Discriminant of an integral lattice (awaiting review)

For an integral lattice $L$, the discriminant $D$ of $L$ is defined as the determinant of the Gram matrix associated to any basis of $L$.

There are several different conventions for the term "discriminant" in the literature; we give them different names here for clarity. The definition we take above is used in Cassels' Rational Quadratic Forms and Conway-Sloane's Sphere Packings, Lattices and Groups.

• The Witt discriminant of $L$ is the quantity $(-1)^{\binom{n}{2}} D$. This choice makes the Witt discriminant of the hyperbolic plane $U$ equal to $1$ so that the Witt discriminant is stable under adding copies of $U$. It also simplifies the connection with Clifford algebras: using this definition, the center of the even Clifford algebra associated to the lattice will be the quadratic étale algebra generated by the square root of the Witt discriminant. This convention is used in Milnor–Husemoller's Symmetric Bilinear Forms and Lam's Introduction to Quadratic Forms over Fields.
• The absolute discriminant of $L$ is the quantity $\lvert D \rvert$. This removes the sign ambiguity completely, and is used in Minkowski's Geometrie der Zahlen and Schmidt's Diophantine Approximations.
• The geometric discriminant of $L$ is the quantity $\sqrt{\lvert D \rvert}$. For positive definite lattices of rank $n$, the geometric discriminant is the covolume of any embedding of the lattice as a subgroup of $\mathbb{R}^n$, ie the volume of a fundamental parallelotope. Borevich and Shafarevich's Number Theory uses this definition.
• The quadratic discriminant of $L$ is the quantity $2^{-n}D$. While we define the quadratic form associated to $L$ to be $Q(\mathbf{v}) = \langle \mathbf{v}, \mathbf{v} \rangle$, some authors scale by $\frac{1}{2}$ so that integer valued quadratic forms can be obtained from integer matrices. Doing so leads to the extra factor of $2^{-n}$. This convention is used in Dickson's History of the Theory of Numbers, Vol. II and Serre's Course in Arithmetic.
• The half discriminant of $L$ is $2^{-n}D$ when $n$ is even and $2^{-n-1}D$ when $n$ is odd. This gives better behavior upon reduction to characteristic $2$. See Voight's Quaternion Algebras and Milnor–Husemoller's Symmetric Bilinear Forms.
• The 2-adic unit discriminant of $L$ is the the quantity $2^{-v_2(D)} D \pmod{8}$. This is used in the $2$-adic classification of quadratic spaces, and gives the $2$-adic square class of the odd part of the discriminant. It is defined in O'Meara's Introduction to Quadratic Forms and Earnest–Hsia's $2$-adic Quadratic Forms.