Equivalence of Unimodular (Quadratic)forms on Abelian groups

In summary, the conversation discusses the definition of equivalence for unimodular quadratic forms on Abelian groups. The speaker has found the definition for equivalence on vector spaces using matrices in Sl_n(Z) and is unsure how to define it for forms on Abelian groups. The use of group algebras, such as ##\mathbb{Z}[A]##, is suggested as a possible solution.
  • #1
WWGD
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Hi, everyone:

I have been looking for a while without success, for the definition of equivalence for
unimodular quadratic forms defined on Abelian groups .
I have found instead ,t the def. of equivalence in the more common case where the two forms Q,Q' are defined on vector spaces , and the definition has to see with matrices
in Sl_n(Z) . It makes sense that the equivalence of unimodular forms has to see with
matrices in Sl_n(Z) , since these have determinant +/- 1 . But it is not too clear to me
how we would define this equivalence if instead we had Q,Q' unimodular ,defined on Abelian groups A,A' respectively. Anyone know?.

Thanks.
 
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  • #2
I'm not sure, but I would consider group algebras like ##\mathbb{Z}[A]## to get the two different operations needed.
 

FAQ: Equivalence of Unimodular (Quadratic)forms on Abelian groups

What is the concept of equivalence of unimodular (quadratic) forms on abelian groups?

The concept of equivalence of unimodular (quadratic) forms on abelian groups is a mathematical concept that relates to the study of quadratic forms on abelian groups. An equivalence between two forms means that they have the same number of variables and coefficients, and that the corresponding value of the forms are equal for any given input of the variables.

How is the equivalence of unimodular (quadratic) forms on abelian groups established?

The equivalence of unimodular (quadratic) forms on abelian groups is established through the use of a linear transformation. This transformation is applied to both forms, and if the resulting forms are equal, then the original forms are considered equivalent. This process is also known as a "change of basis" in linear algebra.

What is the significance of studying equivalence of unimodular (quadratic) forms on abelian groups?

Studying equivalence of unimodular (quadratic) forms on abelian groups is important in various mathematical fields such as number theory and algebraic geometry. It allows for the comparison of different forms and helps in finding solutions to equations involving these forms. It also has applications in coding theory and cryptography.

What are some examples of unimodular (quadratic) forms on abelian groups?

Examples of unimodular (quadratic) forms on abelian groups include the sum of squares function, which gives the sum of squares of all the elements in a group, and the sum of products function, which gives the sum of products of all the elements in a group. Other examples include the quadratic forms over finite fields and the theta series of a lattice.

How does the equivalence of unimodular (quadratic) forms on abelian groups relate to the study of lattices?

The equivalence of unimodular (quadratic) forms on abelian groups is closely related to the study of lattices. In fact, every unimodular (quadratic) form on an abelian group corresponds to a lattice, and the equivalence between two forms is equivalent to the isomorphism of the corresponding lattices. This connection is useful in the study of lattices and their properties.

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