We introduce the p-adic analogue of Arakelov intersection theory on arithmetic surfaces. The intersection pairing in an extension of the p-adic height pairing for divisors of degree 0 in the form described by Coleman and Gross. It also uses Coleman integration and is related to work of Colmez on p-adic Green functions. We introduce the p-adic version of a metrized line bundle and define the metric on the determinant of its cohomology in the style of Faltings. It is possible to prove in this theory analogues of the Adjunction formula and the Riemann-Roch formula.
We prove that the p-adic height pairing of Nekovar, considered for algebraic curves, gives the p-adic height pairing of Coleman and Gross, defined using Coleman integration.
We give a direct proof that the Mazur-Tate and Coleman-Gross heights on elliptic curves coincide. The main ingredient is to extend the Coleman-Gross height to the case of divisors with non-disjoint support and, doing some $p$-adic analysis, show that, in particular, its component above $p$ gives, in the special case of an ordinary elliptic curve, the $p$-adic sigma function. We use this result to give a short proof of a theorem of Kim characterizing integral points on elliptic curves in some cases under weaker assumptions. As a further application, we give new formulas to compute double Colema…
In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the ``closed fibers at infinity''. Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody endowed with a Schottky uniformization. In this paper we consider arithmetic surfaces over the ring of integers in a number field, with fibers of genus $g\geq 2$. We use Connes' theory of spectral triples to relate the hyperbolic geometry of the handlebody to Deninger's Archimedean cohomology…
We describe an algorithm to compute the local component at p of the Coleman-Gross p-adic height pairing on divisors on hyperelliptic curves. As the height pairing is given in terms of a Coleman integral, we also provide new techniques to evaluate Coleman integrals of meromorphic differentials and present our algorithms as implemented in Sage.
We extend the result of a previous work to the case of curves with semi-stable reduction. In this case, one can replace Coleman integration by Vologodsky integration to extend the Coleman-Gross definition of a $p$-adic height pairing. we show that this pairing still equals the one defined by Nekovar.
The Coleman integral is a $p$-adic line integral that plays a key role in computing several important invariants in arithmetic geometry. We give an algorithm for explicit Coleman integration on curves, using the algorithms of the second author to compute the action of Frobenius on $p$-adic cohomology. We present a collection of examples computed with our implementation. This includes integrals on a genus 55 curve, where other methods do not currently seem practical.
In this paper, we aimed at constructing a two-variable Coleman map for a given $p$-adic family of eigen cuspforms with a fixed non-zero slope (Coleman family). A Coleman map is a machinary which transforms a hypothetical $p$-adic family of zeta elements to a $p$-adic $L$-function. The result would be a non-ordinary generalization of a two-variable Coleman map for a given Hida deformation obtained by the second-named author.
In this article, we discuss Iwasawa Main Conjecture for $p$-adic families of elliptic modular cuspforms. After the overview on the situation of the ordinary case of Hida family, we will introduce a Coleman map for Coleman family for the non-ordinary case (Coleman family) which was obtained as a joint work with Filippo Nuccio [NO16] and we give some results on Iwasawa Theory for Coleman families as applications of . First, we give a construction of a two-variable $p$-adic $L$-function for a Coleman family thanks to the ingredients given in [NO16] (Theorem 5.1). Combining this result and Coleman…
The Coleman-Ihara formula expresses Soule's $p$-adic characters restricted to $p$-local Galois group as the Coates-Wiles homomorphism multiplied by $p$-adic $L$-values at positive integers. In this paper, we show an analogous formula that $\ell$-adic polylogarithmic characters for $\ell=p$ restrict to the Coates-Wiles homomorphism multiplied by Coleman's $p$-adic polylogarithms at any roots of unity of order prime to $p$.
Results in $p$-adic transcendence theory are applied to two problems in the Chabauty-Coleman method. The first is a question of McCallum and Poonen regarding repeated roots of Coleman integrals. The second is to give lower bounds on the $p$-adic distance between rational points in terms of the heights of a set of Mordell-Weil generators of the Jacobian. We also explain how, in some cases, a conjecture on the 'Wieferich statistics' of Jacobians of curves implies a bound on the height of rational points of curves of small rank, in terms of the usual invariants of the curve and the height of Mord…
Coleman's theory of p-adic integration figures prominently in several number-theoretic applications, such as finding torsion and rational points on curves, and computing p-adic regulators in K-theory (including p-adic heights on elliptic curves). We describe an algorithm for computing Coleman integrals on hyperelliptic curves, and its implementation in Sage.
