The functional equation for L-functions of hyperelliptic curves. Michel Börner
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1 The functional equation for L-functions of hyperelliptic curves Jahrestagung SPP 1489, Osnabrück Michel Börner Ulm University September 30, 2015 Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
2 Setup Y smooth projective curve over Q (number field) Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
3 Setup Y smooth projective curve over Q (number field) for fixed p Q and a model Y of Y define: Ȳ special fiber, Ȳ = Y Zp F p Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
4 Setup Y smooth projective curve over Q (number field) for fixed p Q and a model Y of Y define: Ȳ special fiber, Ȳ = Y Zp F p L(Y, s) a n L series, L(Y, s) = n n N s = p L p (Y, s) Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
5 Setup Y smooth projective curve over Q (number field) for fixed p Q and a model Y of Y define: Ȳ special fiber, Ȳ = Y Zp F p L(Y, s) g N a n L series, L(Y, s) = n n N s = p genus g = g(y) conductor, N = p p f p L p (Y, s) Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
6 Functional Equation Conjectured functional equation Λ(Y, s) = ±Λ(Y, 2 s) (FEq) where Λ(Y, s) = N s (2π) gs Γ(s) g L(Y, s) For elliptic curves /Q this is the modularity theorem (Wiles) Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
7 Functional Equation Conjectured functional equation Λ(Y, s) = ±Λ(Y, 2 s) (FEq) where Λ(Y, s) = N s (2π) gs Γ(s) g L(Y, s) For elliptic curves /Q this is the modularity theorem (Wiles) Using Dokchitser package in sage [D04] (FEq) may be verified numerically (to a high probability) using all a n M N. Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
8 Functional Equation Conjectured functional equation Λ(Y, s) = ±Λ(Y, 2 s) (FEq) where Λ(Y, s) = N s (2π) gs Γ(s) g L(Y, s) For elliptic curves /Q this is the modularity theorem (Wiles) Using Dokchitser package in sage [D04] (FEq) may be verified numerically (to a high probability) using all a n M N. n a n n s = p L p (Y, s) need sufficiently many local factors L p Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
9 Good and bad primes Good, bad and semistable reduction at a prime good reduction: model Y of Y with Ȳ smooth Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
10 Good and bad primes Good, bad and semistable reduction at a prime good reduction: model Y of Y with Ȳ smooth bad reduction: no such Y exists Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
11 Good and bad primes Good, bad and semistable reduction at a prime good reduction: model Y of Y with Ȳ smooth bad reduction: no such Y exists semistable reduction: Ȳ has at most ordinary double points Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
12 Good and bad primes Good, bad and semistable reduction at a prime good reduction: model Y of Y with Ȳ smooth bad reduction: no such Y exists semistable reduction: Ȳ has at most ordinary double points Good p: Ȳ is curve given by equation mod p Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
13 Good and bad primes Good, bad and semistable reduction at a prime good reduction: model Y of Y with Ȳ smooth bad reduction: no such Y exists semistable reduction: Ȳ has at most ordinary double points Good p: Ȳ is curve given by equation mod p L p (Y, s) 1 = numerator of zeta fctn. of Ȳ = P(Ȳ, T = p s ) Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
