Suppose we take the "even" indefinite lattice from page 50 in Serre A Course in Arithmetic (1973) $$ U \; = \; \left( \begin{array}{cc} 0 & 1 \\\ 1 & 0 \end{array} \right),$$ called $H$ in pages 189-191 of Larry J. Gerstein Basic Quadratic Forms.

What I cannot find in any detail is a proof of this arithmetic statement in SPLAG by Conway and Sloane, page 378 in the first edition(1988), anyway chapter 15 section 7, that quadratic forms $f,g$ are in the same genus if and only if $f \oplus H$ and $g \oplus H$ are integrally equivalent. Then they say this follows from properties of the spinor genus, presumably including Eichler's theorem that indefinite rank at least 3 means spinor genus and class coincide. Also, if f and g do not correspond to "even lattices," I'm not entirely sure what is being claimed. Oh, I absolutely cannot assume $f,g$ are in any way "unimodular." Very popular, that unimodular. Matter of taste, though. I'm not sure it matters, but my $f,g$ are going to be positive, which is surely the difficult case here.

Everybody with whom I have discussed this regards this as either obvious or, essentially, an axiom. I would very much like a reference for this, plus an explanation of what is meant if $f,g$ correspond to "odd" lattices. For example, it would be wonderful if somewhere this claim and the words Theorem or Proposition or Lemma happened in the same sentence. I think I am making progress on the other bits I need, essentially ch. 26,27 in SPLAG, but this claim has me snowed, or perhaps buffaloed, thrown, stumped. As far as books that I own, I do not see the claim being discussed in Jones, Watson, O'Meara, Serre, Cassels, Kitaoka, Ebeling, Gerstein. I stopped by the office of R. Borcherds and discussed related matters for a while, the relevant articles are 1985 The Leech Lattice and 1990 Lattices Like the Leech Lattice, but I don't see the SPLAG claim in an explicit manner.

Thanks for any help,

Suppose we take the "even" indefinite lattice from page 50 in Serre A Course in Arithmetic (1973) $$ U \; = \; \left( \begin{array}{cc} 0 & 1 \\\ 1 & 0 \end{array} \right),$$ called $H$ in pages 189-191 of Larry J. Gerstein Basic Quadratic Forms.

What I cannot find in any detail is a proof of this arithmetic statement in SPLAG by Conway and Sloane, page 378 in the first edition(1988), anyway chapter 15 section 7, that quadratic forms $f,g$ are in the same genus if and only if $f \oplus H$ and $g \oplus H$ are integrally equivalent. Then they say this follows from properties of the spinor genus, presumably including Eichler's theorem that indefinite rank at least 3 means spinor genus and class coincide. Also, if f and g do not correspond to "even lattices," I'm not entirely sure what is being claimed. Oh, I absolutely cannot assume $f,g$ are in any way "unimodular." Very popular, that unimodular. Matter of taste, though. I'm not sure it matters, but my $f,g$ are going to be positive, which is surely the difficult case here.

Everybody with whom I have discussed this regards this as either obvious or, essentially, an axiom. I would very much like a reference for this, plus an explanation of what is meant if $f,g$ correspond to "odd" lattices. For example, it would be wonderful if somewhere this claim and the words Theorem or Proposition or Lemma happened in the same sentence. I think I am making progress on the other bits I need, essentially ch. 26,27 in SPLAG, but this claim has me snowed, or perhaps buffaloed, thrown, stumped. As far as books that I own, I do not see the claim being discussed in Jones, Watson, O'Meara, Serre, Cassels, Kitaoka, Ebeling, Gerstein. I stopped by the office of R. Borcherds and discussed related matters for a while, the relevant articles are 1985 The Leech Lattice and 1990 Lattices Like the Leech Lattice, but I don't see the SPLAG claim in an explicit manner.

EDIT... Sexy application: the Leech lattice and all the Niemeier lattices are in the same genus. Pointed out in an MO comment by Noam Elkies, who knows things.

