Using approximate functional equations to build L functions

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Using approximate functional equations to build L functions Pascal Molin Université Paris 7 Clermont-Ferrand 20 juin 2017

Example : elliptic curves Consider an elliptic curve E /Q of conductor N and root number ε = 1. The associated modular form f = a n q n, q = e 2iπz. (1) satifies Wf = f and vanishes at q = e 2π N, f (q) = 0 = q + a 2 q 2 + a 3 q 3 +.... (2)

Example : elliptic curves Consider an elliptic curve E /Q of conductor N and root number ε = 1. The associated modular form f = a n q n, q = e 2iπz. (1) satifies Wf = f and vanishes at q = e 2π N, f (q) = 0 = q + a 2 q 2 + a 3 q 3 +.... (2) By Hasse, a n σ 0 (n) n n, so the equality is possible only if q nq n q = (1 q) k 2 2 q = q2 (2 q) (1 q) 2. (3)

b(q)=q*(2-q)/(1-q)^2; plot(n=.5,40,b(exp(-2*pi/sqrt(n)))) 0.52188-0.99972 0.5 40

b(q)=q*(2-q)/(1-q)^2; plot(n=.5,40,b(exp(-2*pi/sqrt(n)))) solve(n=.5,40,b(exp(-2*pi/sqrt(n)))-1) 26.181852174699964975652391885916899331 Theorem If an elliptic curve has rank r 1, its conductor satisfies N 27.

Least conductor : generalize Degree 2 L-function L(s), one gamma factor Γ C (s), conductor N, weight k and sign ε. The (symmetrized) inverse Mellin transform F (x) = e x 2 a n e 2π e x n N (4) satisfies F (x) = εf ( x). (5) In particular for all 0 < y < π 2, F (iy) ϵf ( iy) = 0. Let t = e iy k 2, q = e 2π e iy N, one must have a n (tq n ϵtq n ) = 0 n 1

Least conductor : generalize Using a n σ 0 (n)n k 1 2 3n k 2 n>k n k 2 q n 1 K +1 k k 2αy (K+1) one must have (t = e iy k 2, q = e 2π N e iy ) tq ϵtq K n=2 q K = B K (q) if 2α y (K + 1) > k σ 0 (n)n k 1 2 tq n ϵtq n + 2 3B K (q)

Odd elliptic curves 0.46634-1.1072 10 40 y = 0.1 N 25.63 better!

Odd elliptic curves 0.41199-1.272 10 40 y = 0.2 N 27.40. better!

Odd elliptic curves 0.33746-1.5123 10 40 y = 0.3 N 29.77.. better!

Odd elliptic curves 0.28407-1.6234 10 40 y = 0.4 N 31.58... better!

Odd elliptic curves 0.27575-1.7467 10 40 y = 0.5 N 32.53... better!

Odd elliptic curves 0.31813-1.7217 10 40 y = 0.6 N 29.57... too bad!

Odd elliptic curves 0.48487-1.1718 10 40 y = 0.7 N 26.48.. and worse

Odd elliptic curves 0.67103-0.70002 10 40 y = 0.8 N 20.81. and worse

Odd elliptic curves 0.8971-0.37453 10 40 y = 0.9 N 17.95 and worse

Equation F (0) = 0 N 27. Equation F (iy) + F ( iy) = 0, y =.5 N 33 Theorem If an elliptic curve has rank r 1, its level satisfies N 33. (was N 27 previously)

Automatic results Plot on y for the least value of N satisfying inequality. 32.943 20.806 0.1 0.8 k = 2, ε = 1 N 33

Automatic results Plot on y for the least value of N satisfying inequality. 10.46 8.9465 0.1 0.8 k = 2, ε = +1 N 11

Automatic results Plot on y for the least value of N satisfying inequality. 4.6318 4.1107 0.1 0.8 k = 4, ε = +1 N 5

Automatic results Plot on y for the least value of N satisfying inequality. 2.8652 1.9185 0.1 0.9 k = 6, ε = +1 N 3

Automatic results Plot on y for the least value of N satisfying inequality. 1.9603 1.2878 0.1 0.9 k = 8, ε = +1 N 2

Automatic results Plot on y for the least value of N satisfying inequality. 1.7469 0.93798 0.1 0.9 k = 10, ε = +1...hum

Bifurcation the ratio is not monotonous and crosses the 1-axis several times. 5.2946-4.9175 0.5 5 Still the result N > 1.7 is correct, combining several plots.