14 Good and bad primes Good, bad and semistable reduction at a prime good reduction: model Y of Y with Ȳ smooth bad reduction: no such Y exists semistable reduction: Ȳ has at most ordinary double points Good p: Ȳ is curve given by equation mod p L p (Y, s) 1 = numerator of zeta fctn. of Ȳ = P(Ȳ, T = p s ) P(Ȳ, T) = 1 + c 1 T c 2g T 2g (Weil conjectures) Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
15 Good and bad primes Good, bad and semistable reduction at a prime good reduction: model Y of Y with Ȳ smooth bad reduction: no such Y exists semistable reduction: Ȳ has at most ordinary double points Good p: Ȳ is curve given by equation mod p L p (Y, s) 1 = numerator of zeta fctn. of Ȳ = P(Ȳ, T = p s ) P(Ȳ, T) = 1 + c 1 T c 2g T 2g (Weil conjectures) c i : point counting on Ȳ Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
16 Good and bad primes Good, bad and semistable reduction at a prime Good p: good reduction: model Y of Y with Ȳ smooth bad reduction: no such Y exists semistable reduction: Ȳ has at most ordinary double points Ȳ is curve given by equation mod p L p (Y, s) 1 = numerator of zeta fctn. of Ȳ = P(Ȳ, T = p s ) P(Ȳ, T) = 1 + c 1 T c 2g T 2g c i : point counting on Ȳ f p = 0 no contribution to conductor N (Weil conjectures) Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
17 Good and bad primes ctd. Bad p: L p, f p depend on Galois representation on H 1 et (Y Q, Q l) Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
18 Good and bad primes ctd. Bad p: L p, f p depend on Galois representation on Het 1 (Y Q, Q l) First compute semistable reduction of Y at p Fact (Deligne Mumford): Y has semistable reduction over some finite extension of Q p Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
19 Good and bad primes ctd. Bad p: L p, f p depend on Galois representation on Het 1 (Y Q, Q l) First compute semistable reduction of Y at p Fact (Deligne Mumford): Y has semistable reduction over some finite extension of Q p Algorithm for all bad L p, f p for certain types of curves using methods in [BW15] Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
20 Good and bad primes ctd. Bad p: L p, f p depend on Galois representation on Het 1 (Y Q, Q l) First compute semistable reduction of Y at p Fact (Deligne Mumford): Y has semistable reduction over some finite extension of Q p Algorithm for all bad L p, f p for certain types of curves using methods in [BW15] Next: Class of hyperelliptic curves over Q with semistable reduction everywhere Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
21 Hyperelliptic curves with semistable red. at all p Hyperelliptic setup (p = 2 case) g, h Z p [x], g monic, deg g = 2g(Y) + 1, deg h g(y), gcd(f, f ) = 1 Y : y 2 + h(x)y = g(x) y 2 = f (x) = 4g(x) + h(x) 2 Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
22 Hyperelliptic curves with semistable red. at all p Hyperelliptic setup (p = 2 case) g, h Z p [x], g monic, deg g = 2g(Y) + 1, deg h g(y), gcd(f, f ) = 1 Y : y 2 + h(x)y = g(x) y 2 = f (x) = 4g(x) + h(x) 2 Fact (p = 2 case) Y has semistable reduction already over Q p Ȳ : y 2 = f (x) = r 2 (x) s(x) with r, s F p [x] separable & coprime Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
23 Hyperelliptic curves with semistable red. at all p Hyperelliptic setup (p = 2 case) g, h Z p [x], g monic, deg g = 2g(Y) + 1, deg h g(y), gcd(f, f ) = 1 Y : y 2 + h(x)y = g(x) y 2 = f (x) = 4g(x) + h(x) 2 Fact (p = 2 case) Y has semistable reduction already over Q p Ȳ : y 2 = f (x) = r 2 (x) s(x) with r, s F p [x] separable & coprime p = 2 : A little more work Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
24 Computations at bad primes p = 2 Ȳ : y 2 = r 2 (x) s(x) Ȳ : y 2 = r(x) 2 s(x) X Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