Suppose we take the "even" indefinite lattice from page 50 in Serre A Course in Arithmetic (1973) $$ U \; = \; \left( \begin{array}{cc} 0 & 1 \\\ 1 & 0 \end{array} \right),$$ called $H$ in pages 189-191 of Larry J. Gerstein Basic Quadratic Forms.

  What I cannot find in any detail is a proof of this arithmetic statement in

SPLAG by Conway and Sloane, page 378 in the first edition(1988), anyway chapter 15 section 7, that quadratic forms $f,g$ are in the same genus if and only if $f \oplus H$ and $g \oplus H$ are integrally equivalent. Then they say this follows from properties of the spinor genus, presumably including Eichler's theorem that indefinite rank at least 3 means spinor genus and class coincide. Also, if f and g do not correspond to "even lattices," I'm not entirely sure what is being claimed. Oh, I absolutely cannot assume $f,g$ are in any way "unimodular." Very popular, that unimodular. Matter of taste, though. I'm not sure it matters, but my $f,g$ are going to be positive, which is surely the difficult case here.

 Everybody with whom I have discussed this regards this as either

obvious or, essentially, an axiom. I would very much like a reference for this, plus an explanation of what is meant if $f,g$ correspond to "odd" lattices. For example, it would be wonderful if somewhere this claim and the words Theorem or Proposition or Lemma happened in the same sentence. I think I am making progress on the other bits I need, essentially ch. 26,27 in SPLAG, but this claim has me snowed, or perhaps buffaloed, thrown, stumped. As far as books that I own, I do not see the claim being discussed in Jones, Watson, O'Meara, Serre, Cassels, Kitaoka, Ebeling, Gerstein. I stopped by the office of R. Borcherds and discussed related matters for a while, the relevant articles are 1985 The Leech Lattice and 1990 Lattices Like the Leech Lattice, but I don't see the SPLAG claim in an explicit manner.

Thanks for any help,

Suppose we take the "even" indefinite lattice from page 50 in Serre A Course in Arithmetic (1973) $$ U \; = \; \left( \begin{array}{cc} 0 & 1 \\\ 1 & 0 \end{array} \right),$$ called $H$ in pages 189-191 of Larry J. Gerstein Basic Quadratic Forms.

What I cannot find in any detail is a proof of this arithmetic statement in SPLAG by Conway and Sloane, page 378 in the first edition(1988), anyway chapter 15 section 7, that quadratic forms $f,g$ are in the same genus if and only if $f \oplus H$ and $g \oplus H$ are integrally equivalent. Then they say this follows from properties of the spinor genus, presumably including Eichler's theorem that indefinite rank at least 3 means spinor genus and class coincide. Also, if f and g do not correspond to "even lattices," I'm not entirely sure what is being claimed. Oh, I absolutely cannot assume $f,g$ are in any way "unimodular." Very popular, that unimodular. Matter of taste, though. I'm not sure it matters, but my $f,g$ are going to be positive, which is surely the difficult case here.

Everybody with whom I have discussed this regards this as either obvious or, essentially, an axiom. I would very much like a reference for this, plus an explanation of what is meant if $f,g$ correspond to "odd" lattices. For example, it would be wonderful if somewhere this claim and the words Theorem or Proposition or Lemma happened in the same sentence. I think I am making progress on the other bits I need, essentially ch. 26,27 in SPLAG, but this claim has me snowed, or perhaps buffaloed, thrown, stumped. As far as books that I own, I do not see the claim being discussed in Jones, Watson, O'Meara, Serre, Cassels, Kitaoka, Ebeling, Gerstein. I stopped by the office of R. Borcherds and discussed related matters for a while, the relevant articles are 1985 The Leech Lattice and 1990 Lattices Like the Leech Lattice, but I don't see the SPLAG claim in an explicit manner.

Thanks for any help,