Gauss sums of an even lattice #
Let L be an even lattice in a real inner product space E, write e(x) = exp (2 π i x), and
let c be a natural number. For a : ℤ the Gauss sum of L is the finite sum
G_L(a, c) = ∑_{y ∈ L / cL} e(a ‖y‖² / (2c)),
and for m in the dual lattice L^∨ the twisted Gauss sum is
G_L(a, c; m) = ∑_{y ∈ L / cL} e(a ‖y‖² / (2c) + ⟪y, m⟫ / c).
The summand is a function on the quotient L / cL because L is even and m ∈ L^∨: replacing
y by y + c w with w ∈ L changes the argument of e by
a ⟪y, w⟫ + a c ‖w‖² / 2 + ⟪w, m⟫ ∈ ℤ. Both hypotheses are needed: on the odd lattice ℤ ⊆ ℝ,
with a = c = 1 and m = 0, the representatives 0 and 1 of the single class of ℤ / ℤ give
e(0) = 1 and e(1/2) = -1. They are therefore arguments of the summand ZLattice.gaussTerm:
the evenness of L as a proof, and m as an element of L^∨ rather than of E. Here cL is
the lattice (c : ℝ) • L, read in L through Submodule.submoduleOf, so that the classes of
L / cL are those used to split a theta series of L into theta series of translates of cL.
For a full lattice of rank n and c ≠ 0 the quotient is finite of order c ^ n
(ZLattice.card_quotient_smul_submoduleOf).
These sums are the arithmetic input to the transformation law of the theta series of an even
lattice under Γ₀(N), where N = ZLattice.level L is the level of L: the least N > 0 with
N ‖x‖² ∈ 2ℤ for every x ∈ L^∨. Splitting L into classes modulo c expresses the theta series
at a cusp a / c through the twisted Gauss sums, and two facts proved here reduce them to the
untwisted sum. For a multiple c of the level, G_L(a, c; m) vanishes unless m ∈ L, by the
shift y ↦ y + c w, w ∈ L^∨, which multiplies the sum by e(⟪w, m⟫); and for m ∈ L and
a d ≡ 1 (mod c), completing the square gives G_L(a, c; m) = e(-a d² ‖m‖² / (2c)) G_L(a, c).
The reciprocity law for these sums exchanges the modulus and the coefficient and passes to the dual lattice: its other side is the sum
∑_{y ∈ L^∨ / cL} e(a ‖y‖² / (2c))
over the classes of L^∨ modulo cL, again well defined because L is even. For an integral
full lattice of rank n and c ≠ 0 the quotient L^∨ / cL is finite with
cⁿ |L^∨ / L| = cⁿ det L elements (ZLattice.IsIntegral.card_quotient_smul_submoduleOf_dual).
When the coefficient is a multiple N b of the level, the summand is invariant under the larger
lattice c L^∨, and the scaling y ↦ √N • y turns the sum into |L^∨ / L| times the Gauss sum
G_{√N • L^∨}(b, c) of the Fricke partner √N • L^∨, which is again an even lattice.
At an odd modulus both sums are evaluated by the Jacobi symbol of the signed discriminant
D_L = (-1)^k det L of an even lattice of rank 2k, where det L = |L^∨ / L|. In a ℤ-basis of
L with Gram matrix B, the quotient L / cL is (ℤ / c)^{2k}
(Module.Basis.quotientSmulEquivPiZMod) and a ‖y‖² / 2 is the quadratic form xᵀ (a B / 2) x.
Since 2 is invertible modulo an odd c, G_L(a, c) is the Gauss sum of a quadratic form modulo
c (Matrix.IsSymm.sum_stdAddChar_dotProduct_mulVec_eq_jacobiSym), and
G_L(a, c) = (D_L / c) c^k when c is prime to a and to det L. The Fricke partner has
determinant N^{2k} / det L, which differs from det L by the square (N^k / det L)², so the
descent gives ∑_{y ∈ L^∨ / aL} e(t ‖y‖² / (2a)) = det L · (D_L / a) · a^k for an odd a and a
multiple t of the level prime to a.
Main definitions #
ZLattice.gaussTerm L hL a c m: the summande(a ‖y‖² / (2c) + ⟪y, m⟫ / c)onL / cL.ZLattice.gaussSumTwisted L hL a c m: the twisted Gauss sumG_L(a, c; m).ZLattice.gaussSum L hL a c: the Gauss sumG_L(a, c).ZLattice.dualGaussTerm L hL a c: the summande(a ‖y‖² / (2c))onL^∨ / cL.ZLattice.dualGaussSum L hL a c: the Gauss sum∑_{y ∈ L^∨ / cL} e(a ‖y‖² / (2c)).
