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Analytic subgroup theorem

A significant result in modern transcendence theory, a generalisation of Baker's theorem on linear forms in logarithms.

Let $G$ be a commutative algebraic group defined over a number field $K$ and let $B$ be a subgroup of the complex points $G(\mathbb{C})$ defined over $K$. There are points of $B$ defined over the field of algebraic numbers if and only if there is a non-trivial analytic subgroup $H$ of $G$' defined over a number field such that $H(\mathbb{C})$ is contained in $B$.

References

• Alan Baker and Gisbert Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs 9, Cambridge University Press (2007) ISBN 978-0-521-88268-2. Chapter 6, pp.109-146.

Conductor of an abelian variety

In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is. It is connected to the ramification in the field generated by the division points.

For an abelian variety A defined over a field F with ring of integers R, consider the Néron model of A, which is a 'best possible' model of A defined over R. This model may be represented as a scheme over

Spec(R)

(cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism

Spec(F) → Spec(R)

gives back A. Let A⁰ denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a residue field k, A⁰_k is a group variety over k, hence an extension of an abelian variety by a linear group. This linear group is an extension of a torus by a unipotent group. Let u be the dimension of the unipotent group and t the dimension of the torus. The order of the conductor is

\[ f = 2u + t + \delta , \, \]

where δ is a measure of wild ramification.

Properties

• If A has good reduction then f = u = t = δ = 0.
• If A has semistable reduction or, more generally, acquires semistable reduction over a Galois extension of F of degree prime to p, the residue characteristic, then δ = 0.
• If p > 2d + 1, where d is the dimension of A, then δ = 0.

References

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Cubic reciprocity

Various results connecting the solvability of two related cubic equations in modular arithmetic.

The law of cubic reciprocity is most naturally expressed in terms of the Eisenstein integers, that is, the ring $E$ of complex numbers of the form $$ \alpha = a + b\,\omega $$ where and $a$ and $b$ are integers and $\omega = \frac{1}{2}(-1 + i\sqrt 3) = e^{2\pi i/3}$ is a complex cube root of unity. The field norm $\mathbf{N}(\alpha) = \alpha \bar\alpha$.

If $\pi$ is a prime element of $E$ of norm $P$ and $\alpha$ is an element coprime to $\pi$, we define the cubic residue symbol $\left(\frac{\alpha}{\pi}\right)_3$ to be the cube root of unity (power of $\omega$) satisfying $$ \alpha^{(P-1)/3} \equiv \left(\frac{\alpha}{\pi}\right)_3 \pmod \pi $$

We further define a primary prime to be one which is congruent to $-1 \pmod 3$. Then for distinct primary primes $\pi$ and $\theta$ the law of cubic reciprocity is simply $$ \left(\frac{\pi}{\theta}\right)_3 = \left(\frac{\theta}{\pi}\right)_3 $$ with the supplementary laws for the units and for the prime $1-\omega$ of norm 3 that if $\pi = -1 + 3(m+n\omega)$ then $$ \left(\frac{\omega}{\pi}\right)_3 = \omega^{m+n}\,, $$ $$ \left(\frac{1-\omega}{\pi}\right)_3 = \omega^{2m} \ . $$

References

• David A. Cox, Primes of the form $x^2+ny^2$, Wiley, 1989, ISBN 0-471-50654-0.
• K. Ireland and M. Rosen, A classical introduction to modern number theory, 2nd ed, Graduate Texts in Mathematics 84, Springer-Verlag, 1990.
• Franz Lemmermeyer, Reciprocity laws: From Euler to Eisenstein, Springer Verlag, 2000, ISBN 3-540-66957-4.

Genus field

In algebraic number theory, the genus field $G$ of a number field $K$ is the maximal abelian extension of $K$ which is obtained by composing an absolutely abelian field with $K$ and which is unramified at all finite primes of $K$. The genus number of $K$ is the degree $[G:K]$ and the genus group is the Galois group of $G$ over $K$.

If $K$ is itself absolutely abelian, the genus field may be described as the maximal absolutely abelian extension of $K$ unramified at all finite primes: this definition was used by Leopoldt and Hasse.

If $K = \mathbb{Q}(\sqrt m)$ ($m$ squarefree) is a quadratic field of discriminant $D$, the genus field of $K$ is a composite of quadratic fields. Let $p_i$ run over the prime factors of $D$. For each such prime $p$, define $p^*$ as follows:

$$p^* = \pm p \equiv 1 \pmod 4 \text{ if } p \text{ is odd} ; $$ $$2^* = -4, 8, -8 \text{ according as } m \equiv 3 \pmod 4, 2 \pmod 8, -2 \pmod 8 . $$

Then the genus field is the composite $K(\sqrt{p^*_i})$.

See also

• Hilbert class field

References

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Group isomorphism problem

In abstract algebra, the group isomorphism problem is the decision problem of determining whether two group presentations present isomorphic groups.

The isomorphism problem was identified by Max Dehn in 1911 as one of three fundamental decision problems in group theory; the other two being the word problem and the conjugacy problem.

