A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory solutions can be defined in algebraically closed p-adic fields. As an application we show that for at least two arithmetic directions every elliptic curve possesses a non-zero arithmetic PDE Manin map of order 1; such maps do not exist in the arithmetic ODE case. Similarly we construct and study "genuinely PDE" differential modular forms. As further applications we derive a Theorem of the Kernel and a Reciprocity Theorem for arithmetic PDE Manin maps and also a finiteness Diophantine result for modular parameterizations. We also prove structure results for the spaces of "PDE differential modular forms defined on the ordinary locus." We also produce a system of differential equations satisfied by our PDE modular forms based on Serre and Euler operators.
We study the $Λ$-module structure of the ordinary parts of the arithmetic cohomology groups of modular Jacobians made out of various towers of modular curves. We prove that the ordinary parts of $Λ$-adic Selmer groups coming from two different towers have "almost same" $Λ$-module structures. We also prove the cotorsionness of $Λ$-adic Tate-Shafarevich group under mild assumptions.
In this paper, we prove that the derivative of the Iwasawa power series associated to p-adic L-functions of $\mathbb Q(ζ\_p)$ are not divisible by p. This extends previous results obtained by Ferrero and Washington in 1979.
We study the derived Hecke action at $p$ on the ordinary $p$-adic cohomology of arithmetic subgroups of semisimple groups $\mathrm G(\mathbb Q)$, i.e., we study the derived version of Hida's theory for ordinary Hecke algebras. This is the analog at $\ell=p$ of derived Hecke actions studied by Venkatesh in the tame case. We show that properties of the derived Hecke action at $p$ are related to deep conjectures in Galois cohomology which are higher analogs of the classical Leopoldt conjecture.
Let $B$ be a simple CM abelian variety over a CM field $E$, $p$ a rational prime. Suppose that $B$ has potentially ordinary reduction above $p$ and is self-dual with root number $-1$. Under some further conditions, we prove the generic non-vanishing of (cyclotomic) $p$-adic heights on $B$ along anticyclotomic $\Z_{p}$-extensions of $E$. This provides evidence towards Schneider's conjecture on the non-vanishing of $p$-adic heights. For CM elliptic curves over $\Q$, the result was previously known as a consequence of work of Bertrand, Gross--Zagier and Rohrlich in the 1980s. Our proof is based o…
Let p be prime and Zpn the degree n unramified extension of the ring of p-adic integers Zp. In this paper we give an overview of some very fast algorithms for common operations in Zpn modulo p^N. Combining existing methods with recent work of Kedlaya and Umans about modular composition of polynomials, we achieve quasi-linear time algorithms in the parameters n and N, and quasi-linear or quasi-quadratic time in log p, for most basic operations on these fields, including Galois conjugation, Teichmuller lifting and computing minimal polynomials.
We extend the work of Ash and Stevens [Ash-Stevens 97] on p-adic analytic families of p-ordinary arithmetic cohomology classes for GL(N,Q) by introducing and investigating the concept of p-adic rigidity of arithmetic Hecke eigenclasses. An arithmetic eigenclass is said to be "rigid" if (modulo twisting) it does not admit a nontrivial p-adic deformation containing a Zariski dense set of arithmetic specializations. This paper develops tools for explicit investigation into the structure of eigenvarieties for GL(N). We use these tools to prove that known examples of non-sefldual cohomological cusp…
We study the relationship between recent conjectures on slopes of overconvergent p-adic modular forms "near the boundary" of p-adic weight space. We also prove in tame level 1 that the coefficients of the Fredholm series of the U_p operator never vanish modulo p, a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo p using a modular interpretation of the mod p reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.
We show the non-existence of sections of arithmetic fundamental groups of open p-adic annuli of small radii, this implies the non-existence of sections of arithmetic fundamental groups of formal boundaries of formal germs of p-adic curves.
We prove the arithmetic fundamental lemma conjecture over a general $p$-adic field with odd residue cardinality $q\geq \dim V$. Our strategy is similar to the one used by the second author during his proof of the AFL over $\mathbb{Q}_p$ (arXiv:1909.02697), but only requires the modularity of divisor generating series on the Shimura variety (as opposed to its integral model). The resulting increase in flexibility allows us to work over an arbitrary base field. To carry out the strategy, we also generalize results of Howard (arXiv:1303.0545) on CM-cycle intersection and of Ehlen--Sankaran (arXiv…
Let K/Q be Galois and let eta in K* be such that the multiplicative Z[G]-module generated by eta is of Z-rank n.We define the local theta-regulators Delta\_p^theta(eta) in F\_p for the Q\_p-irreducible characters theta of G=Gal(K/Q). Let V\_theta be the theta-irreducible representation. A linear representation L^theta=delta.V\_theta is associated withDelta\_p^theta(eta) whose nullity is equivalent to delta$\ge$1 (Theorem 3.9). Each Delta\_p^theta(eta) yields Reg\_p^theta(eta) modulo p in the factorization $\prod$\_theta (Reg\_p^theta(eta))^phi(1) of Reg\_p^G(eta) := Reg\_p(eta)/p^[K : Q] (norm…
We establish a valuative version of Grothendieck's section conjecture for curves over p-adic local fields. The image of every section is contained in the decomposition subgroup of a valuation which prolongs the p-adic valuation to the function field of the curve.
