Following Apery's proof of the irrationality of zeta(3), Beukers found an elegant reinterpretation of Apery's arguments using modular forms. We show how Beukers arguments can be adapted to a p-adic setting. In this context, certain functional equations arising from Eichler integrals are replaced by the notion of overconvergent p-adic modular forms, and the periods themselves arise not as coefficients of period polynomials but as constant terms of p-adic Eisenstein series. We prove that the analogue of zeta(3) is irrational for p = 2 and 3, as well as the 2-adic analogue of Catalan's constant.
We give a proof of the irrationality of the $p$-adic zeta-values $ζ_p(k)$ for $p=2,3$ and $k=2,3$. Such results were recently obtained by F.Calegari as an application of overconvergent $p$-adic modular forms. In this paper we present an approach using classical continued fractions discovered by Stieltjes. In addition we show irrationality of some other $p$-adic $L$-series values, and values of the $p$-adic Hurwitz zeta-function.
We prove that the sum of the series $\sum_{n=0}^{\infty}\, p^{v_p(n!)}$ is a $p$-adic irrational for all primes $p$, where $v_p(n!)$ denotes the exponent of the highest power of $p$ dividing $n!$.
We show how the theory of linear forms in two logarithms allows one to get effective irrationality measures for $n$-th roots of rational numbers ${a \over b}$, when $a$ is very close to $b$. We give a $p$-adic analogue of this result under the assumption that $a$ is $p$-adically very close to $b$, that is, that a large power of $p$ divides $a-b$. As an application, we solve completely certain families of Thue-Mahler equations. Our results illustrate, admittedly in a very special situation, the strength of the known estimates for linear forms in logarithms.
The knowledge on irrationality of p-adic zeta values has recently progressed. The irrationality of zeta_2(2), ζ_2(3) and of a few other p-adic series of Dirichlet was obtained by F. Calegari. F. Beukers gave a more elementary proof of this result. In parallel, T. Rivoal has just obtained, in the complex case, some Pade approximants of Lerch functions. It is this work which, transposed to C_p, enables us to obtain results of irrationality and linear independence.
We prove a variant of Oesterlé's conjecture about $p$-adic periods of the modular curve $X_0(p)$, with an additional $Γ(2)$-structure. We use de Shalit's techniques and $p$-adic uniformization of curves whose reduction is semi-stable.
Presenting p-adic numbers as {\em deformations} of finite fields allows a better understanding of Frobenius lifts and their connection with p-derivations in the sense of Buium \cite{Buium-Main}. In this way "numbers {\em are} functions", as recognized before \cite{Manin:Numbers}, allowing to view initial structure deformation problems as arithmetic differential equations as in \cite{Buium-Manin}, and providing a cohomological interpretation to Buium calculus via Hochschild cohomology which controls deformations of algebraic structures. Applications to p-adic periods are considered, including t…
In this paper, we will apply the ideas from the mirror symmetry of Calabi-Yau threefolds to study the modular forms and one-parameter family of K3 surfaces found by Beukers and Peters, which provide enlightenment to the two mysterious sequences constructed by Apéry in his proof of the irrationality of $ζ(3)$. We will construct a fourth order differential operator and a prepotential from the canonical solutions of this differential operator. The third derivative of this prepotential with respect to the mirror map defines a Yukawa coupling that is a weight-4 modular form. The instanton expansion…
In this paper, we prove the equality between the transcendental degree of the field generated by the v-adic periods of a t-motive M and the dimension of the Tannakian Galois group for M, where v is a "finite" place of the rational function field over a finite field. As an application, we prove the algebraic independence of certain "formal" polylogarithms.
