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In this paper, we proved a local arithmetic Siegel-Weil formula for a U(1, 1)-Shimura variety at a ramified prime, a.k.a. a Kudla-Rapoport conjecture at a ramified case. The formula needs to be modified from the original Kudla-Rapoport conjecture. In the process, we also give an explicit decomposition of the special divisors of the Rapoport-Zink space of unitary type (1, 1) (Krämer model). A key ingredient is to relate the Rapoport-Zink space to the Drinfeld upper plane. [ABSTRACT FROM AUTHOR]

In this short note we correct the statement of the main result of [Trans. Amer. Math. Soc. 373 (2020), no. 3, 1909-1940]. That paper presented the rational cohomology ring of a toric arrangement by generators and relations. One of the series of relations given in the paper is indexed over the set circuits in the arrangement's arithmetic matroid. That series of relations should however be indexed over all sets X with |X| = rk(X)+1. Below we give the complete and correct presentation of the rational cohomology ring. [ABSTRACT FROM AUTHOR]

In this paper we provide the first examples of arithmetic hyperbolic 3–manifolds that are rational homology spheres and bound geometrically either compact or cusped hyperbolic 4–manifolds. [ABSTRACT FROM AUTHOR]

In this paper we introduce the local Nori fundamental group scheme of a reduced scheme or algebraic stack over a perfect field k. We give particular attention to the case of fields: to any field extension K/k we attach a pro-local group scheme over k. We show how this group has many analogies, but also some crucial differences, with the absolute Galois group. We propose two conjectures, analogous to the classical Neukirch-Uchida Theorem and Abhyankar Conjecture, providing some evidence in their favor. Finally we show that the local fundamental group of a normal variety is a quotient of the local fundamental group of an open, of its generic point (as it happens for the étale fundamental group) and even of any smooth neighborhood. [ABSTRACT FROM AUTHOR]

Zeta values in Tate algebras were introduced by Pellarin in 2012. They are generalizations of Carlitz zeta values and play an increasingly important role in function field arithmetic. In this paper we prove a conjecture of Pellarin on identities for these zeta values. The proof is based on arithmetic properties of Carlitz zeta values and an explicit formula for Bernoulli-type polynomials attached to zeta values in Tate algebras. [ABSTRACT FROM AUTHOR]

In this paper, we present a variant of the Balog-Szemerédi-Gowers theorem for the Vinogradov system. We then use our result to deduce a higher degree analogue of the sum-product phenomenon. [ABSTRACT FROM AUTHOR]

Let Γ0(n,N) denote the usual congruence subgroup of type Γ0and level N in SL(n,Z). Suppose for i = 1,2 that we have an irreducible odd n-dimensional Galois representation ρi attached to a homology Hecke eigenclass in H*(Γ0(n,Ni),Mi), where the level N_i and the weight and nebentype making up Miare as predicted by the Serre-style conjecture of Ash, Doud, Pollack, and Sinnott. We assume that n is odd, that N1N2 is squarefree, and that ρ1 ⊕ ρ2 is odd. We prove two theorems that assert that ρ1 ⊕ ρ2 is attached to a homology Hecke eigenclass in H*(Γ0(2n,N),M), where N and M are as predicted by the Serre-style conjecture. The first theorem requires the hypothesis that the highest weights of M1 and M2 are small in a certain sense. The second theorem requires the truth of a conjecture as to what degrees of homology can support Hecke eigenclasses with irreducible Galois representations attached, but no hypothesis on the highest weights of M1 and M2. This conjecture is known to be true for n = 3, so we obtain unconditional results for GL(6). A similar result for GL(4) appeared in an earlier paper. [ABSTRACT FROM AUTHOR]

A dominant rational self-map on a projective variety is called p-cohomologically hyperbolic if the p-th dynamical degree is strictly larger than other dynamical degrees. For such a map defined over \overline {\mathbb {Q}}, we study lower bounds of the arithmetic degrees, existence of points with Zariski dense orbit, and finiteness of preperiodic points. In particular, we prove that, if f is an 1-cohomologically hyperbolic map on a smooth projective variety, then (1) the arithmetic degree of a \overline {\mathbb {Q}}-point with generic f-orbit is equal to the first dynamical degree of f; and (2) there are \overline {\mathbb {Q}}-points with generic f-orbit. Applying our theorem to the recently constructed rational map with transcendental dynamical degree, we confirm that the arithmetic degree can be transcendental. [ABSTRACT FROM AUTHOR]

