We show that the decidability of an amplification of Hilbert's Tenth Problem in three variables implies the existence of uncomputably large integral points on certain algebraic curves. We obtain this as a corollary of a new positive complexity result: the Diophantine prefixes EAE and EEAE are generically decidable. This means, taking the former prefix as an example, that we give a precise geometric classification of those polynomials f in Z[v,x,y] for which the question...
``Does there exists a v in N such that for all x in N, there exists a y in N with f(v,x,y)=0?''
...may be undecidable, and we show that this set of polynomials is quite small in a rigourous sense. (The decidability of EAE was previously an open question.) The analogous result for the prefix EEAE is even stronger. We thus obtain a connection between the decidability of certain Diophantine problems, height bounds for points on curves, and the geometry of certain complex surfaces and 3-folds.
We present a rectilinearization theorem for p-adic semi-algebraic sets depending on parameters. As an application of our main theorem we present an alternative proof of a rationality result for parametric p-adic inte- grals, due to Denef.
We prove a transcendence theorem concerning values of holomorphic maps from a disk to a quasi-projective variety over $\overline{\mathbf{Q}}$ that are integral curves of some algebraic vector field (defined over $\overline{\mathbf{Q}}$). These maps are required to satisfy some integrality property, besides a growth condition and a strong form of Zariski-density that are natural for integral curves of algebraic vector fields. This result generalizes a theorem of Nesterenko concerning algebraic independence of values of the Eisenstein series $E_2,E_4,E_6$. The main technical improvement in our a…
In this paper consisting of two parts, we study the integral of a logarithmic differential form on a compact semi-algebraic set in R^n or C^n. In Part I, we prove the convergence of the integral when the semi-algebraic set satisfies allowability (or admissibility), a condition on the dimension of the intersection of the set and the pole divisor of the differential form.
In this paper we prove certain algebraic identities, which correspond to differentiations of the shuffle relation, the stuffle relation, and the relations which arise from Möbius transformations of iterated integrals. These formulas provide fundamental and useful tools in the study of iterated integrals on a punctured projective line.
Following the work of Gangl, Goncharov and Levin in [GGL], we will give a combinatorial framework for motivic study of iterated integrals on the affine line. We will show that under a certain genericity condition these combinatorial objects yield to elements in the motivic Hopf algebra constructed in Bloch-Kriz [BK]. It will be shown that the Hodge realization of these elements coincides with the Hodge structure induced from the fundamental torsor of path of punctured affine line.
A classical theorem of Siegel asserts that the set of S-integral points of an algebraic curve C over a number field is finite unless C has genus 0 and at most two points at infinity. In this paper we give necessary and sufficient conditions for C to have infinitely many S-integral points.
We show that for a variety which admits a quasi-finite period map, finiteness (resp.~non-Zariski-density) of $S$-integral points implies finiteness (resp.~non-Zariski-density) of points over all $\mathbb{Z}$-finitely generated integral domains of characteristic zero. Our proofs rely on foundational results in Hodge theory due to Deligne, Griffiths, and Schmid, and Bakker-Brunebarbe-Tsimerman. We give straightforward applications to Shimura varieties, locally symmetric varieties, the moduli space of smooth hypersurfaces in projective space, and the moduli of smooth divisors in an abelian variet…
In an earlier paper, the notion of integrality known from algebraic number fields and fields of algebraic functions has been extended to D-finite functions. The aim of the present paper is to extend the notion to the case of P-recursive sequences. In order to do so, we formulate a general algorithm for finding all integral elements for valued vector spaces and then show that this algorithm includes not only the algebraic and the D-finite cases but also covers the case of P-recursive sequences.