In 2006, Fukaya and Kato formulated a general conjecture about $p$-adic $L$-functions for a large class of motives and derived a precise conjectural interpolation formula from the ETNC. We study this conjectural $p$-adic $L$-function in the setting of Hida families and compare it to the $p$-adic $L$-function constructed by Kitagawa. A priori it is not clear whether the interpolation formulas coincide, mainly because of the rather inexplicit definition of periods (via comparison isomorphisms) that appear in the general formula. Using $p$-adic Eichler-Shimura isomorphisms and their interpolation…
As an example of relative p-adic Hodge theory, we sketch the construction of the universal admissible filtration of an isocrystal (ϕ$-module) over the completion of the maximal unramified extension of Q_p, together with the associated universal crystalline local system.
We construct the anticyclotomic $p$-adic $L$-function that interpolates a square root of central values of twisted spinor $L$-functions of a quadratic base change of a Siegel cusp form of genus $2$ with respect to a paramodular group of square-free level, assuming the Böcherer conjecture for the central $L$-values with anticyclotomic twists.
We compute the p-torsion and p-adic etale cohomologies with compact support of period domains over local fields in the case of basic isocrystals for quasi-split reductive groups. For the p-torsion case, we follow the method used by Orlik in his computations of the l-torsion etale cohomology using as a key new ingredient the computation of Ext groups between mod p generalized Steinberg representations of p-adic groups. For the p-adic case, we don't use Huber's definition of etale cohomology with compact support as Orlik did since it seems to give spaces that are much too big; instead we use con…
In this article, we study $p$-adic torus periods for certain $p$-adic valued functions on Shimura curves coming from classical origin. We prove a $p$-adic Waldspurger formula for these periods, generalizing the recent work of Bertolini, Darmon, and Prasanna. In pursuing such a formula, we construct a new anti-cyclotomic $p$-adic $L$-function of Rankin-Selberg type. At a character of positive weight, the $p$-adic $L$-function interpolates the central critical value of the complex Rankin-Selberg $L$-function. Its value at a Dirichlet character, which is outside the range of interpolation, essent…
Manin's relations characterize period polynomials of a modular form. In this paper, we propose a finite set of relations checked by bi-period polynomials of a couple of forms. Then we construct an Eisenstein part of depth 2 which essentially completed the set of bi-periods polynomials among polynomials canceled by the relations. By giving a computational description of that part, we prove a result of irrationality of ratio of the periods for some weights.
In this note we describe our personal encounters with the $p$-adic Stark conjecture. Gross describes the period between $1977$ and $1986$ when he came to formulate these conjectures (§1-8), and Dasgupta describes the period between $1998$ and $2021$, when he worked with others to finally prove them (§9-12).
Let $E$ be a modular elliptic curve over a totally real number field $F$. We prove the weak exceptional zero conjecture which links a (higher) derivative of the $p$-adic $L$-function attached to $E$ to certain $p$-adic periods attached to the corresponding Hilbert modular form at the places above $p$ where $E$ has split multiplicative reduction. Under some mild restrictions on $p$ and the conductor of $E$ we deduce the exceptional zero conjecture in the strong form (i.e.\ where the automorphic $p$-adic periods are replaced by the $\cL$-invariants of $E$ defined in terms of Tate periods) from a…
Let $f$ be a continuous ring endomorphism of $\mathbf{Z}_p[[x]]/\mathbf{Z}_p$ of degree $p.$ We prove that if $f$ acts on the tangent space at $0$ by a uniformizer and commutes with an automorphism of infinite order, then it is necessarily an endomorphism of a formal group over $\mathbf{Z}_p.$ The proof relies on finding a stable embedding of $\mathbf{Z}_p[[x]]$ in Fontaine's crystalline period ring with the property that $f$ appears in the monoid of endomorphisms generated by the Galois group of $\mathbf{Q}_p$ and crystalline Frobenius. Our result verifies, over $\mathbf{Z}_p,$ the height one…
Let F be a number field and N an integral ideal in its ring of integers. Let f be a modular newform over F of level Gamma0(N) with rational Fourier coefficients. Under certain additional conditions, Guitart-Masdeu-Sengun constructed a p-adic lattice which is conjectured to be the Tate lattice of an elliptic curve E whose L-function equals that of f. The aim of this note is to generalize this construction when the Hecke eigenvalues of f generate a number field of degree d >= 1, in which case the geometric object associated to f is expected to be, in general, an abelian variety A of dimension d.…
There is a notion of $p$-adic period coming from the crystalline Frobenius automorphism of the de Rham cohomology of an algebraic variety. In this paper, we consider sequences of $p$-adic periods, one for each prime. We study the sequences using motivic periods, and we formulate an analogue of the Grothendieck period conjecture.