In this paper we prove that there is a direct relationship between Salem numbers and translation lengths of hyperbolic elements of arithmetic hyperbolic groups that are determined by a quadratic form over a totally real number field. As an application we determine a sharp lower bound for the length of a closed geodesic in a noncompact arithmetic hyperbolic n-orbifold for each dimension n. We also discuss a short geodesic conjecture, and prove its equivalence with Lehmer's conjecture for Salem numbers. [ABSTRACT FROM AUTHOR]

The authors note that certain height formulas in their paper had been proven earlier by Bertrand. [ABSTRACT FROM AUTHOR]

If in a given rank r, there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number \ell, there is an absolutely irreducible \ell-adic local system of the same rank, with the same determinant and monodromies at infinity, up to semi-simplification. A finitely presented group is said to be weakly integral with respect to a torsion character and a rank r if once there is an irreducible rank r complex linear representation with determinant isomorphic to this torsion character, then for any \ell, there is an absolutely irreducible one of rank r and determinant this given character, which is defined over \bar { \mathbb {Z}}_\ell. We prove that this property is a new obstruction for a finitely presented group to be the fundamental group of a smooth quasi-projective complex variety. The proofs rely on the arithmetic Langlands program via the existence of Deligne's companions (L. Lafforgue, Drinfeld) and the geometric Langlands program via de Jong's conjecture (Gaitsgory for \ell \ge 3). We also define weakly arithmetic complex local systems and show they are Zariski dense in the Betti moduli. Finally we show that our method gives an arithmetic proof of the Corlette-T. Mochizuki theorem, proved using tame pure imaginary harmonic metrics, which shows the pull-back by a morphism between two smooth complex algebraic varieties of a semi-simple complex local system is semi-simple. [ABSTRACT FROM AUTHOR]

We use recent advances in the local evaluation of Brauer elements to study the role played by odd torsion elements of the Brauer group in the arithmetic of diagonal quartic surfaces over arbitrary number fields. We show that over a local field if the order of the Brauer element is odd and coprime to the residue characteristic then the evaluation map it induces on the local points is constant. Over number fields we give a sufficient condition on the coefficients of the equation, which is mild and easy to check, under which the odd torsion does not obstruct weak approximation. We also note a systematic way to produce K3 surfaces over \mathbb {Q}_2 with good reduction and a non-trivial 2-torsion element of the Brauer group with Swan conductor zero. [ABSTRACT FROM AUTHOR]

We prove that cocompact arithmetic lattices in a simple Lie group are uniformly discrete if and only if Salem numbers are uniformly bounded away from 1. We also prove an analogous result for semisimple Lie groups. Finally, we shed some light on the structure of the bottom of the length spectrum of an arithmetic orbifold \Gamma \backslash X by showing the existence of a positive constant \delta (X)>0 such that squares of lengths of closed geodesics shorter than \delta must be pairwise linearly dependent over \mathbb {Q}. [ABSTRACT FROM AUTHOR]

We define a family of arithmetic zero cycles in the arithmetic Chow group of a modular curve X_0(N), for N>3 odd and squarefree, and identify the arithmetic degrees of these cycles as q-coefficients of the central derivative of a Siegel Eisenstein series of genus two. This parallels work of Kudla-Rapoport-Yang for Shimura curves. [ABSTRACT FROM AUTHOR]

We introduce a new arithmetic invariant for hermitian line bundles on arithmetic varieties. We use this invariant to measure the variation of the volume function with respect to the metric. We apply the theory developed here to the study of the arithmetic geometry of toric varieties. As an application, we obtain a generalized Hodge index theorem for hermitian line bundles which are not necessarily toric. When the metrics are toric, we recover some results due to Burgos, Phillippon, Sombra and Moriwaki. [ABSTRACT FROM AUTHOR]

