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Let p ∈ N be a prime with p ≠ 2 , and let K be a p -adic field. Let F be the function field of a curve over K. Let Ω F be the set of all divisorial discrete valuations of F. In this paper, we ask whether the Hasse principle holds for semisimple adjoint linear algebraic groups over F. We give a positive answer to this question for a class of adjoint classical groups. [ABSTRACT FROM AUTHOR]

Let R be the henselization of a local ring of a semistable family over the spectrum of a discrete valuation ring of mixed characteristic (0, p) and k the residue field of R. In this paper, we prove an isomorphism of étale hypercohomology groups Hétn+1(R, Tr(n) ≃ Hét¹(k, WrΩlogn) for any integers n ≥ 0 and r > 0 where Tr(n) is the p-adic Tate twist and WrΩlogn is the logarithmic Hodge-Witt sheaf. As an application, we prove the local-global principle for Galois cohomology groups over function fields of curves over an excellent henselian discrete valuation ring of mixed characteristic. [ABSTRACT FROM AUTHOR]

In this paper we inspect from closer the local and global points of the twists of the Klein quartic. For the local ones we use geometric arguments, while for the global ones we strongly use the modular interpretation of the twists. The main result is providing families with (conjecturally infinitely many) twists of the Klein quartic that are counterexamples to the Hasse Principle. [ABSTRACT FROM AUTHOR]

Let ℓ be a prime number. We classify the subgroups G of Sp 4 (F ℓ) and GSp 4 (F ℓ) that act irreducibly on F ℓ 4 , but such that every element of G fixes an F ℓ -vector subspace of dimension 1. We use this classification to prove that a local-global principle for isogenies of degree ℓ between abelian surfaces over number fields holds in many cases—in particular, whenever the abelian surface has non-trivial endomorphisms and ℓ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes ℓ for which some abelian surface A / Q fails the local-global principle for isogenies of degree ℓ . [ABSTRACT FROM AUTHOR]

For any number field, we prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group has a nontrivial 2-torsion subgroup. [ABSTRACT FROM AUTHOR]

An integral quadratic lattice is called indefinite k-universal if it represents all integral quadratic lattices of rank k for a given positive integer k. For k ≥ 3 , we prove that the indefinite k-universal property satisfies the local–global principle over number fields. For k = 2 , we show that a number field F admits an integral quadratic lattice which is locally 2-universal but not indefinite 2-universal if and only if the class number of F is even. Moreover, there are only finitely many classes of such lattices over F. For k = 1 , we prove that F admits a classic integral lattice which is locally classic 1-universal but not classic indefinite 1-universal if and only if F has a quadratic unramified extension where all dyadic primes of F split completely. In this case, there are infinitely many classes of such lattices over F. All quadratic fields with this property are determined. [ABSTRACT FROM AUTHOR]

For any number field not containing Q (i) , we give an explicit construction to prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group is nontrivial. [ABSTRACT FROM AUTHOR]

Let n ∈ N and let K be a field with a henselian discrete valuation of rank n with hereditarily euclidean residue field. Let F / K be a function field in one variable. It is known that every sum of squares is a sum of 3 squares. We determine the order of the group of nonzero sums of 3 squares modulo sums of 2 squares in F in terms of equivalence classes of certain discrete valuations of F of rank at most n. In the case of function fields of hyperelliptic curves of genus g , K.J. Becher and J. Van Geel showed that the order of this quotient group is bounded by 2 n (g + 1). We show that this bound is optimal. Moreover, in the case where n = 1 , we show that if F / K is a hyperelliptic function field such that the order of this quotient group is 2 g + 1 , then F is nonreal. [ABSTRACT FROM AUTHOR]

We study an analogue of the Brauer–Manin obstruction to the local–global principle for embedding problems over global fields. We will prove the analogues of several fundamental structural results. In particular we show that the (algebraic) Brauer–Manin obstruction is the only one to strong approximation when the embedding problem has abelian kernel. [ABSTRACT FROM AUTHOR]

We prove a local-global principle for torsors under the prosolvable geometric fundamental group of an affine curve over a number field. [ABSTRACT FROM AUTHOR]