Let $K$ be a discrete valuation field with perfect residue field, we study the functor from weakly admissible filtered $(φ,N,G_K)$-modules over $K$ to the isogeny category of Breuil-Kisin-Fargues $G_K$-modules. This functor is the composition of a functor defined by Fargues-Fontaine from weakly admissible filtered $(φ,N,G_K)$-modules to $G_K$-equivariant modifications of vector bundles over the Fargues-Fontaine curve $X_{FF}$, with the functor of Fargues-Scholze that between the category of admissible modifications of vector bundles over $X_{FF}$ and the isogeny category of Breuil-Kisin-Fargue…
Let $f_{\mathrm{new}}$ be a classical newform of weight $\geq 2$ and prime to $p$ level. We study the arithmetic of $f_{\mathrm{new}}$ and its unique $p$-stabilisation $f$ when $f_{\mathrm{new}}$ is $p$-irregular, that is, when its Hecke polynomial at $p$ admits a single repeated root. In particular, we study $p$-adic weight families through $f$ and its base-change to an imaginary quadratic field $F$ where $p$ splits, and prove that the respective eigencurves are both Gorenstein at $f$. We use this to construct a two-variable $p$-adic $L$-function over a Coleman family through $f$, and a three…
We study the $p$-adic (generalized) hypergeometric equations by using the theory of multiplicative convolution of arithmetic $\mathscr{D}$-modules. As a result, we prove that the hypergeometric isocrystals with suitable rational parameters have a structure of overconvergent $F$-isocrystals.
Let $p$ be a prime. We discuss $p$-adic properties of various arithmetical functions related to the coefficients of modular form and generating functions. Modular forms are considered as a tool of solving arithmetical problems. Examples of congruences between modular forms modulo $p$ and modulo $p^n$ are given, and the use of a computer for the study of modular forms is illistrated.
We propose in this paper an approach to Breuil's conjecture on a Langlands correspondence between $p$-adic Galois representations and representations of $p$-adic Lie groups in $p$-adic topological vector spaces. We suggest that Berthelot's theory of arithmetic $D$-modules should give a $p$-adic analogue of Kashiwara's theory of $D$-modules for real Lie groups i.e. it should give a realization of the $p$-adic representations of a $p$-adic Lie group as spaces of overconvergent solutions of arithmetic $D$-modules which will come equipped with an action of the Galois group. We shall discuss the ca…
By analyzing the coefficients of the power series defining the Kubota--Leopoldt $p$-adic $L$-function associated to the non-trivial character of a real quadratic field, we prove a congruence of Ankeny--Artin--Chowla-type for prime power modulus. Additionally, we show how some classical congruences relating Bernoulli numbers and Wilson quotients fit naturally into the theory of the $p$-adic Riemann zeta function.
Let $p$ be a prime. We discuss methods of solution of congruences modulo $p^n$ using $p$-adic numbers; these methods are similar to computations with real numbers (local methods). Examples of relations between local and global methods are given producing a passage from congruences to solutions in integers (global methods). The use of a computer is illistrated for the study of $p$-adic numbers and algebraic curves.
If one proposes to use the theory of Eisenstein cohomology to prove algebraicity results for the special values of automorphic L-functions as in my work with Harder for Rankin-Selberg L-functions, or its generalizations as in my work with Bhagwat for L-functions for orthogonal groups and independently with Krishnamurthy on Asai L-functions, then in a key step, one needs to prove that the normalised standard intertwining operator between induced representations for p-adic groups has a certain arithmetic property. The principal aim of this article is to address this particular local problem in t…
This paper investigates Galois branched covers of the open $p$-adic disc and their reductions to characteristic $p$. Using the field of norms functor of Fontaine and Wintenberger, we show that the special fiber of a Galois cover is determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations. As applications, we derive a criterion for good reduction in the abelian case, and give an arithmetic reformulation of the local Oort Conjecture concerning the liftability of cyclic covers of germs of curves.