In this article we address the following question, whose interest was recently renewed by problems arising in arithmetic dynamics: under which conditions does there exist a local biholomorphism between the Julia sets of two given one-dimensional rational maps? In particular we find criteria ensuring that such a local isomorphism is induced by an algebraic correspondence. This extends and unifies classical results due to Baker, Beardon, Eremenko, Levin, Przytycki and others. The proof involves entire curves and positive currents. [ABSTRACT FROM AUTHOR]

We establish an asymptotic formula for the weighted quantum variance of dihedral Maass forms on \Gamma _0(D) \backslash \mathbb H in the large eigenvalue limit, for certain fixed D. As predicted in the physics literature, the resulting quadratic form is related to the classical variance of the geodesic flow on \Gamma _0(D) \backslash \mathbb H, but also includes factors that are sensitive to underlying arithmetic of the number field \mathbb Q(\sqrt {D}). [ABSTRACT FROM AUTHOR]

Enumerating integral orbits in prehomogeneous vector spaces plays an important role in arithmetic statistics. We describe a method of proving subconvexity of the zeta function enumerating the integral orbits, illustrated by proving a subconvex estimate for the Shintani \zeta function enumerating class numbers of binary cubic forms. [ABSTRACT FROM AUTHOR]

We prove the expected algebraicity property for the critical values of character twists of the standard L-function associated to vector-valued holomorphic Siegel cusp forms of archimedean type (k_1, k_2, ..., k_n), where k_n \ge n+1 and all k_i are of the same parity. For the proof, we use an explicit integral representation to reduce to arithmetic properties of differential operators on vector-valued nearly holomorphic Siegel cusp forms. We establish these properties via a representation-theoretic approach. [ABSTRACT FROM AUTHOR]

Let G be a finite group and K a number field. We construct a G-extension E/F, with F of transcendence degree 2 over K, that specializes to all G-extensions of K_\mathfrak {p}, where \mathfrak {p} runs over all but finitely many primes of K. If furthermore G has a generic extension over K, we show that the extension E/F has the so-called Hilbert–Grunwald property. These results are compared to the notion of essential dimension of G over K, and its arithmetic analogue. [ABSTRACT FROM AUTHOR]

Using the discriminant modular form and the Noether formula it is possible to write the admissible self-intersection of the relative dualising sheaf of a semistable hyperelliptic curve over a number field or function field as a sum, over all places, of a certain adelic invariant $ \chi$, based on the arithmetic of symmetric roots. We propose the conjecture that the invariant $ \chi$ introduced in a recent paper by S.-W. Zhang. This conjecture is true in the genus $ 2$ over number fields. [ABSTRACT FROM AUTHOR]

The arithmetic nature of Euler's constant $gamma $ is still unknown and even getting good rational approximations to it is difficult. Recently, Aptekarev managed to find a third order linear recurrence with polynomial coefficients which admits two rational solutions $a_n$ and $b_n$ such that $a_n/b_n$ converges sub-exponentially to~$gamma $, viewed as $-Gamma '(1)$, where $Gamma $ is the usual Gamma function. Although this is not yet enough to prove that $gamma not in mathbb {Q}$, it is a major step in this direction. par In this paper, we present a different, but related, approach based on simultaneous Padé approximants to Euler's functions, from which we construct and study a new third order recurrence that produces a sequence in $mathbb {Q}(z)$ whose height grows like the factorial and that converges sub-exponentially to $log (z)+gamma $ for any complex number $zin mathbb {C}setminus (-infty ,0]$, where $log $ is defined by its principal branch. We also show how our approach yields in theory rational approximations of numbers related to $Gamma ^{(s)}(1)$ for any integer $sge 1$. In particular, we construct a sixth order recurrence which provides simultaneous rational approximations (of factorial height) converging sub-exponentially to the numbers $gamma $ and $Gamma ''(1)-2Gamma '(1)^2=zeta (2)-gamma ^2.$ [ABSTRACT FROM AUTHOR]

We study the average behaviour of the Iwasawa invariants for the Selmer groups of elliptic curves, setting out new directions in arithmetic statistics and Iwasawa theory. [ABSTRACT FROM AUTHOR]