Main statements #
ZLattice.norm_gaussSumTwisted_le:‖G_L(a, c; m)‖ ≤ |L / cL|.ZLattice.gaussSumTwisted_eq_of_modEq,ZLattice.gaussSum_eq_of_modEq: the Gauss sums depend onaonly moduloc.ZLattice.gaussSumTwisted_one,ZLattice.gaussSum_one:G_L(a, 1; m) = G_L(a, 1) = 1.ZLattice.gaussSum_comap: the Gauss sums are invariant under linear isometry equivalences.ZLattice.gaussSumTwisted_orthSum,ZLattice.gaussSum_orthSum: the Gauss sums are multiplicative under orthogonal direct sums,G_{L ⊕ M}(a, c) = G_L(a, c) G_M(a, c).ZLattice.gaussSumTwisted_eq_zero: if the level ofLdividesc, thenG_L(a, c; m) = 0form ∈ L^∨ \ L.ZLattice.gaussSumTwisted_eq_of_mem: form ∈ Landa d ≡ 1 (mod c),G_L(a, c; m) = e(-a d² ‖m‖² / (2c)) G_L(a, c).ZLattice.dualGaussSum_one: forc = 1the Gauss sum overL^∨ / cLis the sum∑_{γ ∈ L^∨ / L} e(a ‖γ‖² / 2)over the discriminant group.ZLattice.conj_dualGaussSum: complex conjugation negates the coefficient of the Gauss sum overL^∨ / cL.ZLattice.dualGaussSum_level_mul: for an even full lattice of levelN,∑_{y ∈ L^∨ / cL} e(N b ‖y‖² / (2c)) = |L^∨ / L| · G_{√N • L^∨}(b, c).ZLattice.dualGaussSum_eq_pow_mul_card_of_dvd: for an even full lattice of ranknand levelN, ifa N ∣ tthen∑_{y ∈ L^∨ / aL} e(t ‖y‖² / (2a)) = aⁿ |L^∨ / L|.ZLattice.IsEven.gaussSum_eq_jacobiSym: for oddcprime toaand todet L,G_L(a, c) = (D_L / c) c^k.ZLattice.IsEven.dualGaussSum_eq_jacobiSym: for oddaand a multipletof the level prime toa,∑_{y ∈ L^∨ / aL} e(t ‖y‖² / (2a)) = det L · (D_L / a) · a^k.
References #
- B. Schoeneberg, Elliptic Modular Functions, Springer, 1974, Chapter IX.
- J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Springer, 1973, Appendix 4.
The summand #
The summand e(a ‖y‖² / (2c) + ⟪y, m⟫ / c) of the twisted Gauss sums of an even lattice L,
as a function of the class y ∈ L / cL. It is well defined because L is even and m ∈ L^∨;
both are therefore arguments.
Instances For
The Gauss summand of the class of v ∈ L is e((a ‖v‖² / 2 + ⟪v, m⟫) / c).
The Gauss summand has absolute value 1.
Gauss sums #
The twisted Gauss sum G_L(a, c; m) = ∑_{y ∈ L / cL} e(a ‖y‖² / (2c) + ⟪y, m⟫ / c) of an even
lattice L, for m ∈ L^∨. The quotient L / cL is finite for a full lattice and c ≠ 0.
Instances For
The Gauss sum G_L(a, c) = ∑_{y ∈ L / cL} e(a ‖y‖² / (2c)) of an even lattice L.
Instances For
The twisted Gauss sum is bounded by the number of classes of L / cL.
The twisted Gauss sum at m = 0 is the Gauss sum.
The Gauss summand depends on a only modulo c, because ‖y‖² / 2 ∈ ℤ on the even
lattice L.
The twisted Gauss sums depend on a only modulo c.
The Gauss sums depend on a only modulo c.
For c = 1 the quotient L / cL is trivial and the twisted Gauss sum is 1.
For c = 1 the Gauss sum is 1.
Invariance under isometries #
The twisted Gauss sums are invariant under linear isometry equivalences: pulling L back
along e and the twisting vector along with it does not change G_L(a, c; m).
The Gauss sums are invariant under linear isometry equivalences.
Orthogonal direct sums #
The twisted Gauss sums are multiplicative under orthogonal direct sums: for m ∈ (L ⊕ M)^∨
with components m₁ ∈ L^∨ and m₂ ∈ M^∨, G_{L ⊕ M}(a, c; m) = G_L(a, c; m₁) G_M(a, c; m₂).
