Hypergeometric motives and an unusualapplication of the Guinand-Weil-Mestre
explicit formula
David P. Roberts
University of Minnesota, MorrisJanuary 17, 2017

Sections of today’s talk
1. Hypergeometric motives, illustrated by1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
M14 = H 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 , 1 .
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
A key feature is that there is a decomposition M14 = M8 ⊕ M6 .
2. A sketch of the Guinand-Weil-Mestre explicit formula and how it
gives lower bounds on conductors for general motives.
3. Applying the explicit formula to understand the factorization
L(M14 , s) = L(M8 , s)L(M6 , s).
Note: After initializing the variable x throughR<x>:=PolynomialRing(Integers());, all brown commands canby typed directly into Magma. To speed up (necessary for the free
online calculator), reduce precision from 10 to 5.

1. Hypergeometric motives. Why study them?
Many people here are studying aspects of hypergeometric motives
M = H(α1 , . . . , αd ; β1 , . . . , βd ; t).
An attractive feature is that for certain (α, β) these are classicalobjects, coming from number fields, Artin representations, genus onecurves, genus two curves, K3 surfaces, etc.
For general (α, β), these motives come from more exotic algebraic
varieties. However source varieties can often be subjugated to a
background role, as many standard questions are answerable directly
and uniformly in terms of the parameters (α, β, t).
A goal of an ongoing joint project with Fernando Rodriguez Villegasand Mark Watkins is to use hypergeometric motives to illustrate thegeneral theory of motives.

Hypergeometric L-functions
In particular, Magma currently goes far towards the goal of allowing
one to input a rational (α, β, t) and receive its complete L-function
Λ(H(α, β, t), s) = N s/2 L∞ (s)Yp1.
fp (p −s )This L-function is computed factor-by-factor. Different techniques,due to a wide range of people, are relevant for different factors.
Like all self-dual motivic L-functions, these hypergeometric motivic
L-functions conjecturally satisfy
Λ(H(α, β, t), s) = ±Λ(H(α, β, t), w + 1 − s),with w being the weight of H(α, β, t).

The Magma package illustrated by today’s example
Getting Magma’s guess at the L-function:
H:=HypergeometricData([1/2: i in [1..16]], [0: i in [1..16]]);
L := LSeries(H,1:Precision:=10);
Magma warns you that its guesses at 2 may be wrong, but that is nota concern for us yet.
HodgeStructure(L:PHV); returns[1,1,1,1,1,1,1,0,0,1,1,1,1,1,1,1]
This gives the list (h15,0 , h14,1 , . . . , h1,14 , h0,15 ). In particular thismotive can only appear in the cohomology of varieties of dimension≥ 15. In this particular case, the Hodge vector can easily becalculated mentally!

A decomposition, f3(x), and Hodge numbers
In general, if d is even and the αi ’s and the βj ’s are obtained fromone another by adding 1/2, then H(α, β, 1) decomposes as a sum of
two motives. In our case, we know a priori that M14 = M8 ⊕ M6 .
Factorization(EulerFactor(L,3));then tells us (in two seconds!) that f3 (x) =(1 − 268 · 3x + 204193 · 34 x 2 − 1001800 · 39 x 3 + 204193 · 319 x 4−268 · 331 x 5 + 345 x 6 )(1 + 2992 · x + 39116 · 34 x 2 − 7596496 · 36 x 3 − 203836426 · 312 x 4−7596496 · 321 x 5 + 39116 · 334 x 6 + 2992 · 345 x 7 + 360 x 8 )Thus, M6 and M8 are both irreducible. Moreover Newton-over-Hodge
forces the Hodge vector of M14 to decompose nicely:
M6 :
M8 :

(0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0) =: h6(1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1) =: h8 .

f5(x) and motivic Galois groups
Factorization(EulerFactor(L,5)); takes 30 seconds and tells
us that f5 (x) =(1 + 1614 · 53 x + 28284579 · 54 x 2 + 1394686516 · 59 x 3 + 28284579 · 519 x 4+1614 · 533 x 5 + 545 x 6 )(1 − 41208 · x − 44999364 · 53 x 2 − 22376708712 · 56 x 3 + 3926679014806 · 512 x 4−22376708712 · 521 x 5 − 44999364 · 533 x 6 − 41208 · 545 x 7 + 560 x 8 )The two factors define completely different number fields from thoseof f3 (x) as
Gal(f3a (x)f5a (x)) = W3 × W3 ,Gal(f3b (x)f5b (x)) = W4 × W4 .
This fact implies that the Mk each have motivic Galois group as large
as possible, namely GSp2k .

Behavior at 2
EulerFactor(L,2); returns 1, telling us that Magma is guessing atrivial Euler factor at 2.
Conductor(L); returns 16384, which we recognize as 214 .
Are these right? CFENew(L); takes four minutes and returns
0.05909621133, so no!
What are the right factors? After some experimentation we redefine
L := LSeries(H,1:Precision:=10, BadPrimes:=[<2,15,1>]);
Then CFENew(L) takes eight minutes and returns 0.0000000000, so
we proceed under the assumption yes! We will likewise trust similar
analytic computations in the sequel.

Analytic computations
Sign(L); instantaneously returns 1, so L has even analytic rank.
Evaluate(L,8); takes four seconds and returns 0.000000000, so L
has analytic rank > 0.
Evaluate(L,8:Derivative:=2); takes fourteen seconds andreturns 7.851654518, so L has analytic rank 2.
The Hardy Z -function is a rescaled version of L(M, 8 + ti). On [0, 7]
it graphs out to64
2
1234567-2

The double zero at t = 0 is visible. The next three roots are
γ1 ≈ 1.93195000805, γ2 ≈ 3.00559765, γ3 ≈ 3.61679.