Coleman calculated the absolute Frobenius on Fermat curves explicitly. In this paper we show that a kind of $p$-adic continuity implies a large part of his formula. To do this, we study a relation between functional equations of the ($p$-adic) gamma function and monomial relations on ($p$-adic) CM-periods.
In this paper, we introduce an algorithm for computing p-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally; and abelian integrals with desirable number-theoretic properties. By covering a bad reduction hyperelliptic curve by annuli and basic wide open sets, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on good reduction hyperelliptic curves. These are due to Balakrishnan, Bradshaw, and Kedlaya, and to Balakrishnan and Besser fo…
We will introduce a regularization for $p$-adic multiple zeta values and show that the generalized double shuffle relations hold. This settles a question raised by Deligne, given as a project in Arizona winter school 2002. Our approach is to use the theory of Coleman functions on the moduli space of genus zero curves with marked points and its compactification. The main ingredients are the analytic continuation of Coleman functions to the normal bundle of divisors at infinity and definition of a special tangential base point on the moduli space.
This paper is a sequel to our earlier paper "Wach modules and Iwasawa theory for modular forms" (arXiv: 0912.1263), where we defined a family of Coleman maps for a crystalline representation of the Galois group of Qp with nonnegative Hodge-Tate weights. In this paper, we study these Coleman maps using Perrin-Riou's p-adic regulator L_V. Denote by H(Γ) the algebra of Qp-valued distributions on Γ= Gal(Qp(μ(p^\infty) / Qp). Our first result determines the H(Γ)-elementary divisors of the quotient of D_{cris}(V) \otimes H(Γ) by the H(Γ)-submodule generated by (ϕ* N(V))^{ψ= 0}, where N(V) is the Wac…
Nearly higher Coleman theory and p-adic L-functions for $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ and $\mathrm{GSp}(4) \times \mathrm{GL}(2) \times \mathrm{GL}(2)$
We describe a more efficient algorithm to compute p-adic Coleman integrals on odd degree hyperelliptic curves for large primes p. The improvements come from using fast linear recurrence techniques when reducing differentials in Monsky-Washnitzer cohomology, a technique introduced by Harvey arXiv:math/0610973 when computing zeta functions. The complexity of our algorithm is quasilinear in $\sqrt p$ and is polynomial in the genus and precision. We provide timings comparing our implementation with existing approaches.
In this paper, we study $λϕ^4$ scalar field theory defined on the unramified extension of p-adic numbers ${\mathbb Q}_{p^n}$. For different ``space-time'' dimensions $n$, we compute one-loop quantum corrections to the effective potential. Surprisingly, despite the unusual properties of non-Archimedean geometry, the Coleman-Weinberg potential of p-adic field theory has a structure very similar to that of its real cousin. We also study two formal limits of the effective potential, $p \rightarrow 1$ and $p \rightarrow \infty$. We show that the $p\rightarrow 1$ limit allows to reconstruct the cano…
We give the first explicit examples beyond the Chabauty-Coleman method where Kim's nonabelian Chabauty program determines the set of rational points of a curve defined over $\mathbb{Q}$ or a quadratic number field. We accomplish this by studying the role of $p$-adic heights in explicit nonabelian Chabauty.
The purpose of this paper is to prove integrality for certain $p$-adic iterated Coleman integrals. As underlying geometry we will take the complement of a divisor $D\subset X$ with good reduction, where $X$ is the projective line or an elliptic curve over the Witt vectors of a perfect characteristic $p$ field. As a corollary we prove a lower bound for the valuations of $p$-adic multiple zeta values.