Let $K$ be a number field of degree $n$ with ring of integers $O_K$. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $h\in K[X]$ maps every element of $O_K$ of degree $n$ to an algebraic integer, then $h(X)$ is integral-valued over $O_K$, that is $h(O_K)\subset O_K$. A similar property holds if we consider the set of all algebraic integers of degree $n$ and a polynomial $f\in\mathbb{Q}[X]$: if $f(α)$ is integral over $\mathbb{Z}$ for every algebraic integer $α$ of degree $n$, then $f(β)$ is integral over $\mathbb{Z}$ …
Given systems of two (inhomogeneous) quadratic equations in four variables, it is known that the Hasse principle for integral points may fail. Sometimes this failure can be explained by some integral Brauer-Manin obstruction. We study the existence of a non-trivial algebraic part of the Brauer group for a family of such systems and show that the failure of the integral Hasse principle due to an algebraic Brauer-Manin obstruction is rare, as for a generic choice of a system the algebraic part of the Brauer-group is trivial. We use resolvent constructions to give quantitative upper bounds on the…
We study the algebra $\mathcal{I}^{QM}$ of iterated integrals of quasimodular forms for $\operatorname{SL}_2(\mathbb{Z})$, which is the smallest extension of the algebra $QM_{\ast}$ of quasimodular forms, which is closed under integration. We prove that $\mathcal{I}^{QM}$ is a polynomial algebra in infinitely many variables, given by Lyndon words on certain monomials in Eisenstein series. We also prove an analogous result for the $M_{\ast}$-subalgebra $\mathcal{I}^{M}$ of $\mathcal{I}^{QM}$ of iterated integrals of modular forms.
Sketch of proof of a theorem relating the two subjects of the title. It can be thought as an extension of results of Landau for the classical hypergeometric function. It relies on the characterization of algebraic hypergeometric functions of Beukers and Heckman. In the process we also show that a variant of a classical construction of Bezout (producing a quadratic form, the Bezoutian, out of two polynomials in one variable) gives the Hermitian form fixed by the monodromy group, up to scaling.
For an elliptic curve $E$ defined over a field $k\subset \mathbb C$, we study iterated path integrals of logarithmic differential forms on $E^\dagger$, the universal vectorial extension of $E$. These are generalizations of the classical periods and quasi-periods of $E$, and are closely related to multiple elliptic polylogarithms and elliptic multiple zeta values. Moreover, if $k$ is a finite extension of $\mathbb Q$, then these iterated integrals along paths between $k$-rational points are periods in the sense of Kontsevich--Zagier.
We study the algebraic and analytic structure of Feynman integrals by proposing an operation that maps an integral into pairs of integrals obtained from a master integrand and a corresponding master contour. This operation is a coaction. It reduces to the known coaction on multiple polylogarithms, but applies more generally, e.g. to hypergeometric functions. The coaction also applies to generic one-loop Feynman integrals with any configuration of internal and external masses, and in dimensional regularization. In this case, we demonstrate that it can be given a diagrammatic representation pure…
Let $Z \to X$ be a finite branched Galois cover of normal projective geometrically integral varieties of dimension $d \geq 2$ over a perfect field $k$. For such a cover, we prove a Chebotarev-type density result describing the decomposition behaviour of geometrically integral Cartier divisors. As an application, we classify Galois covers among all finite branched covers of a given normal geometrically integral variety $X$ over $k$ by the decomposition behaviour of points of a fixed codimension $r$ with $0 < r < \dim X$.
