Hurwitz-type Euler zeta functions will appear in the studying for a special case of p-adic analogue of the ( S , T ) -version of the abelian rank one Stark conjecture. Video For a video summary of this paper, please click here or visit http://youtu.be/DW77g3aPcFU .

PERFECT POWERS WITH THREE DIGITS

We solve the equation xa + xb + 1 = yq in positive integers x, y, a, b and q with a > b and q ≥ 2 coprime to φ(x). This requires a combination of a variety of techniques from effective Diophantine approximation, including lower bounds for linear forms in complex and p-adic logarithms, the hypergeometric method of Thue and Siegel applied p-adically, local methods, and the algorithmic resolution of Thue equations.

On the solutions of the exponential Diophantine equation a x + b y = (m 2 + 1) z

In this paper, we consider the Diophantine equation a x + b y = c z . Combining Laurent's result on lower bounds for linear forms in two logarithms, Bugeaud's result on upper bounds for the p-adic logarithms, and Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers, we obtain a sharper computable upper bound for the number of positive integer solutions of the equation ∣v r ∣ x + ∣u r∣ y = c z , where and r is an odd number. Moreover, if r is a prime and r ≡ 5 (mod 8) or r ≡ 19 (mod 24), , we prove that the equation has only the solution (x, y, z) = (2, 2, r).

Faster [formula omitted]-adic feasibility for certain multivariate sparse polynomials

We present algorithms revealing new families of polynomials admitting sub-exponential detection of p-adic rational roots, relative to the sparse encoding. For instance, we prove NP-completeness for the case of honest n-variate (n+1)-nomials and, for certain special cases with p exceeding the Newton polytope volume, constant-time complexity. Furthermore, using the theory of linear forms in p-adic logarithms, we prove that the case of trinomials in one variable can be done in NP. The best previous complexity upper bounds for all these problems were EXPTIME or worse. Finally, we prove that detecting p-adic rational roots for sparse polynomials in one variable is NP-hard with respect to randomized reductions. The last proof makes use of an efficient construction of primes in certain arithmetic progressions. The smallest n where detecting p-adic rational roots for n-variate sparse polynomials is NP-hard appears to have been unknown.

Hilbert modular forms and the Gross-Stark conjecture

Let F be a totally real eld and an abelian totally odd character of F . In 1988, Gross stated a p-adic analogue of Stark’s conjecture that relates the value of the derivative of the p-adic L-function associated to and the p-adic logarithm of a p-unit in the extension of F cut out by . In this paper we prove Gross’s conjecture when F is a real quadratic eld and is a narrow ring class character. The main result also applies to general totally real elds for which Leopoldt’s conjecture holds, assuming that either there are at least two primes above p in F , or that a certain condition relating the L -invariants of and 1 holds. This condition on L -invariants is always satised when is quadratic.