Let E/Q be an optimal elliptic curve, −D be a negative fundamental discriminant coprime to the conductor N of E/Q and let E−D/Q be the twist of E/Q by −D. A conjecture of Agashe predicts that if E−D/Q has analytic rank 0, then the square of the order of the torsion subgroup of E−D/Q divides the product of the order of the Shafarevich-Tate group of E−D/Q and the orders of the arithmetic component groups of E−D/Q, up to a power of 2. This conjecture can be viewed as evidence for the second part of the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero. We provide a proof of a slightly more general statement without using the optimality hypothesis. [ABSTRACT FROM AUTHOR]

In this paper, we give an explicit formula for the first two terms of the Taylor expansion of a classical Eisenstein series of weight $2k+1$ for $\Gamma_{0}(q)$. Both the first term and the second term have interesting arithmetic interpretations. We apply the result to compute the central derivative of some Hecke $L$-functions. [ABSTRACT FROM AUTHOR]

Given a Riemannian space N of dimension n and a field D of symmetric endomorphisms on N, we define the extension M of N by D to be the Riemannian manifold of dimension n+1 obtained from N by a construction similar to extending a Lie group by a derivation of its Lie algebra. We find the conditions on N and D which imply that the extension M is Einstein. In particular, we show that in this case, D has constant eigenvalues; moreover, they are all integer (up to scaling) if det D ≠ 0. They must satisfy certain arithmetic relations which imply that there are only finitely many eigenvalue types of D in every dimension (a similar result is known for Einstein solvmanifolds). We give the characterisation of Einstein extensions for particular eigenvalue types of D, including the complete classification for the case when D has two eigenvalues, one of which is multiplicity free. In the most interesting case, the extension is obtained, by an explicit procedure, from an almost Kähler Ricci flat manifold (in particular, from a Calabi-Yau manifold). We also show that all Einstein extensions of dimension four are Einstein solvmanifolds. A similar result holds valid in the case when N is a Lie group with a left-invariant metric, under some additional assumptions. [ABSTRACT FROM AUTHOR]

We prove a weak version of the Siegel-Weil formula on SL2 for the dual pair (SL2, O2,2), where O2,2 is the split orthogonal group. By this formula and the Siegel-Weil formula, we give explicit formulas for Hecke correspondence's degree and average representation numbers over genus associated to Eichler orders. At last, we give explicit formulas for representations of a number as sums of three squares and four squares by local Whittaker functions, and it turns out that these functions are exactly the local factors of Hardy's singular series. [ABSTRACT FROM AUTHOR]

We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field k, this enrichment counts the number of lines meeting four lines defined over k in P3k, with such lines weighted by their fields of definition together with information about the cross-ratio of the intersection points and spanning planes. We generalize this example to an infinite family of such enrichments, obtained using an Euler number in A1-homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms. [ABSTRACT FROM AUTHOR]

Using the arithmetic Schottky uniformization theory, we show the arithmeticity of PSL2(C) Chern-Simons invariant. In terms of this invariant, we give an explicit formula of the Deligne-Riemann-Roch isomorphism as the Zograf-McIntyre-Takhtajan infinite product for families of algebraic curves. Applying this formula to the Liouville theory, we determine the unknown constant which appears in the holomorphic factorization formula of determinants of Laplacians on Riemann surfaces. [ABSTRACT FROM AUTHOR]

This paper presents a construction method for a class of codimension three arithmetically Gorenstein subschemes of projective space. These schemes are obtained from degeneracy loci of sections of certain specially constructed rank three reflexive sheaves. In contrast to the structure theorem of Buchsbaum and Eisenbud, we cannot obtain every arithmetically Gorenstein codimension three subscheme by our method. However, certain geometric applications are facilitated by the geometric aspect of this construction, and we discuss several examples of this in the final section. [ABSTRACT FROM AUTHOR]

Let S be a finite dimensional noetherian scheme. For any proper morphism between smooth S-schemes, we prove a Riemann-Roch formula relating higher algebraic K-theory and motivic cohomology, thus with no projective hypotheses either on the schemes or on the morphism. We also prove, without projective assumptions, an arithmetic Riemann-Roch theorem involving Arakelov's higher K-theory and motivic cohomology as well as an analogous result for the relative cohomology of a morphism. These results are obtained as corollaries of a motivic statement that is valid for morphisms between oriented absolute spectra in the stable homotopy category of S. [ABSTRACT FROM AUTHOR]