Let $X$ be a proper, smooth, and geometrically connected curve of genus $g(X)\ge 1$ over a $p$-adic local field. We prove that there exists an effectively computable open affine subscheme $U\subset X$ with the property that $period (X)=1$, and $index (X)$ equals $1$ or $2$ (resp. $period(X)=index (X)=1$, assuming $period (X)=index (X)$), if (resp. if and only if) the exact sequence of the geometrically abelian fundamental group of $U$ splits. We compute the torsor of splittings of the exact sequence of the geometrically abelian absolute Galois group associated to $X$, and give a new characteri…
In this brief note, we consider p-adic unit roots or poles of L-functions of exponential sums defined over finite fields. In particular, we look at the number of unit roots or poles, and a congruence relation on the units. This raises a question in arithmetic mirror symmetry.
A main theme of the paper is a conjecture of Bloch-Kato on the image of $p$-adic regulator maps for a proper smooth variety $X$ over an algebraic number field $k$. The conjecture for a regulator map of particular degree and weight is related to finiteness of two arithmetic objects: One is the $p$-primary torsion part of the Chow group in codimension 2 of $X$. Another is an unramified cohomology group of $X$. As an application, for a regular model ${\mathscr X}$ of $X$ over the integer ring of $k$, we show an injectivity result on torsion of a cycle class map from the Chow group in codimension …
We provide lower bounds for p-adic valuations of multisums of factorial ratios which satisfy an Apéry-like recurrence relation: these include Apéry, Domb, Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In particular, we prove Beukers' conjectures on the p-adic valuation of Apéry numbers. Furthermore, we give an effective criterion for a sequence of factorial ratios to satisfy the p-Lucas property for almost all primes p.
We refine and extend previous constructions of $p$-adic $L$-functions for Rankin-Selberg convolutions on $\GL(n)\times\GL(n-1)$ for regular algebraic representations over totally real fields. We also prove an intrinsic functional equation for this $p$-adic $L$-function, which will be of interest in further study of its arithmetic properties.
We complete the picture of local and global arithmetic duality theorems for short complexes of finite Galois modules and tori over $p$-adic function fields. In view of the duality theorems, we deduce a $12$-term Poitou--Tate exact sequence which relates global Galois cohomology groups to restricted topological products of local Galois cohomology groups.
Deligne conjectured that a single l-adic lisse sheaf on a normal variety over a finite field can be embedded into a compatible system of l'-adic lisse sheaves with various l'. Drinfeld used Lafforgue's result as an input and proved this conjecture when the variety is smooth. We consider an analogous existence problem for a regular flat scheme over Z and prove some cases using Lafforgue's result and the work of Barnet-Lamb, Gee, Geraghty, and Taylor.
Let $k$ be a complete discretely valued field of equal characteristic $p > 0$ with possibly imperfect residue field and let $G_k$ be its Galois group. We prove that the conductors computed by the arithmetic ramification filtrations on $G_k$ coincide with the differential Artin conductors and Swan conductors of Galois representations of $G_k$. As a consequence, we give a Hasse-Arf theorem for arithmetic ramification filtrations in this case. As applications, we obtain a Hasse-Arf theorem for finite flat group schemes; we also give a comparison theorem between the differential Artin conductors a…
We extend to $p=2$ and $p=3$ the result of Gillard and Schneps which says that the $μ$-invariant of Katz $p$-adic $L$ functions attached to imaginary quadratic fields is zero for $p>3$. The arithmetic interpretation of this fact is given in the introduction.
Let $τ$ be the primitive Dirichlet character of conductor 4, let $χ$ be the primitive even Dirichlet character of conductor 8 and let $k$ be an integer. Then the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $2k+1$ and character $τ$ has slopes in the arithmetic progression ${2,4,...,2n,...}$, and the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $k$ and character $χ\cdot τ^k$ has slopes in the arithmetic progression ${1,2,...,n,...}$. We then show that the characteristic polynomials of the Hecke operators $U_2$ and $T_p$ acting on the space…
This paper studies geometric and spectral properties of $S$-adic shifts and their relation to continued fraction algorithms. These shifts are symbolic dynamical systems obtained by iterating infinitely many substitutions. Pure discrete spectrum for $S$-adic shifts and tiling properties of associated Rauzy fractals are established under a generalized Pisot assumption together with a geometric coincidence condition. These general results extend the scope of the Pisot substitution conjecture to the $S$-adic framework. They are applied to families of $S$-adic shifts generated by Arnoux-Rauzy as we…
We consider stacks of filtered phi-modules over rigid analytic spaces and adic spaces. We show that these modules parametrize p-adic Galois representations of the absolute Galois group of a p-adic field with varying coefficients over an open substack containing all classical points. Further we study a period morphism (defined by Pappas and Rapoport) from a stack parametrizing integral data and determine the image of this morphism.