For a prime number $p\ge 5$, we consider three classical cusp eigenforms $f_j(z)$ of weights $k_1, k_2, k_3$, of conductors $N_1, N_2, N_3$, and of nebentypus characters $ψ_j \bmod N_j$. According to H.Hida and R.Coleman, one can include each $f_j$ into a {$p$-adic analytic family} $k_j \mapsto \{f_{j,k_j}\}$ of cusp eigenforms $f_{j,k_j}$ of weights $k_j$ in such a way that $f_{j,k_j}=f_j$, and that all their Fourier coefficients $a_n(f_{j, k_j})$ are given by certain $p$-adic analytic functions $k_j{}\mapsto a_{n, j}(k_j{})$. The purpose of this paper is to describe a four variable $p$-adic …
The Chabauty-Coleman method is a $p$-adic method for finding all rational points on curves of genus $g$ whose Jacobians have Mordell-Weil rank $r < g$. Recently, Edixhoven and Lido developed a geometric quadratic Chabauty method that was adapted by Spelier to cover the case of geometric linear Chabauty. We compare the geometric linear Chabauty method and the Chabauty-Coleman method and show that geometric linear Chabauty can outperform Chabauty-Coleman in certain cases. However, as Chabauty-Coleman remains more practical for general computations, we discuss how to strengthen Chabauty-Coleman t…
Let p be a rational prime and N a positive integer which is prime to p. Let W be the p-adic weight space for GL_{2,Q}. Let C_N be the p-adic Coleman-Mazur eigencurve of tame level N. In this paper, we prove that any irreducible component of C_N which is of finite degree over W is in fact finite over W. Combined with an argument of Chenevier and a conjecture of Coleman-Mazur-Buzzard-Kilford (which has been proven in special cases, and for general quaternionic eigencurves) this shows that the only finite degree components of the eigencurve are the ordinary components.
We discuss modular forms as objects of computer algebra and as elements of certain p-adic Banach modules. Problem-solving approach in number theory is discussed which is based on the use of generating functions and their links with modular forms. In particular, the critical values of various L-functions of modular forms produce non-trivial but computable solutions of arithmetical problems. Namely, for a prime number $p\ge 5$, we consider three classical cusp eigenforms $$f_j(z)=\sum_{n=1}^\infty a_{n,j}e(nz)\in \Sr_{k_j}(N_j, ψ_j),\ (j=1, 2,3)$$ of weights k_1, k_2, k_3, of conductors N_1, N_2…
The eigencurve is a powerful tool introduced by Coleman and Mazur to study $p$-adic families of overconvergent modular forms. In this article, we introduce an analogous set of tools for understanding families of "overconvergent" $p$-adic representations of $π_1(X)$, where $X$ is a smooth affine variety over a finite field of characteristic $p$. Our main theorem is a trace formula relating the $L$-function of such a family to the geometry of a sequence of associated eigenvarieties. In the case of a single $p$-adic representation, our result reduces to the well known trace formula of Monsky. We …
CM Elliptic Analogue of the Coleman--Ihara Formula
Graduate School of Science and Technology, Keio University（庆应义塾大学理工学部）；RIKEN Center for Advanced Intelligence Project (AIP)（理化学研究所先进智能研究中心）；Research Institute for Mathematical Sciences (RIMS), Kyoto University（京都大学数理科学研究科）
Our main aim in this paper is to give a foundation of the theory of $p$-adic multiple zeta values. We introduce (one variable) $p$-adic multiple polylogarithms by Coleman's $p$-adic iterated integration theory. We define $p$-adic multiple zeta values to be special values of $p$-adic multiple polylogarithms. We consider the (formal) $p$-adic KZ equation and introduce the $p$-adic Drinfel'd associator by using certain two fundamental solutions of the $p$-adic KZ equation. We show that our $p$-adic multiple polylogarithms appear as coefficients of a certain fundamental solution of the $p$-adic KZ…
We develop a variant of Coleman and Perrin Riou's methods giving, for a de Rham $p$-adic Galois representation, a construction of $p$-adic $L$ functions from a compatible system of global elements. As a result, we construct analytic functions on an open set of the $p$-adic weight space containing all locally algebraic characters of large enough conductor. Applied to Kato's Euler system, this gives $p$-adic $L$-functions for elliptic curves with additive bad reduction and, more generally, for modular forms which are supercuspidal at $p$. In the case of dimension $2$, we prove a functional equat…
Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\geq 5$ and has complex multiplication by the full ring of integers $\mathcal{O}_K$ of $K$. In this paper, we construct $p$-adic analogues of the Eisenstein-Kronecker series for such elliptic curve as Coleman functions on the elliptic curve. We then prove $p$-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.