Let (S,E) be a log Calabi-Yau surface pair with E a smooth divisor. We define new conjecturally integer-valued counts of A1-curves in (S,E). These log BPS numbers are derived from genus 0 log Gromov-Witten invariants of maximal tangency along E via a formula analogous to the multiple cover formula for disk counts. A conjectural relationship to genus 0 local BPS numbers is described and verified for del Pezzo surfaces and curve classes of arithmetic genus up to 2. We state a number of conjectures and provide computational evidence. [ABSTRACT FROM AUTHOR]

We give an effective upper bound, for certain arithmetic hyperbolic 3-manifold groups obtained from a quadratic form construction, on the minimal index of a subgroup that embeds in a fixed 6-dimensional right-angled reflection group, stabilizing a totally geodesic subspace. In particular, for manifold groups in any fixed commensurability class we show that the index of such a subgroup is asymptotically smaller than any fractional power of the volume of the manifold. We also give effective bounds on the geodesic residual finiteness growths of closed hyperbolic manifolds that totally geodesically immerse in non-compact right-angled reflection orbifolds, extending work of the third author from the compact case. The first result gives examples to which the second applies, and for these we give explicit bounds on geodesic residual finiteness growth. [ABSTRACT FROM AUTHOR]

Let F ∈ Z[x] be a diagonal, non-singular quadratic form in four variables. Let λ (n) be the normalised Fourier coefficients of a holomorphic Hecke form of full level. We give an upper bound for the problem of counting integer zeros of F with |x| ≤ X, weighted by λ(x1). [ABSTRACT FROM AUTHOR]

We first develop some basic facts about hypergeometric sheaves on the multiplicative group Gm in characteristic p > 0. Specializing to some special classes of hypergeometric sheaves, we give relatively ''simple" formulas for their trace functions, and a criterion for them to have finite monodromy. We then show that one of our local systems, of rank 24 in characteristic p = 2, has the big Conway group 2.Co1, in its irreducible orthogonal representation of degree 24 as the automorphism group of the Leech lattice, as its arithmetic and geometric monodromy groups. Each of the other two, of rank 12 in characteristic p = 3, has the Suzuki group 6.Suz, in one of its irreducible representations of degree 12 as the Q(ζ3)-automorphisms of the Leech lattice, as its arithmetic and geometric monodromy groups. We also show that the pullback of these local systems by x → xN mappings to the affine line A1 yields the same arithmetic and geometric monodromy groups. [ABSTRACT FROM AUTHOR]

Double covers of a generic genus four curve C are in bijection with Cayley cubics containing the canonical model of C. The Prym variety associated to a double cover is a quadratic twist of the Jacobian of a genus three curve X. The curve X can be obtained by intersecting the dual of the corresponding Cayley cubic with the dual of the quadric containing C. We take this construction to its limit, studying all smooth degenerations and proving that the construction, with appropriate modifications, extends to the complement of a specific divisor in moduli. We work over an arbitrary field of characteristic different from two in order to facilitate arithmetic applications. [ABSTRACT FROM AUTHOR]

We prove the existence of a module for the largest Mathieu group, whose trace functions are weight 2 quasimodular forms. Restricting to the subgroup fixing a point, we see that the integrality of these functions is equivalent to certain divisibility conditions on the number of Fp points on Jacobians of modular curves. Extending such expressions to arbitrary primes, we find trace functions for modules of cyclic groups of prime order with similar connections. Moreover, for cyclic groups we give an explicit vertex operator algebra construction whose trace functions are given only in terms of weight 2 Eisenstein series. [ABSTRACT FROM AUTHOR]

We obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on T³ = R³/Z³ (three-dimensional “arithmetic random waves”). We prove that, as the multiplicity of the eigenspace goes to infinity, the nodal area converges to a universal, non-Gaussian distribution. Universality follows from the equidistribution of lattice points on the sphere. Our arguments rely on the Wiener chaos expansion of the nodal area: we show that, analogous to the two-dimensional case addressed by Marinucci et al., [Geom. Funct. Anal. 26 (2016), pp. 926–960] the fluctuations are dominated by the fourth-order chaotic component. The proof builds upon recent results from Benatar and Maffiucci [Int. Math. Res. Not. IMRN (to appear)] that establish an upper bound for the number of nondegenerate correlations of lattice points on the sphere. We finally discuss higher-dimensional extensions of our result. [ABSTRACT FROM AUTHOR]