This is a sequel to our previous paper (joint with Furusho). It will give a more natural framework for constructing elements in the Hopf algebra of framed mixed Tate motives according to Bloch and Kriz. This framework allows us to extend our previous results to interpret all multiple zeta values (including the divergent ones) and the multiple polylogarithms in one variable as elements of this Hopf algebra. It implies that the pro-unipotent completion of the torsor of paths on projective line minus three points, is a mixed Tate motive in the sense of Bloch-Kriz. Also It allows us to interpret t…
We consider conjugacy of integral matrices by elements in $\text{GL}_{n}(R)$ for certain rings $R$ with subring $\mathbb{Z}$. We note that a Hasse principal does not hold in the context of matrix conjugacy because matrices which are $\text{GL}_{n}(\mathbb{Z}_p)$-conjugate for all $p$ are not necessarily $\text{GL}_{n}(\mathbb{Z})$-conjugate. By a theorem of Guralnick, we know that integral $n \times n$ matrices are $\text{GL}_{n}(\mathbb{Z}_p)$-conjugate for all primes $p$ if and only if they are conjugate by an element in $\text{GL}_{n}(E)$ for some algebraic integral extension $E$ of $\mathb…
We develop an explicit algebriac de Rham theory for relative completion of $\mathrm{SL}_2(\mathbb{Z})$. This allows the construction of iterated integrals involving modular forms of the second kind, generalizing iterated integrals of holomorphic modular forms that were previously studied by Brown and Manin. These newly constructed iterated integrals provide all 'multiple modular values' defined by Brown.
We construct a finite subgroup of Brauer-Manin obstruction for detecting the existence of integral points on integral models of homogeneous spaces of linear algebraic groups of multiplicative type. As application, the strong approximation theorem for linear algebraic groups of multiplicative type is established. Moreover, the sum of two integral squares over some quadratic fields is discussed.
In this paper we construct arbitrarily large families of smooth projective varieties and closed Riemannian manifolds that share many algebraic and analytic invariants. For instance, every non-arithmetic, closed hyperbolic $3$--manifold admits arbitrarily large collections of non-isometric finite covers which are strongly isospectral, length isospectral, and have isomorphic integral cohomology where the isomorphisms commute with restriction and co-restriction. We can also construct arbitrarily large collections of pairwise non-isomorphic smooth projective surfaces where these isomorphisms in co…
Under suitable, fairly weak hypotheses on an elliptic curve $E/\mathbb{Q}$ and a primitive non-trivial Dirichlet character $χ$, we show that the algebraic $L$-value $\mathscr{L}(E,χ)$ at $s=1$ is an algebraic integer. For instance, for semistable curves $\mathscr{L}(E,χ)$ is integral whenever $E$ admits no isogenies defined over $\mathbb{Q}$. Moreover we give examples illustrating that our hypotheses are necessary for integrality to hold.
We prove that the coefficients of certain weight -1/2 harmonic Maass forms are traces of singular moduli for weak Maass forms. To prove this theorem, we construct a theta lift from spaces of weight -2 harmonic weak Maass forms to spaces of weight -1/2 vector-valued harmonic weak Maass forms on Mp_2(Z), a result which is of independent interest. We then prove a general theorem which guarantees (with bounded denominator) when such Maass singular moduli are algebraic. As an example of these results, we derive a formula for the partition function p(n) as a finite sum of algebraic numbers which lie…
We describe an algorithm which verifies whether linear algebraic cycles of the Fermat variety generate the lattice of Hodge cycles. A computer implementation of this confirms the integral Hodge conjecture for quartic and quintic Fermat fourfolds. Our algorithm is based on computation of the list of elementary divisors of both the lattice of linear algebraic cycles, and the lattice of Hodge cycles written in terms of vanishing cycles, and observing that these two lists are the same.
Let $K = \Q(θ)$ be an algebraic number field with $θ$ satisfying an irreducible polynomial $x^{9} - a$ over the field $\Q$ of rationals and $\Z_K$ denote the ring of algebraic integers of $K$. In this article, we provide the exact power of each prime which divides the index of the subgroup $\Z[θ]$ in $\Z_K$. Further, we give a $p$-integral basis of $K$ for each prime $p$. These $p$-integral bases lead to a construction of an integral basis of $K$ which is illustrated with examples.
We prove a cohomological formula for non-critical residues of degree eight automorphic $L$-functions of $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ in the spirit of Beilinson conjecture. We rely on the cohomological interpretation of an automorphic period integral and on the study of Novodvorsky's integral representation of the $L$-functions.