Questions about the factorization
L(M, s) = L(M6, s)L(M8, s)
Our main focus:
Q1. Since f2 (x) = 1, there are only two possibilities for(cond(M6 ), cond(M8 )), namely (26 , 29 ) or (27 , 28 ). Which is it?
Q2. There are only three possibilities for (rank(M6 ), rank(M8 )),namely (2, 0), (1, 1), or (0, 2). Which one is correct?
Closely related questions:• Are the factorizations of f7 (x), f11 (x), . . . obtainable?
• In the factorization Z (t) = Z6 (t)Z8 (t), which γj are roots of Z6 (t)and which are roots of Z8 (t)?

2. Quick sketch of the GWM explicit formula
In this section, we sketch the Guinand-Weil-Mestre explicit formula as
it appears in Mestre’s 1988 Compositio paper Formules explicites etminorations de conducteurs de variétés algébriques.
Throughout, we assume the Riemann hypothesis for all L-functions.
Without this assumption, the final lower bounds obtained areconsiderably weaker.
Mestre emphasizes the Hodge vectors (g , g ) for abelian varieties and(1, 0, . . . , 0, 1) for modular forms. We emphasize here its applicability
to general Hodge vectors h, although restrict to odd weight motives
for simplicity.
We use the analytic normalization where the functional equation has
the form s 7→ 1 − s. In this kill-Tate-twists spirit, we write hp−qinstead of hp,q .

The formula
For any odd weight motive M, and any allowed test function F , theHodge vector h, the conductor N, the analytic rank r , the Frobenius
traces cpe = Tr(frep |M), and the critical 1/2 + γk i in the upper halfplane are related bylog N = 2πr Fˆ(0) + 4πXFˆ(γk ) + 2Z ∞k+2XXpe

cpe0Fˆ(t)Xhj Ej (t)dt

jlog p log pF().
p e/22π

Today we are thinking of this explicit formula as an infinite family ofexact formulas for log N which can be used to get lower bounds onlog N. The Ej (t) are built in a simple way from the digammafunction Γ0 (t)/Γ(t).

The Fourier transform and test functions
We require F (x) to be even, compactly supported with F (0) = 1,and have two continuous derivatives. Its Fourier transform is thenZ ∞ˆF (t) =F (x)e −2πitx dx.
−∞

Among many standard properties is the scaling property: the Fouriertransform of F (x/z) is z Fˆ(zt).
In this talk, we used only scaled versions of the Odlyzko function:
sin |πx|FOd (x) = χ[−1,1] (1 − |x|) cos(πx) +, (in C 2 but not C 3 ).
πIts Fourier transform is8 cos2 (πt)ˆFOd (t) = 2(quartic decay at ∞).
π (1 − 4t 2 )2For brevity, we write Fz (x) = FOd (2πx/ log z).

Plots of typical test functions
One would like both F and Fˆ to very localized, but this is impossible
because of the uncertainty principle. F2 and F13 :
1.0
0.8
0.6
0.4
0.2
0.1

0.2

0.3

0.4

Fb2 and Fb13 :
0.30
0.25
0.20
0.15
0.10
0.05
24

6810

The pair (F2 , Fˆ2 ) can be used to give lower bounds for conductors ofarbitrary motives with given Hodge vectors. For example for h6 andh8 these lower bounds are 1.96 and 2.91.

3. Understanding L(M14, s) = L(M6, s)L(M8, s)
well enough to answer Q1 and Q2
We have tons of cpe for our motive M14 . However, to get thedecomposition cpe = cp6e + cp8e , even for just e = 1, we need to factor
all of fp (x). The next two (8 minutes and 2.5 hours):
F7 (x)

=


4 2
9 3
19 4
1 + 248232 · 7x + 36864645 · 7 x − 12114440144 · 7 x + 36864645 · 7 x +31 5
45 6248232 · 7 x + 7 x·
2 26 313 4
1 + 667104x + 92084011804 · 7 x + 107704347009888 · 7 x + 216772203079210 · 7 x

21 532 645 760 8+107704347009888 · 7 x + 92084011804 · 7 x + 667104 · 7 x + 7 x

F11 (x)

=


4 2
9 3
19 4
1 − 883812 · 11x + 86399921193 · 11 x − 113266524342552 · 11 x + 86399921193 · 11 x

31 545 6−883812 · 11 x + 11 x

2 2
7 3
12 4
1 + 34438544x + 7563161639884 · 11 x − 5931371880123984 · 11 x + 1164681420132811670 · 11 x

22 532 645 760 8−5931371880123984 · 11 x + 7563161639884 · 11 x + 34438544 · 11 x + 11 x=

Applying the explicit formula to M6 and M8
Plugging into the explicit formula using (F13 , Fˆ13 ), dividing all terms
by log 2 for greater clarity, and keeping track of partial sums:

h357
911r
γ1γ2..
.
Total

term6
3.113240.17011−0.35472−0.07386−0.02269
0.00028
2.99946
0.13610..
.

(Tends to6 or 7)
total6
3.11324
3.28335
2.928642.85477
2.83209
2.83237
5.831835.83183
5.96793..
.
6.00000

term8
4.86171−0.63306
0.07245−0.02836
0.00183−0.001012.999461.68061..
.

(Tends to8 or 9)
total84.86171
4.22866
4.301114.27275
4.27458
4.27357
7.273038.953648.95364..
.
9.00000

Comments

from the successively
harder factorizationsof Frobenius
polynomials fp (x)Forced! A2 = (1, 1)
Forced! A1 = (26 , 29 )Forced!