A Sturmian sequence is an infinite nonperiodic string over two letters with minimal subword complexity. In two papers, the first written by Morse and Hedlund in 1940 and the second by Coven and Hedlund in 1973, a surprising correspondence was established between Sturmian sequences on one side and rotations by an irrational number on the unit circle on the other. In 1991 Arnoux and Rauzy observed that an induction process (invented by Rauzy in the late 1970s), related with the classical continued fraction algorithm, can be used to give a very elegant proof of this correspondence. This process, …
In this note, we study the arithmetic function $f : \mathbb{Z}_+^* \to \mathbb{Q}_+^*$ defined by $f(2^k \ell) = \ell^{1 - k}$ ($\forall k, \ell \in \mathbb{N}$, $\ell$ odd). We show several important properties about that function and then we use them to obtain some curious results involving the 2-adic valuation.
In this paper, we study arithmetic properties of weighted Catalan numbers. Previously, Postnikov and Sagan found conditions under which the $2$-adic valuations of the weighted Catalan numbers are equal to the $2$-adic valutations of the Catalan numbers. We obtain the same result under weaker conditions by considering a map from a class of functions to $2$-adic integers. These methods are also extended to $q$-weighted Catalan numbers, strengthening a previous result by Konvalinka. Finally, we prove some results on the periodicity of weighted Catalan numbers modulo an integer and apply them to t…
We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms. Specifically, we conjecture that for fixed N, m, and prime p with p not dividing N, there is only a finite number of reductions modulo p^m of normalized eigenforms on Γ_1(N). We consider various variants of our basic finiteness conjecture, prove a weak version of it, and give some numerical evidence.
Let M be an imaginary quadratic field, f a Hecke eigenform on GL2(Q) and πthe unitary base-change to M of the automorphic representation associated to f. Take a unitary arithmetic Hecke character χof M inducing the inverse of the central character of f. The celebrated formula of Waldspurger relates the square of an integral L(f) of f and χover the idele class group of M to the central critical value L(1/2,π\otimesχ). In this paper, we present a construction of a new p-adic L-function that interpolates L(f) over arithmetic characters for a cusp form f in the spirit of the landmark result of Kat…
For each prime p and each embedding of the multiplicative group of an algebraic closure of F_p as complex roots of unity, we construct a p-adic indecomposable representation of the integral BC-system as additive endomorphisms of the big Witt ring of an algebraic closure of F_p. The obtained representations are the p-adic analogues of the complex, extremal KMS states at zero temperature of the BC-system. The role of the Riemann zeta function, as partition function of the BC-system over complex numbers is replaced, in the p-adic case, by the p-adic L-functions and the polylogarithms whose 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 …
Using the description of Paileve' VI family of differential equations in terms of a universal elliptic curve, going back to R. Fuchs (cf. [Ma96]), we translate it into the realm of Arithmetic Differential Equations (cf. [Bu05]), where the role of derivative "in the $p$--adic direction" is played by a version of Fermat quotient
Let X be a smooth proper curve over a finite field of characteristic p. We prove a product formula for p-adic epsilon factors of arithmetic D-modules on X. In particular we deduce the analogous formula for overconvergent F-isocrystals, which was conjectured previously. The p-adic product formula is the equivalent in rigid cohomology of the Deligne-Laumon formula for epsilon factors in l-adic étale cohomology (for a prime l different from p). One of the main tools in the proof of this p-adic formula is a theorem of regular stationary phase for arithmetic D-modules that we prove by microlocal te…
S-arithmetic Khintchine-type theorem for products of non-degenerate analytic p-adic manifolds is proved for the convergence case. In the p-adic case the divergence part is also obtained.
We show that for arithmetic weights with a fixed finite order character, the slopes of $U_p$ (for $p=2$) acting on overconvergent Hilbert modular forms of level $U_0(4)$ are independent of the (algebraic part of the) weight and can be obtained by a simple recipe from the classical slopes in parallel weight $3$.
Let K be a complete discretely valued field of mixed characteristic (0, p) with possibly imperfect residue field. We prove a Hasse-Arf theorem for the arithmetic ramification filtrations on G_K, except possibly in the absolutely unramified and non-logarithmic case, or p=2 and logarithmic case. As an application, we obtain a Hasse-Arf theorem for filtrations on finite flat group schemes over O_K.