In this article, we introduce a systematic and uniform construction of non-singular plane curves of odd degrees n \geq 5 which violate the local-global principle. Our construction works unconditionally for n divisible by p^{2} for some odd prime number p. Moreover, our construction also works for n divisible by some p \geq 5 which satisfies a conjecture on a p-adic property of the fundamental unit of \mathbb {Q}(p^{1/3}) and \mathbb {Q}((2p)^{1/3}). This conjecture is a natural cubic analogue of the classical Ankeny-Artin-Chowla-Mordell conjecture for \mathbb {Q}(p^{1/2}) and easily verified numerically. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, I an ideal of R and M an R-module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and C of local cohomology modules. We show that if M is a minimax R-module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if n is a nonnegative integer such that (HIi(M))m is a minimax Rm-module for all m ∈ Max(R) and for all i < n, then the set AssR(HIn(M)) is finite. Also, if HIi(M) is minimax for all i ⩾ n ⩾ 1, then HIi(M) is Artinian for i ⩾ n. It is shown that if M is a C − minimax module over a local ring such that HIi(M) are C − minimax modules for all i < n (or i ⩾ n), where n ⩾ 1, then they must be minimax. Consequently, a vanishing theorem is proved for local cohomology modules. [ABSTRACT FROM AUTHOR]

We prove that the ring of integers in the totally real cubic subfield K (49) of the cyclotomic field Q (ζ 7) has Pythagoras number equal to 4. This is the smallest possible value for a totally real number field of odd degree. Moreover, we determine which numbers are sums of integral squares in this field, and use this knowledge to construct a diagonal universal quadratic form in five variables. [ABSTRACT FROM AUTHOR]

For quadratic forms in 4 variables defined over the rational function field in one variable over C ((t)) , the validity of the local-global principle for isotropy with respect to different sets of discrete valuations is examined. [ABSTRACT FROM AUTHOR]

We provide examples of abelian surfaces over number fields K whose reductions at almost all good primes possess an isogeny of prime degree ℓ rational over the residue field, but which themselves do not admit a K-rational ℓ -isogeny. This builds on work of Cullinan and Sutherland. When K = Q , we identify certain weight-2 newforms f with quadratic Fourier coefficients whose associated modular abelian surfaces A f exhibit such a failure of a local–global principle for isogenies. [ABSTRACT FROM AUTHOR]

Given a positive integer n , a sufficient condition on a field is given for bounding its Pythagoras number by 2 n + 1. The condition is satisfied for n = 1 by function fields of curves over iterated formal power series fields over R , as well as by finite field extensions of R ((t 0 , t 1)). In both cases, one retrieves the upper bound 3 on the Pythagoras number. The new method presented here might help to establish more generally 2 n + 1 as an upper bound for the Pythagoras number of function fields of curves over R ((t 1 , ... , t n)) and for finite field extensions of R ((t 0 , ... , t n)). [ABSTRACT FROM AUTHOR]

In this article, we consider the Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal subgroup of the orthogonal group of a non-degenerate quadratic space with a hyperbolic summand over a commutative ring, introduced by Roy. We prove a set of commutator relations among the elementary generators of the DSER elementary orthogonal group. As an application, we prove that this group is perfect and an action version of the Quillen's local-global principle for this group is proved. This affirmatively answers a question of Rao in his Ph.D. thesis. [ABSTRACT FROM AUTHOR]

We prove a generalization of the Ahlswede-Cai local-global principle for the edge-isoperimetric problems. A new technique to handle edge-isoperimetric problems is introduced which we call the pull-push method. Our main result includes all previously published results in this area as special cases with the only exception of the edge-isoperimetric problem for grids. With this we partially answer a question of Harper on local-global principles. We also describe a strategy for further generalization of our results so that the case of grids would be covered, which would completely settle Harper's question. [ABSTRACT FROM AUTHOR]

The concept of Faltings’ local-global principle for the in dimension <n of local cohomology modules over a Noetherian ring R is introduced, and it is shown that this principle holds at levels 1, 2. We also establish the same principle at all levels over an arbitrary Noetherian ring of dimension not exceeding 3. These generalize the main results of Brodmann et al. [8]. Moreover, as a generalization of Raghavan’s result, we show that the Faltings’ local-global principle for the in dimension <n of local cohomology modules holds at all levels r∈ℕ whenever the ring R is a homomorphic image of a Noetherian Gorenstein ring. Finally, it is shown that if M is a finitely generated R-module, 픞 an ideal of R and r a non-negative integer such that is in dimension < 2 for all i<r and for some positive integer t, then for any minimax submodule N of , the R-module is finitely generated. As a consequence, it follows that the associated primes of are finite. This generalizes the main results of Brodmann-Lashgari [7] and Quy [24]. [ABSTRACT FROM AUTHOR]