Results for degree 2 weight k and level N, root number ϵ. k ϵ bound LMFDB object accurate 2 1 N > 10.45 11a -1 N > 32.95 37a 4 1 N > 4.63 5.4.1a -1 N > 12.24 13.4.1a 6 1 N > 2.85 3.6.1a -1 N > 6.55 7.6.1b 8 1 N > 1.95 2.8.1a -1 N > 4.09 5.8.1a 10 1 N > 1.37 2.10.1a -1 N > 2.75 3.10.1b 12 1-1 N > 1.97 4.12.1a

k=12 : Ramanujan function 6.43-4.2363 0.4 3 x = 0.05i, N 0.86

k=12 : Ramanujan function 7.6495-5.8338 0.4 3 x = 0.20i, N 0.88

k=12 : Ramanujan function 8.1186-9.573 0.4 3 x = 0.25i, N 0.90

k=12 : Ramanujan function 5.922-7.5415 0.4 3 x = 0.35i, N 0.93

k=12 : Ramanujan function 5.3859-6.2267 0.4 3 x = 0.40i, N 0.94

k=12 : Ramanujan function 7.6747-5.5377 0.4 3 x = 0.45i, N 0.96

k=12 : Ramanujan function 3.9056-9.1369 0.4 3 x = 0.50i, N 0.99

k=12 : Ramanujan function 4.7161-8.0303 0.4 3 x = 0.55i, 0.90 N 1.01 or N 1.18

k=12 : Ramanujan function 4.61-6.8765 0.4 3 x = 0.60i, 0.84 N 1.04 or N 1.27

k=12 : Ramanujan function 6.2605-5.23 0.4 3 x = 0.70i, 0.78 N 1.12 or N 1.37

k=12 : Ramanujan function 7.0654-5.8119 0.4 3 x = 0.80i, N 0.75

k=12 : Ramanujan function 5.5883-4.5893 0.4 3 x = 0.90i, N 0.74

k=12 : Ramanujan function 5.4563-3.078 0.4 3 x = 1.00i, N 0.55

k=12 : Ramanujan function From an analytic point of vue, it s a miracle that exists. It could not for N <.9999, nor 1.00001 < N < 1.37.

General L functions Dirichlet series L(s) = n 1 a n n s gamma factor of level N and degree d γ(s) = N s 2 d j=1 functional equation of weight k Γ R (s + λ i ) Λ(s) = L(s)γ(s) = ϵλ(k s) Ramanujan bound a n σ 0 (n) d 1 n k 1 2 + Λ meromorphic. Here assume holomorphic.

Theta equations Fourier form Λ(s) = ϵλ(k s) if and only if for all x R+] dπ 4, dπ 4 [, where F (x) = ϵf ( x) F (x) = e k 2 x a n M 1 [γ(s); e x n] (6) = 1 2π n R Λ( k 2 + it)eixt dt (7)

Inverse Mellin transforms γ(s) N 2 s Γ R (s) N 2 s Γ C (s) N 2 s Γ C (s)γ C (s + ν) M 1 [γ; x] e N π x2 e 2π x N Bessel K ν More gamma factors (real shifts) now available in Pari/gp g=gammamellininvinit([0,0,1,1,1]) gammamellininv(g,x) Conclusion For any L function, easy to produce equations n a n x n satisfied by the Dirichlet series, with x n 0 exponentially.

Results for weight k=1 Theorem (Smallest discriminants of number fields) Let K /Q be a number field of signature r, s. if r, s = 3, 0, then 25 ; if r, s = 1, 1, then 15 ; if r, s = 4, 0, then 105 ; if r, s = 2, 1, then 64 ; if r, s = 0, 2, then 40 ; if r, s = 5, 0, then 356.