References

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Hall algebra

The Hall polynomials in mathematics were developed by Philip Hall in the 1950s in the study of group representations.

A finite abelian p-group M is a direct sum of cyclic p-power components \(C_{p^\lambda_i}\) where \(\lambda=(\lambda_1,\lambda_2,\ldots)\) is a partition of \(n\) called the type of M. Let \(g^\lambda_{\mu,\nu}(p)\) be the number of subgroups N of M such that N has type \(\nu\) and the quotient M/N has type \(\mu\). Hall showed that the functions g are polynomial functions of p with integer coefficients: these are the Hall polynomials.

Hall next constructs an algebra \(H(p)\) with symbols \(u_\lambda\) a generators and multiplication given by the \(g^\lambda_{\mu,\nu}\) as structure constants

\[ u_\mu u_\nu = \sum_\lambda g^\lambda_{\mu,\nu} u_\lambda \]

which is freely generated by the \(u_{\mathbf1_n}\) corresponding to the elementary p-groups. The map from \(H(p)\) to the algebra of symmetric functions \(e_n\) given by \(u_{\mathbf 1_n} \mapsto p^{-n(n-1)}e_n\) is a homomorphism and its image may be interpreted as the Hall-Littlewood polynomial functions. The theory of Schur functions is thus closely connected with the theory of Hall polynomials.

References

• I.G. Macdonald, Symmetric functions and Hall polynomials, (Oxford University Press, 1979) ISBN 0-19-853530-9

Hall–Littlewood polynomials

In mathematics, the Hall–Littlewood polynomials encode combinatorial data relating to the representations of the general linear group. They are named for Philip Hall and Dudley E. Littlewood.

See also

• Littlewood polynomial

References

• I.G. Macdonald; Symmetric Functions and Hall Polynomials, (1979), pp. 101-104, Oxford University Press ISBN: 0-19-853530-9
• D.E. Littlewood; On certain symmetric functions, Proc. London Math. Soc., 43 (1961), pp. 485–498, DOI: 10.1112/plms/s3-11.1.485

Hutchinson operator

In mathematics, in the study of fractals, a Hutchinson operator is a collection of functions on an underlying space E. The iteration on these functions gives rise to an iterated function system, for which the fixed set is self-similar.

Definition

Formally, let f_i be a finite set of N functions from a set X to itself. We may regard this as defining an operator H on the power set P X as

\[H : A \mapsto \bigcup_{i=1}^N f_i[A],\,\]

where A is any subset of X.

A key question in the theory is to describe the fixed sets of the operator H. One way of constructing such a fixed set is to start with an initial point or set S₀ and iterate the actions of the f_i, taking S_n+1 to be the union of the images of S_n under the operator H; then taking S to be the union of the S_n, that is,

\[S_{n+1} = \bigcup_{i=1}^N f_i[S_n] \]

and

\[S = \bigcup_{n=0}^\infty S_n . \]

Properties

Hutchinson (1981) considered the case when the f_i are contraction mappings on a Euclidean space X = R^d. He showed that such a system of functions has a unique compact (closed and bounded) fixed set S.

The collection of functions \(f_i\) together with composition form a monoid. With N functions, then one may visualize the monoid as a full N-ary tree or a Cayley tree.

References

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Kneser theorem

A statement about set addition in finite groups. It may be regarded as an extension of the Cauchy-Davenport theorem on sumsets in groups of prime order.

Let $G$ be a non-trivial abelian group and $A,B$ finite non-empty subsets. If $|A| + |B| \le |G|$ then there is a finite subgroup $H$ of $G$ such that $$ |A+B| \ge |A+H| + |B+H| - |H| \ge |A| + |B| - |H| \ . $$

The subgroup $H$ can be taken to be the stabiliser of $A+B$ $$ H = \lbrace g \in G : g + (A+B) = (A+B) \rbrace \ . $$

References

• Combinatorial number theory and additive group theory, ser. Advanced Courses in Mathematics CRM Barcelona (2009), Birkhäuser, Zbl 1177.11005 ISBN: 978-3-7643-8961-1
• Kneser, Martin; Abschätzungen der asymptotischen Dichte von Summenmengen, Math. Zeitschr., 58 (1953), pp. 459–484, Zbl 0051.28104
• Nathanson, Melvyn B.; Additive Number Theory: Inverse Problems and the Geometry of Sumsets, ser. Graduate Texts in Mathematics 165 (1996), pp. 109–132, Springer-Verlag, Zbl 0859.11003 ISBN: 0-387-94655-1
• Tao, Terence; Vu, Van H.; Additive Combinatorics, (2010), Cambridge University Press, Zbl 1179.11002 ISBN: 978-0-521-13656-3

Linked field

A field for which the quadratic forms attached to quaternion algebras have a common property.

Let $F$ be a field of characteristic not equal to 2. Let $A = (a_1,a_2)_F$, $B = (b_1,b_2)_F$ be quaternion algebras over F. The algebras $A$ and $B$ are linked quaternion algebras over $F$ if there is $x \in F$ such that $A$ is equivalent to $(x,y)_F$ and $B$ is equivalent to $(x,z)_F$.