A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local-global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi-Brauer varieties and mod 2 Galois representations, and prove that the local-global principle holds for conics and cubics. We also construct counterexamples to the local-global principle for quartics using the results of Mumford, Harris, and Shioda on theta characteristics. [ABSTRACT FROM AUTHOR]

In this paper, we study the action of special n × n linear (resp. symplectic) matrices which are homotopic to identity on the right invertible n × m matrices. We also prove that the commutator subgroup of O 2n (R [ X ]) is two stably elementary orthogonal for a local ring R with 1 2 ∈ R and n ≥ 3. [ABSTRACT FROM AUTHOR]

In a recent paper, Colliot-Thélène, Parimala and Suresh conjectured that a local–global principle holds for projective homogeneous spaces under connected linear algebraic groups over function fields of p-adic curves. In this paper, we show that the conjecture is true for any linear algebraic group whose almost simple factors of its semisimple part are isogenous to unitary groups or special unitary groups of hermitian or skew-hermitian spaces over central simple algebras with involutions. The proof implements patching techniques of Harbater, Hartmann and Krashen. As an application, we obtain a Springer-type theorem for isotropy of hermitian spaces over odd degree extensions of function fields of p-adic curves. [ABSTRACT FROM AUTHOR]

In this paper we give a version of Krivine-Stengle's Positivstellensatz, Schweighofer's Positivstellensatz, Scheiderer's local-global principle, Scheiderer's Hessian criterion and Marshall's boundary Hessian conditions for polynomial matrices, i.e. matrices with entries from the ring of polynomials in the variables $$x_1,\ldots ,x_d$$ with real coefficients. Moreover, we characterize Archimedean quadratic modules of polynomial matrices, and study the relationship between the compactness of a subset in $$\mathbb R^{d}$$ with respect to a subset $$\mathcal {G}$$ of polynomial matrices and the Archimedean property of the preordering and the quadratic module generated by $$\mathcal {G}$$ . [ABSTRACT FROM AUTHOR]

The concept of Faltings’ local-global principle for the minimaxness of local cohomology modules over a commutative Noetherian ringRis introduced, and it is shown that this principle holds at level 2. We also establish the same principle at all levels over an arbitrary commutative Noetherian ring of dimension not exceeding 3. These generalize the main results of Brodmann et al. in [6]. Moreover, it is shown that ifMis a finitely generatedR-module, \bm𝔞 an ideal ofRandra non-negative integer such thatis skinny for alli < rand for some positive integert, then for any minimax submoduleNof, theR-moduleis finitely generated. As a consequence, it follows that the associated primes ofare finite. This generalizes the main results of Brodmann-Lashgari [5] and Quy [16]. [ABSTRACT FROM AUTHOR]

Suslin's local-global principle asserts that if a matrix over a polynomial ring vanishes modulo the independent variable and is locally elementary then it is elementary. In this article we prove Suslin's local-global principle for principal congruence subgroups of Chevalley groups. This result is a common generalization of the result of Abe for the absolute case and Apte, Chattopadhyay and Rao for classical groups. For the absolute case the localglobal principle was recently obtained by Petrov and Stavrova in the more general settings of isotropic reductive groups. [ABSTRACT FROM AUTHOR]

Abstract: We define and investigate a family of local–global principles for fields involving both orderings and p-valuations. This family contains the PAC, PRC and PpC fields and exhausts the class of pseudo classically closed fields. We show that the fields satisfying such a local–global principle form an elementary class, admit diophantine definitions of holomorphy domains, and their orderings satisfy the strong approximation property. [Copyright &y& Elsevier]