Proof (case 4,0) ζ K (s) = ζ K (s)/ζ(s) is degree 4, weight 1, holomorphic L function, with γ(s) = N s 2 Γ R (s) 4. 104.43 48.449 0.2 1.3

Example : weight k=1 First weight 1 L functions gamma N formulas ( ) ( ) ( ) [0] 5, 8, 12,... ( 5 ), ( 8, ) ( 12 ),... [1] 3, 4, 7,... ( -3 ), -4, -7,... [0, 0] 25, 40, 49,... 5 2, ( ) 5 8, ζ ( x 3 -x 2-2 x+1,... [0, 1] 15, 20, 23,... -3 5 ) (, -4 5 ), ζ ( ) x 3 -x 2 +1,... [1, 1] (-3) 2 9, 12, 16,..., ( ) ( -3-4, -4-4 ),... ( ) ( ) [0, 0, 0] 125, 200, 245,... 5 3, 5 2 8, ( ) 5 ζ ( ) ( ) x 3 -x 2-2 x+1,... [0, 0, 1] 75, 100, 115,... -3 5 2, -4 5 2, ( ) 5 ζ ( ) x 3 -x 2 +1,... [0, 1, 1] (-3) 45, 60, 69,... 2 5, ( ) ( ) -3-4 5, -3 ζ ( ) ) ( x 3 )-x 2 +1,... [1, 1, 1] (-3) 3-3 (-4) 2 27, 36, 48,...,,,... ( (-3) 2-4

Enumerate Dirichlet series Recall : from the beginning we use q k q b k, where a k b k. (8) k 2 Could push the game further trying values of a 2 in the left-hand side... q + a2 q 2 q k b k, where a k b k. (9) k 3 for all values a 2 [ b 2, b 2 ].

Enumerate Dirichlet series goal find all L-functions having specified invariants input an approximate functional equation n a n x n = 0 hypothesis a n Z + Ramanujan bounds a n b n. 1 a 1 = 1 2 a 2 [ b 2, b 2 ] s.t. (a 1 x 1 ) + a 2 x 2 B 2 = b 3 x 3 + b 4 x 4 + b 5 x 5 + b 6 x 6 + b 7 x 7 + 3 a 3 [ b 3, b 3 ] s.t. (a 1 x 1 + a 2 x 2 ) + a 3 x 3 B 3 = b 4 x 4 + b 5 x 5 + b 6 x 6 + b 7 x 7 +. 4...

Enumerate Dirichlet series goal find all L-functions having specified invariants input an approximate functional equation n a n x n = 0 hypothesis a n Z + Ramanujan bounds a n b n. 1 a 1 = 1 2 a 2 [ b 2, b 2 ] s.t. (a 1 x 1 ) + a 2 x 2 B 2 = b 3 x 3 + b 4 x 4 + b 5 x 5 + b 6 x 6 + b 7 x 7 + 3 a 3 [ b 3, b 3 ] s.t. (a 1 x 1 + a 2 x 2 ) + a 3 (x 3 + a 2 x 6 ) B 3 = b 4 x 4 + b 5 x 5 + + b 7 x 4...

More structure on Dirichlet series Euler product L(s) = F p (p s ) 1 Euler factor F p (T ) = 1 + c p,1 T +... T d local functional equation c p,d j = χ(p)p w 2 (d 2j) c p,j with χ Dirichlet character modulo N Ramanujan bounds : using roots p w 2 c p,j a k k w 2 ( ) d j p j w 2 p e k ( ) e + d 1 d 1

Search Idea Explore a tree of Dirichlet series (or local Euler factors) Search on Euler coefficients c p,e for e d 2 (or e d 1 if p N). Depth-first search (constant in memory) Examples gp > lfunbuild([[],1,[0,1],2,11,1],45,[2]) time = 11 ms. %1 = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] gp > lfunbuild([[],1,[0,1],2,12,1],45,[2]) time = 11 ms. %2 = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

Search Idea Explore a tree of Dirichlet series (or local Euler factors) Search on Euler coefficients c p,e for e d 2 (or e d 1 if p N). Depth-first search (constant in memory) Examples gp> lfunbuild([[],1,[0,1],2,26,1],59,[1]) time = 7 ms. %3 = [1, 3, 3, 3, 2, 26, 4, 15, 5, 2, 2, 2, 2, 12, 2, 2, 2]