Let E be a quadratic extension of a totally real number field. We construct Stickelberger elements for Hilbert modular forms of parallel weight 2 in anticyclotomic extensions of E. Extending methods developed by Dasgupta and Spieß from the multiplicative group to an arbitrary one-dimensional torus we bound the order of vanishing of these Stickelberger elements from below and, in the analytic rank zero situation, we give a description of their leading terms via automorphic L-invariants. If the field E is totally imaginary, we use the p-adic uniformization of Shimura curves to show the equality …
Complex periods are algebraic integrals over complex algebraic domains, also appearing as Feynman integrals and multiple zeta values. The Grothendieck-de Rham period isomorphisms for p-adic algebraic varieties defined via Monski-Washnitzer cohomology, is briefly reviewed. The relation to various p-adic analogues of periods are considered, and their relation to Buium-Manin arithmetic differential equations.
The Langlands Programme predicts that a weight 2 newform f over a number field K with integer Hecke eigenvalues generally should have an associated elliptic curve E_f over K. In our previous paper, we associated, building on works of Darmon and Greenberg, a p-adic lattice to f, under certain hypothesis, and implicitly conjectured that this lattice is commensurable with the p-adic Tate lattice of E_f . In this paper, we present this conjecture in detail and discuss how it can be used to compute, directly from f, an explicit Weierstrass equation for the conjectural E_f . We develop algorithms to…
Periodicity and Period-Length Bounds for Browkin $p$-Adic Continued Fractions
Dipartimento di Matematica, Università di Trento（特伦托大学数学系）；Dipartimento di Scienze Matematiche ”Giuseppe Luigi Lagrange”, Politecnico di Torino（都灵理工学院朱塞佩·路易吉·拉格朗日数学科学系）
We construct $p$-adic triple product $L$-functions that interpolate (square roots of) central critical $L$-values in the balanced region. Thus, our construction complements that of M. Harris and J. Tilouine. There are four central critical regions for the triple product $L$-functions and two opposite settings, according to the sign of the functional equation. In the first case, three of these regions are of interpolation, having positive sign; they are called the unbalanced regions and one gets three $p$% -adic $L$-functions, one for each region of interpolation (this is the Harris-Tilouine se…
We prove the Fargues-Rapoport conjecture for p-adic period domains: for a reductive group G over a p-adic field and a minuscule cocharacter μ of G, the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set B(G,μ) is fully Hodge-Newton decomposable.
We prove that all $p$-adic period domains (and their non-minuscule analogues) are geometrically connected. This answers a question of Hartl [Har13] and has interesting consequences to the geometry of Shimura and local Shimura varieties.
We prove the Fargues-Rapoport conjecture for p-adic period domains in the non-basic case with minuscule cocharacter. More precisely, we give a group theoretical criterion for the cases when the admissible locus and weakly admissible locus coincide.
This paper contributes to compute 2-adic complexity of two classes of Ding-Helleseth generalized cyclotomic sequences. Results show that 2-adic complexity of these sequences is good enough to resist the attack by the rational approximation algorithm.
We give a new and representation theoretic construction of $p$-adic interpolation series for central values of self-dual Rankin-Selberg $L$-functions for $\operatorname{GL}_2$ in dihedral towers of CM fields, using expressions of these central values as automorphic periods. The main novelty of this construction, apart from the level of generality in which it works, is that it is completely local. We give the construction here for a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real field corresponding to a $\mathfrak{p}$-ordinary Hilbert modular forms of parallel …
This paper is the augmented notes of a course I gave jointly with Laurent Berger in Rennes in 2014. Its aim was to introduce the periods rings B crys and B dR and state several comparison theorems between{é}tale and crystalline or de Rham cohomologies for p-adic varieties.