In this paper we construct new derived invariants with integral coefficients using the theory of motifs, and give several applications. Specifically, we obtain the following results: For complex algebraic surfaces, we prove that certain torsion in the abelianized fundamental group is a derived invariant. We prove that the collection of Hodge-Witt cohomology groups is a derived invariant. In particular, Hodge-Witt reduction and ordinary reduction are preserved by derived equivalence when the characteristic is sufficiently large. Finally, using the techniques of non-commutative algebraic geometr…
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.
Feynman integrals are central to all calculations in perturbative Quantum Field Theory. They often give rise to iterated integrals of dlog-forms with algebraic arguments, which in many cases can be evaluated in terms of multiple polylogarithms. This has led to certain folklore beliefs in the community stating that all such integrals evaluate to polylogarithms. Here we discuss a concrete example of a double iterated integral of two dlog-forms that evaluates to a period of a cusp form. The motivic versions of these integrals are shown to be algebraically independent from all multiple polylogarit…
We define the doubling zeta integral for smooth families of representations of classical groups. Following this we prove a rationality result for these zeta integrals and show that they satisfy a functional equation. Moreover, we show that there exists an apropriate normalizing factor which allows us to construct $γ$-factors for smooth families out of the functional equation. We prove that under certain hypothesis, specializing this $γ$-factor at a point of the family yields the $γ$-factor defined by Piateski-Shapiro and Rallis.
We prove that only finitely many $j$-invariants of elliptic curves with complex multiplication are algebraic units. A rephrased and generalized version of this result resembles Siegel's Theorem on integral points of algebraic curves.
A result of Habegger shows that there are only finitely many singular moduli such that $j$ or $j-α$ is an algebraic unit. The result uses Duke's Equidistribution Theorem and is thus not effective. For a fixed $j$-invariant $α\in \bar{\mathbb{Q}}$ of an elliptic curve without complex multiplication, we prove that there are only finitely many singular moduli $j$ such that $j-α$ is an algebraic unit. The difference to the work of Habegger is that we give explicit bounds.
Integer relation algorithms can convert numerical results for Feynman integrals to exact evaluations, when one has reason to suspect the existence of reductions to linear combinations of a basis, with rational or algebraic coefficients. Once a tentative reduction is obtained, confidence in its validity is greatly increased by computing more decimal digits of the terms and verifying the stability of the result. Here we give examples of how the PSLQ and LLL algorithms have yielded remarkable reductions of Feynman integrals to multiple polylogarithms and to the periods and quasi-periods of modula…
We compare the behaviour of entire curves and integral sets, in particular in relation to locally trivial fiber bundles, algebraic groups and finite ramified covers over semi-abelian varieties.
We observe how certain distribution on $\mathrm{GL}(n)$ that generates Godement-Jacquet $γ$-factors appears naturally in the C*-algebraic normalization process for standard intertwining integrals on $\mathrm{SL}(n+1)$.
By Northcott's Theorem there are only finitely many algebraic points in affine $n$-space of fixed degree over a given number field and of height at most $X$. For large $X$ the asymptotics of these cardinalities have been investigated by Schanuel, Schmidt, Gao, Masser and Vaaler, and the author. In this paper we study the case where the coordinates of the points are restricted to algebraic integers, and we derive the analogues of Schanuel's, Schmidt's, Gao's and the author's results.
For a fixed $j$-invariant $j_0$ of an elliptic curve without complex multiplication we bound the number of $j$-invariants $j$ that are algebraic units and such that elliptic curves corresponding to $j$ and $j_0$ are isogenous. Our bounds are effective. We also modify the problem slightly by fixing a singular modulus $α$ and looking at all $j$ such that $j-α$ is an algebraic unit and such that elliptic curves corresponding to $j$ and $j_0$ are isogenous. The number of such $j$ can again be bounded effectively.