The Gauss sums are multiplicative under orthogonal direct sums:
G_{L ⊕ M}(a, c) = G_L(a, c) G_M(a, c).
Vanishing and completing the square #
For a multiple c of the level of L, that is, c ‖x‖² ∈ 2ℤ for every x ∈ L^∨, the
twisted Gauss sum G_L(a, c; m) vanishes unless m ∈ L. Indeed c • L^∨ ≤ L, and shifting
the summation variable by c w, w ∈ L^∨, multiplies the sum by e(⟪w, m⟫); as L = (L^∨)^∨,
some w ∈ L^∨ has ⟪w, m⟫ ∉ ℤ when m ∉ L.
Completing the square. For m ∈ L and a d ≡ 1 (mod c), the twisted Gauss sum is a
multiple of the untwisted one: G_L(a, c; m) = e(-a d² ‖m‖² / (2c)) G_L(a, c). The shift
y ↦ y + d m turns e(a ‖y‖² / (2c)) into e(a d² ‖m‖² / (2c)) times the twisted summand.
Gauss sums over L^∨ / cL #
The summand e(a ‖y‖² / (2c)) of the Gauss sums of an even lattice L over L^∨ / cL, as a
function of the class y. It is well defined because L is even: replacing y ∈ L^∨ by
y + c w with w ∈ L changes the argument of e by the integer a ⟪y, w⟫ + a c ‖w‖² / 2.
Instances For
The dual Gauss summand of the class of v ∈ L^∨ is e(a ‖v‖² / (2c)).
The Gauss sum ∑_{y ∈ L^∨ / cL} e(a ‖y‖² / (2c)) of an even lattice L over the classes of
its dual lattice modulo cL. For a full lattice of rank n and c ≠ 0 the quotient is finite
of order cⁿ |L^∨ / L| (ZLattice.IsIntegral.card_quotient_smul_submoduleOf_dual).
Instances For
For c = 1 the Gauss sum over L^∨ / cL is the sum ∑_{γ ∈ L^∨ / L} e(a ‖γ‖² / 2) over the
discriminant group, computed with any function f on L^∨ / L taking the value
e(a ‖v‖² / 2) at the class of each v ∈ L^∨.
Complex conjugation negates the coefficient of the Gauss sum over L^∨ / cL:
conj (∑_y e(a ‖y‖² / (2c))) = ∑_y e(-a ‖y‖² / (2c)).
Descent to the Fricke partner. Let L be an even full lattice of level N. For a
coefficient N b divisible by the level, the Gauss sum over L^∨ / cL is |L^∨ / L| times the
Gauss sum of the Fricke partner √N • L^∨:
∑_{y ∈ L^∨ / cL} e(N b ‖y‖² / (2c)) = |L^∨ / L| · G_{√N • L^∨}(b, c).
Here |L^∨ / L| = det L (ZLattice.IsIntegral.det_eq_card_discriminantGroup), and the right
side is a sum over L^∨ / c L^∨, read on the scaled lattice √N • L^∨.
The trivial dual Gauss sum. Let L be an even full lattice of rank n and level N. If
a N divides t, then every summand e(t ‖y‖² / (2a)) is 1, since N ‖y‖² / 2 ∈ ℤ for
y ∈ L^∨, and the Gauss sum over L^∨ / aL is its number of classes:
∑_{y ∈ L^∨ / aL} e(t ‖y‖² / (2a)) = aⁿ |L^∨ / L|.
Evaluation at an odd modulus #
Gauss sums of an even lattice at an odd modulus. Let L be an even full lattice of even
rank 2k, and c an odd modulus prime to a and to det L = |L^∨ / L|. With the Jacobi symbol
(· / c),
G_L(a, c) = ∑_{y ∈ L / cL} e(a ‖y‖² / (2c)) = ((-1)^k det L / c) c^k.
The dual Gauss sum at an odd modulus. Let L be an even full lattice of even rank 2k
and level N, a an odd modulus, and t a multiple of N prime to a. With the Jacobi symbol
(· / a) and the signed discriminant D_L = (-1)^k det L, where det L = |L^∨ / L|,
∑_{y ∈ L^∨ / aL} e(t ‖y‖² / (2a)) = det L · (D_L / a) · a^k.
At t = -c, for a multiple c of the level prime to a, this evaluates the side over
L^∨ / aL of the reciprocity law for the Gauss sums G_L(a, c).