We study the orbit of R̂ under the Möbius action of the Bianchi group PSL2(OK) on Ĉ, where OK is the ring of integers of an imaginary quadratic field K. The orbit SK, called a Schmidt arrangement, is a geometric realisation, as an intricate circle packing, of the arithmetic of K. Wegive a simple geometric characterisation of certain subsets of SK generalizing Apollonian circle packings, and show that SK, considered with orientations, is a disjoint union of all primitive integral such K-Apollonian packings. These packings are described by a new class of thin groups of arithmetic interest called KApollonian groups. We make a conjecture on the curvatures of these packings, generalizing the local-to-global conjecture for Apollonian circle packings. [ABSTRACT FROM AUTHOR]

We investigate the arithmetic of special values of a new class of L-functions recently introduced by the second author. We prove that these special values are encoded in some particular polynomials which we call Anderson-Stark units. We then use these Anderson-Stark units to prove that L-functions can be expressed as sums of polylogarithms. [ABSTRACT FROM AUTHOR]

Let K be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over K whose l-power torsion fields are arithmetically constrained for some rational prime l. Such arithmetic constraints are related to an unresolved question of Ihara regarding the kernel of the canonical outer Galois representation on the pro-l fundamental group of P¹ - {0, 1,8}. Under GRH, we demonstrate the set of classes is finite for any fixed K and any fixed dimension. Without GRH, we prove a semistable version of the result. In addition, several unconditional results are obtained when the degree of K/Q and the dimension of abelian varieties are not too large through a careful analysis of the special fiber of such abelian varieties. In some cases, the results (viewed as a bound on the possible values of l) are uniform in the degree of the extension K/Q. [ABSTRACT FROM AUTHOR]

We investigate measure-theoretic aspects of various notions of reducibility by applying analogs of Demuth's Theorem in the hyperarithmetic and set-theoretic settings. [ABSTRACT FROM AUTHOR]

Jeřábek introduced fragments of bounded arithmetic which are axiomatized with weak surjective pigeonhole principles and support a robust notion of approximate counting. We extend these fragments to accommodate modular counting quantifiers. These theories can formalize and prove the relativized versions of Toda's theorem on the collapse of the polynomial hierarchy with modular counting. We introduce a version of the Paris-Wilkie translation for converting formulas and proofs of bounded arithmetic with modular counting quantifiers into constant depth propositional logic with modular counting gates. We also define Paris-Wilkie translations to Nullstellensatz and polynomial calculus refutations. As an application, we prove that constant depth propositional proofs that use connectives AND, OR, and mod p gates, for p a prime, can be translated, with quasipolynomial increase in size, into propositional proofs containing only propositional formulas of depth three in which the top level is Boolean, the middle level consists of mod p gates, and the bottom level subformulas are small conjunctions. These results are improved to depth two by using refutations from the weak surjective pigeonhole principles. [ABSTRACT FROM AUTHOR]

We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every noncomputable real is non-trivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of atoms, i.e. considers only continuous measures, it turns out that every non-hyperarithmetical real is random for a continuous measure. On the other hand, examples of reals not random for any continuous measure can be found throughout the hyperarithmetical Turing degrees. [ABSTRACT FROM AUTHOR]

Ramsey's theorem states that each coloring has an infinite homogeneous set, but these sets can be arbitrarily spread out. Paul Erdős and Fred Galvin proved that for each coloring f, there is an infinite set that is "packed together" which is given "a small number" of colors by f. We analyze the strength of this theorem from the perspective of computability theory and reverse mathematics. We show that this theorem is close in computational strength to the standard Ramsey's theorem by giving arithmetical upper and lower bounds for solutions to computable instances. In reverse mathematics, we show that that this packed Ramsey's theorem is equivalent to Ramsey's theorem for exponents n ≠ 2. When n = 2, we show that it implies Ramsey's theorem and that it does not imply ACA0. [ABSTRACT FROM AUTHOR]