Let $\mathcal S$ be a Serre subcategory of the category of $R$-modules, where $R$ is a commutative Noetherian ring. Let $\mathfrak a$ and $\mathfrak b$ be ideals of $R$ and let $M$ and $N$ be finite $R$-modules. We prove that if $N$ and $H^i_{\mathfrak a}(M,N)$ belong to $\mathcal S$ for all $i\lt n$ and if $n\leq \mathrm {f}$-$\mathrm {grad}({\mathfrak a},{\mathfrak b},N )$, then $\mathrm {Hom}_{R}(R/{\mathfrak b},H^n_{{\mathfrak a}}(M,N))\in \mathcal S$. We deduce that if either $H^i_{\mathfrak a}(M,N)$ is finite or $\mathrm {Supp}\,H^i_{\mathfrak a}(M,N)$ is finite for all $i\lt n$, then $\mathrm {Ass}\,H^n_{\mathfrak a}(M,N)$ is finite. Next we give an affirmative answer, in certain cases, to the following question. If, for each prime ideal ${\mathfrak {p}}$ of $R$, there exists an integer $n_{\mathfrak {p}}$ such that $\mathfrak b^{n_{\mathfrak {p}}} H^i_{\mathfrak a R_{\mathfrak {p}}}({M_{\mathfrak {p}}},{N_{\mathfrak {p}}})=0$ for every $i$ less than a fixed integer $t$, then does there exist an integer $n$ such that $\mathfrak b^nH^i_{\mathfrak a}(M,N)=0$ for all $i\lt t$? A formulation of this question is referred to as the local-global principle for the annihilation of generalised local cohomology modules. Finally, we prove that there are local-global principles for the finiteness and Artinianness of generalised local cohomology modules. [ABSTRACT FROM AUTHOR]

Abstract: Let be a field and a finite group. The question of ‘admissibility’ of over was originally posed by Schacher, who gave partial results in the case . In this paper, we give necessary conditions for admissibility of a finite group over function fields of curves over complete discretely valued fields. Using this criterion, we give an example of a finite group which is not admissible over . We also prove a certain Hasse principle for division algebras over such fields. [Copyright &y& Elsevier]

Abstract: Let G be an abelian group of finite order a field and a ring. Let such that for every character (where and is a primitive nth root of unity). What does D look like? The case where and was settled by Bridges and Mena. Here we obtain a complete characterization for the case where K is a finite extension of the field and R is its valuation ring under the condition that p does not divide n. As an application we obtain the following local-global principle for (where and are distinct primes): If , then for every character if and only if for every prime p and every character . [Copyright &y& Elsevier]

Abstract: For two semi-simple algebras A and B over an arbitrary ground field F, we give a numerical criterion when , the set of F-algebra homomorphisms between them, is non-empty. We also determine when the orbit set is finite and give an explicit formula for its cardinality. A few applications of main results are given. [Copyright &y& Elsevier]

Let R be a commutative Noetherian ring, I a proper ideal of R, and M a finitely generated R-module. We prove that there is a local-global principle for the Artinianness of the local cohomology module , i.e., for any integer n > 0, is Artinian for all i < n if and only if is Artinian for all i < n and all prime ideals 𝔭, which deduces one interesting property of filter depth. [ABSTRACT FROM PUBLISHER]

Let $${\Gamma < {\rm SL}(2, {\mathbb Z})}$$ be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors $${v_{0}, w_{0} \in \mathbb {Z}^{2} \, {\backslash} \, \{0\}}$$. We consider the set $${\mathcal {S}}$$ of all integers occurring in $${\langle v_{0}\gamma, w_{0}\rangle}$$, for $${\gamma \in \Gamma}$$ and the usual inner product on $${\mathbb {R}^2}$$. Assume that the critical exponent δ of Γ exceeds 0.99995, so that Γ is thin but not too thin. Using a variant of the circle method, new bilinear forms estimates and Gamburd's 5/6-th spectral gap in infinite-volume, we show that $${\mathcal {S}}$$ contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we show that the exceptional set $${\mathfrak {E}(N)}$$ of integers | n| < N which are locally admissible $${(n \in \mathcal {S} \, \, ({\rm mod} \, q) \, \, {\rm for\,all} \,\, q \geq 1)}$$ but fail to be globally represented, $${n \notin \mathcal {S}}$$, has a power savings, $${|\mathfrak {E}(N)| \ll N^{1-\varepsilon_{0}}}$$ for some $${\varepsilon_{0} > 0}$$, as N → ∞. [ABSTRACT FROM AUTHOR]