The Albert form for $A,B$ is $$ q = \left\langle{ -a_1, -a_2, a_1a_2, b_1, b_2, -b_1b_2 }\right\rangle \ . $$ It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of $A$ and $B$. The quaternion algebras are linked if and only if the Albert form is isotropic.

The field $F$ is linked if any two quaternion algebras over $F$ are linked. Every global and local field is linked since all quadratic forms of degree 6 over such fields are isotropic.

The following properties of F are equivalent:

• $F$ is linked.
• Any two quaternion algebras over F are linked.
• Every Albert form (dimension six form of discriminant −1) is isotropic.
• The quaternion algebras form a subgroup of the Brauer group of F.
• Every dimension five form over F is a Pfister neighbour.
• No biquaternion algebra over F is a division algebra.

A nonreal linked field has u-invariant equal to 1,2,4 or 8.

References

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Pinch point

A pinch point or cuspidal point is a type of singular point on an algebraic surface. It is one of the three types of ordinary singularity of a surface.

The equation for the surface near a pinch point may be put in the form

\[ f(u,v,w) = u^2 - vw^2 + [4] \, \]

where [4] denotes terms of degree 4 or more.

References

• P. Griffiths; J. Harris; Principles of Algebraic Geometry, ser. Wiley Classics Library (1994), p. 617, Wiley Interscience ISBN: 0-471-05059-8

Quaternionic structure

Q-structure

An axiomatic system that abstracts the concept of a quaternion algebra over a field.

A quaternionic structure is a triple $(G,Q,q)$ where $G$ is an elementary abelian group of exponent 2 with a distinguished element $−1$, $Q$ is a pointed set with distinguished element $1$, and $q$ is a symmetric surjection from $G \times G \rightarrow Q$ satisfying axioms

1. $q(a,(-1)a) = 1$;
2. $q(a,b) = q(a,c) \Leftrightarrow q(q,bc) = 1$;
3. $q(a,b) = q(c,d) \Rightarrow \exists x\in Q\,:\, q(a,b) = q(a,x),\, q(c,d) = q(c,x)$.

Every field $F$ supports a quaternionic structure by taking $G$ to be $F^*/(F^*)^2$, $Q$ the set of Brauer classes of quaternion algebras in the Brauer group of $F$ with the split quaternion algebra as distinguished element and $q(a,b)$ the quaternion algebra $(a,b)_F$.

References

• Tsit-Yuen Lam, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67 , American Mathematical Society (2005) ISBN 0-8218-1095-2 Zbl 1068.11023 MR2104929

Siegel identity

One of two formulae that are used in the resolution of Diophantine equations.

The first formula is $$ \frac{x_3 - x_1}{x_2 - x_1} + \frac{x_2 - x_3}{x_2 - x_1} = 1 \ . $$ The second is $$ \frac{x_3 - x_1}{x_2 - x_1} \cdot\frac{t - x_2}{t - x_3} + \frac{x_2 - x_3}{x_2 - x_1} \cdot \frac{t - x_1}{t - x_3} = 1 \ . $$

The identities are used in translating Diophantine problems connected with integral points on hyperelliptic curves into S-unit equations.

See also: Siegel formula.

References

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Splicing rule

A transformation on formal languages which formalises the action of gene splicing in molecular biology. A splicing language is a language generated by iterated application of a splicing rule: the splicing languages form a proper subset of the regular languages.

Let $A$ be an alphabet and $L$ a language over $A$, that is, a subset of the free monoid $A^*$. A splicing rule is a quadruple $r = (a,b,c,d)$ of elements of $A^*$, and the action of the rule $r$ on $L$ is to produce the language $$ r(L) = \{ xady : xabq, pcdy \in L \} \ . $$ If $R$ is a set of rules then $R(L)$ is the union of the languages produced by the rules of $R$. We say that $R$ respects $L$ if $R(L)$ is a subset of $L$. The $R$-closure of $L$ is the union of $L$ and all iterates of $R$ on$L$: clearly it is respected by $R$. A splicing language is an $R$-closure of a finite language.

A rule set $R$ is reflexive if $(a,b,c,d)$ in $R$ implies that $(a,b,a,b)$ and $(c,d,c,d)$ are in $R$. A splicing language is reflexive if it is defined by a reflexive rule set.

Let $A = \{a,b,c\}$. The rule $(caba,a,cab,a)$ applied to the finite set $\{cabb,cabab,cabaab\}$ generates the regular language $caba^*b$.

Properties

• All splicing languages are regular.
• Not all regular languages are splicing. An example is $(aa)^*$ over $\{a,b\}$.
• If $L$ is a regular language on the alphabet $A$, and $z$ is a letter not in $A$, then the language $\{ zw : w \in L \}$ is a splicing language.
• There is an algorithm to determine whether a given regular language is a reflexive splicing language.
• The set of splicing rules that respect a regular language can be determined from the syntactic monoid of the language.

References

• James A. Anderson, "Automata theory with modern applications" (with contributions by Tom Head) Cambridge University Press (2006) ISBN 0-521-61324-8 Zbl 1127.68049