We give a construction of a new $p$-adic Maass-Shimura operator defined on an affinoid subdomain of the preperfectoid $p$-adic universal cover $\mathcal{Y}$ of a modular curve $Y$. We define a new notion of $p$-adic modular forms as sections of a certain sheaf $\mathcal{O}_Δ$ of "nearly rigid functions" which transform under the action of subgroups of the Galois group $\mathrm{Gal}(\mathcal{Y}/Y)$ by $\mathcal{O}_Δ^{\times}$-valued weight characters. This extends Katz's notion of $p$-adic modular forms as functions on the Igusa tower $Y^{\mathrm{Ig}}$; indeed we may recover Katz's theory by re…
Let A be a modular elliptic curve over a totally real field F, and let E/F be a totally imaginary quadratic extension. In the event of exceptional zero phenomenon, we prove a formula for the derivative of the multivariable anticyclotomic p-adic L-function attached to (A,E), in terms of the Hasse-Weil L-function and certain p-adic periods attached to the respective automorphic forms. Our methods are based on a new construction of the anticyclotomic p-adic L-function by means of the corresponding automorphic representation.
Dirichlet's version of Gauss's reduction theory for indefinite binary quadratic forms includes a map from Gauss-reduced forms to strings of natural numbers. It attaches to a form the minimal period of the continued fraction of a quadratic irrationality associated with the form. When Zagier developed his own reduction theory, parallel to Dirichlet's, he omitted an analogue of this map. We define a new map on Zagier-reduced forms that serves as this analogue. We also define a map from the set of Gauss-reduced forms into the set of Zagier-reduced forms that gives a near-embedding of the structure…
This is an introduction to $p$-adic geometry and $p$-adic analysis focusing on the theme of $p$-adic period mappings. We follow as closely as possible the development of the classical theory of complex period mappings, blending differential equations, group theory and geometry. The text ends with a detailed discussion of $p$-adic triangle groups.
Let $p$ be an odd prime. Let $F$ be the function field of a $p$-adic curve. Let $A$ be a central simple algebra of period 2 over $F$ with an involution $σ$. There are known upper bounds for the $u$-invariant of hermitian forms over $(A, σ)$. In this article we compute the exact values of the $u$-invariant of hermitian forms over $(A, σ)$.
This work is a study of $p$-adic multiple zeta values at roots of unity ($p$MZV$μ_{N}$'s), the $p$-adic periods of the crystalline pro-unipotent fundamental groupoid of $(\mathbb{P}^{1} - \{0,μ_{N},\infty\})/ \mathbb{F}_{q}$. The main tool is new objects which we call $p$-adic pro-unipotent harmonic actions. In this part IV we define and study $p$-adic analogues of some elementary complex analytic functions which interpolate multiple zeta values at roots of unity such as the multiple zeta functions. The indices of $p$MZV$μ_{N}$'s involve sequences of positive integers ; in this IV-1, by consid…
We prove equality of the various $p$-adic period morphisms for smooth, not necessarily proper, schemes. We start with showing that the $K$-theoretical uniqueness criterium we had found for proper smooth schemes extends to proper finite simplicial schemes in the good reduction case and to cohomology with compact support in the semistable reduction case. It yields the equality of the period morphisms for cohomology with compact support defined using the syntomic, almost étale, and motivic constructions. We continue with showing that the $h$-cohomology period morphism agrees with the syntomic and…
We construct the three-variable p-adic triple product L-functions attached to Hida families of ellptic newforms and prove the explicit interpolation formulae at all critical specializations by establishing explicit Ichino's formulae for the trilinear period integrals of automorphic forms. Our formulae perfectly fit the conjectural shape of p-adic L-functions predicted by Coates and Perrin-Riou. As an application, we prove the factorization of certain unbalanced p-adic triple product L-functions into a product of anticyclotomic p-adic L-functions for modular forms. By this factorization, we giv…