This paper continues the investigation of the groups $\mathcal{RF}(G)$ first introduced in the forthcoming book of Chiswell and Müller 'A Class of Groups Universal for Free ℝ-Tree Actions' and in the article by Müller and Schlage-Puchta (Abh. Math. Semin. Univ. Hambg. 79:193-227, ). We establish a criterion for a family $\{\mathcal{H}_{\sigma}\}$ of hyperbolic subgroups $\mathcal{H}_{\sigma}\leq\mathcal{RF}(G)$ to generate a hyperbolic subgroup isomorphic to the free product of the $\mathcal{H}_{\sigma}$ (Theorem 1.2), as well as a local-global principle for local incompatibility (Theorem 4.1). In conjunction with the theory of test functions as developed by Müller and Schlage-Puchta (Abh. Math. Semin. Univ. Hambg. 79:193-227, ), these results allow us to obtain a necessary and sufficient condition for a free product of real groups to embed as a hyperbolic subgroup in $\mathcal{RF}(G)$ for a given group G (Corollary 5.4). As a further application, we show that the centralizers associated with a family of pairwise locally incompatible cyclically reduced functions in $\mathcal{RF}(G)$ generate a hyperbolic subgroup isomorphic to the free product of these centralizers (Corollary 5.2). [ABSTRACT FROM AUTHOR]

Given an algebraic curve C/ℚ having points everywhere locally and endowed with a suitable involution, we show that there exists a positive density family of prime quadratic twists of C violating the Hasse principle. The result applies in particular to wN-Atkin–Lehner twists of most modular curves X0(N) and to wp-Atkin–Lehner twists of certain Shimura curves XD+. [ABSTRACT FROM AUTHOR]

Abstract: We discuss some results on the triviality and finiteness for Galois cohomology of connected unipotent groups over non-perfect (and especially local and global function) fields, and their relation with the closedness of orbits, which extend some well known results of Serre, Raynaud and Oesterlé. As one of the applications, we show that a separable additive polynomial over a global field k of characteristic in two variables is universal over k if and only if it is so over all completions of k. [Copyright &y& Elsevier]

Using sheaves of special groups, we show that a general local-global principle holds for every reduced special group whose associated space of orderings only has a finite number of accumulation points. We also compute the behaviour of the Boolean hull functor applied to sheaves of special groups. [ABSTRACT FROM AUTHOR]

Let K be a function field with an A-algebra structure. The ring A arises in the definition of the Drinfeld module φ over K. By E(K) we denote K together with the A-module structure induced on it by φ. For any principal prime ideal (a)⊂A, we study the question whether an element x∈E(K) which is an a-fold in E(Kν) for every place ν of K, is an a-fold in E(K). In particular, we study the groupS(a,K)≔kerE(K)/aE(K)→∏lower limit ν E(Kν)/aE(Kν)for Drinfeld modules of rank 2. We show that this finite group is trivial in many cases, but can become arbitrarily large. [Copyright &y& Elsevier]

We consider the shadow minimization problem (SMP) for Cartesian powers P n of a Macaulay poset P. Our main result is a local-global principle with respect to the lexicographic order L n. Namely, we show that under certain conditions the shadow of any initial segment of the order L n for n ≥ 3 is minimal iff it is so for n = 2. These conditions include such poset properties as additivity, shadow increasing, final shadow increasing and being rank-greedy. We also show that these conditions are essentially necessary for the lexicographic order to provide nestedness in the SMP. [ABSTRACT FROM AUTHOR]

We study the logical and computational properties of basic theorems of uncountable mathematics, in particular Pincherle's theorem , published in 1882. This theorem states that a locally bounded function is bounded on certain domains, i.e. one of the first 'local-to-global' principles. It is well-known that such principles in analysis are intimately connected to (open-cover) compactness , but we nonetheless exhibit fundamental differences between compactness and Pincherle's theorem. For instance, the main question of Reverse Mathematics, namely which set existence axioms are necessary to prove Pincherle's theorem, does not have an unique or unambiguous answer, in contrast to compactness. We establish similar differences for the computational properties of compactness and Pincherle's theorem. We establish the same differences for other local-to-global principles, even going back to Weierstrass. We also greatly sharpen the known computational power of compactness, for the most shared with Pincherle's theorem however. Finally, countable choice plays an important role in the previous, we therefore study this axiom together with the intimately related Lindelöf lemma. [ABSTRACT FROM AUTHOR]