Idea Search Explore a tree of Dirichlet series (or local Euler factors) Search on Euler coefficients c p,e for e d 2 (or e d 1 if p N). Depth-first search (constant in memory) Examples gp> lfunbuild([[],1,[0,1],2,26,1],59,[1]) time = 7 ms. %3 = [1, 3, 3, 3, 2, 26, 4, 15, 5, 2, 2, 2, 2, 12, 2, 2, 2] the choice of equation is still important! gp> lfunbuild([[],1,[0,1],2,26,1],59,[[.42]]) time = 7 ms. %4 = [1, 3, 3, 2, 3, 2, 4, 2, 3, 2, 2, 2, 2, 2, 4, 2, 2]

Program lfunbuild Two independant functions prepare seach tree compute a family of equations identify search variables, ranges, search levels in the tree craft nice equation for each level compute tails B p prune tree using depth first search (constant memory) start at p = 2 solve polynomial(a p ) B p for each possible value a p propagate value in Dirichlet series recursively descend next level gp > for(n=10,100,print(n,": ",Vec(lfunbuild([[],1,[0,1],2,N,1],31,[2]))))

Framework Write k p e if k is p-smooth and p e k. Assume {a k } known for k p e. Equation for coefficient a p e : ( ) ( ) k p e a k x k + a p e a m x p e m m p Ramanujan bound a n b n on the tail polynomial equation P(a p e ) r solve in integers in [ b p e, b p e ]. = k p e a k x k

Framework Write k p e if k is p-smooth and p e k. Assume {a k } known for k p e. Equation for coefficient a p e : ( ) ( ) k p e a k x k + a p e a m x p e m m p If e d 2, by reciprocity of F p(t ) all a p l ( ) a n x n n p e Ramanujan bound a n b n on the tail polynomial equation P(a p e ) r solve in integers in [ b p e, b p e ]. = k p e a k x k in terms of a p j, j e ( ) + a p l a m x p l m = a n x n l e m p n p

Good and bad tails Case of polynomial equation of degree 1 : S 0 + a p S 1 B p = b p+ x p+ + b p++ x p++ +... S 1 B p at most one solution a p. the ratio B p x p can be studied a priori usually nice at big prime gaps can be horrible for twin primes

Combine equations Combine approximate functional equations : X n = (x n,1,... x n,r ) define S = n p e a n X n, T = m p a m X p e m, and R = (b n X n ) n p. For all λ R n, S.λ + a p T.λ R.λ 1 Choose λ to minimize R.λ W.λ : least absolute deviations. Can be solved with iterated + weighted least squares.

Observations Ramanujan very easy to build (despite k = 12). gp > lfunbuild([[],1,[0,1],12,1,1],100,2) time = 42 ms. %1 = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Observations Ramanujan very easy to build (despite k = 12). Same for higher weight, conductor 1 gp > lfunbuild([[],1,[0,1],20,1,1],20,[2],1) time = 20 ms. %4 = [[1, 456, 50652, -316352, -2377410, 23097312, -16917544, -383331840, 140336

Observations Ramanujan very easy to build (despite k = 12). Same for higher weight, conductor 1 gp > lfunbuild([[],1,[0,1],20,1,1],20,[2],1) time = 20 ms. %4 = [[1, 456, 50652, -316352, -2377410, 23097312, -16917544, -383331840, 140336 gp > mfcoefs(mfsearch([1,20])[1][2],30) %5 = [0, 1, 456, 50652, -316352, -2377410, 23097312, -16917544, -383331840, 1403 For N = 66, k = 2 and central character χ = ( 66 ), a match exists for the 71 first primes, then disapears.

Observations Ramanujan very easy to build (despite k = 12). May need many equations to cancel tail (here 11a 2) For N = 66, k = 2 and central character χ = ( 66 ), a match exists for the 71 first primes, then disapears. ( ) 66 is trivial up to p = 19, and π(19 2 ) = 72

Observations Ramanujan very easy to build (despite k = 12). May need many equations to cancel tail (here 11a 2) For N = 66, k = 2 and central character χ = ( 66 ), a match exists for the 71 first primes, then disapears. Modular forms expansions 805b and 805c start to differ at primes 11 and 13, with values exchanged

Conclusion ToDo work still in progress... [bugs, precision] optimize equations try other sources of equations write tree search in ball arithmetic (Arb) Goals compute interesting examples. Challenges : Γ C (s 6)Γ C (s), k = 20, N = 1, a 2 = 0... prove nothing exists outside what is expected fill missing bad Euler factors rigorously (e.g. symmetric powers)

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