25 Computations at bad primes p = 2 Ȳ : y 2 = r 2 (x) s(x) Ȳ 0 : y 2 = s(x) Ȳ : y 2 = r(x) 2 s(x) X Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
26 Computations at bad primes p = 2 Ȳ : y 2 = r 2 (x) s(x) Ȳ 0 : y 2 = s(x) r 1 r 2 r n Ȳ : y 2 = r(x) 2 s(x) X r = i r i F p [x] Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
27 Algorithm Bad primes Exponent f p = #loops = i deg(r i ) Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
28 Algorithm Bad primes Exponent f p = #loops = i deg(r i ) Nodes have split or non-split reduction and with d i = deg(r i ). L p (Y, T = p s ) = L p(ȳ 0, T) (1±T d i ) i Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
29 Algorithm Bad primes Exponent f p = #loops = i deg(r i ) Nodes have split or non-split reduction and with d i = deg(r i ). L p (Y, T = p s ) = L p(ȳ 0, T) (1±T d i ) i L p (Ȳ 0, T) : Just point counting on Ȳ 0 Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
30 Algorithm Bad primes Exponent f p = #loops = i deg(r i ) Nodes have split or non-split reduction and with d i = deg(r i ). L p (Y, T = p s ) = L p(ȳ 0, T) (1±T d i ) i L p (Ȳ 0, T) : Just point counting on Ȳ 0 Point counting (good/bad p): Just for p k M where M N Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
31 Example Y : y 2 + h(x)y = g(x) g = x 7 + x 6 + 2x 5 + 2x 4 + 2x 3 1 h = x 3 + x 2 + x + 2 Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
32 Example Y : y 2 + h(x)y = g(x) g = x 7 + x 6 + 2x 5 + 2x 4 + 2x 3 1 p =2 = Ȳ : y 2 = r 2 (x) s(x) h = x 3 + x 2 + x + 2 bad p: 2, 3, 11, 37 g(y) = 3 Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
33 Example Y : y 2 + h(x)y = g(x) g = x 7 + x 6 + 2x 5 + 2x 4 + 2x 3 1 p =2 = Ȳ : y 2 = r 2 (x) s(x) h = x 3 + x 2 + x + 2 bad p: 2, 3, 11, 37 g(y) = 3 p = 3: Ȳ : y 2 = (x + 1) 2 (x 2 + 1) 2 x Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
34 Example Y : y 2 + h(x)y = g(x) g = x 7 + x 6 + 2x 5 + 2x 4 + 2x 3 1 p =2 = Ȳ : y 2 = r 2 (x) s(x) h = x 3 + x 2 + x + 2 bad p: 2, 3, 11, 37 g(y) = 3 p = 3: Ȳ : L 3 (T) 1 = i(1 ± T d i) L 3 (Ȳ 0, T) y 2 = (x + 1) 2 (x 2 + 1) 2 x = (1 + T) (1 T 2 )/1 f 3 = deg(r) = 3 Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
35 Example Y : y 2 + h(x)y = g(x) g = x 7 + x 6 + 2x 5 + 2x 4 + 2x 3 1 p =2 = Ȳ : y 2 = r 2 (x) s(x) h = x 3 + x 2 + x + 2 bad p: 2, 3, 11, 37 g(y) = 3 p = 3: Ȳ : L 3 (T) 1 = i(1 ± T d i) L 3 (Ȳ 0, T) y 2 = (x + 1) 2 (x 2 + 1) 2 x = (1 + T) (1 T 2 )/1 f 3 = deg(r) = 3 p = 11, 37: similar Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
36 Example Y : y 2 + h(x)y = g(x) g = x 7 + x 6 + 2x 5 + 2x 4 + 2x 3 1 p =2 = Ȳ : y 2 = r 2 (x) s(x) h = x 3 + x 2 + x + 2 bad p: 2, 3, 11, 37 g(y) = 3 p = 3: Ȳ : L 3 (T) 1 = i(1 ± T d i) L 3 (Ȳ 0, T) y 2 = (x + 1) 2 (x 2 + 1) 2 x = (1 + T) (1 T 2 )/1 f 3 = deg(r) = 3 p = 11, 37: similar p = 2: more work Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
37 Example ctd. Other primes: L 1 2 = (1 T 2 ) (1 T + 2T 2 ) f 2 = 2 L 1 11 = (1 + T)2 (1 4T + 11T 2 ) f 11 = 2 L 1 37 = (1 T) ( T 4 ) f 37 = 1 Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
38 Example ctd. Other primes: Conductor N: L 1 2 = (1 T 2 ) (1 T + 2T 2 ) f 2 = 2 L 1 11 = (1 + T)2 (1 4T + 11T 2 ) f 11 = 2 L 1 37 = (1 T) ( T 4 ) f 37 = 1 N = Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
39 Example ctd. Other primes: Conductor N: Point counting: L 1 2 = (1 T 2 ) (1 T + 2T 2 ) f 2 = 2 L 1 11 = (1 + T)2 (1 4T + 11T 2 ) f 11 = 2 L 1 37 = (1 T) ( T 4 ) f 37 = 1 N = #points of Ȳ/F p k for p k Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
40 Example ctd. Other primes: Conductor N: Point counting: L 1 2 = (1 T 2 ) (1 T + 2T 2 ) f 2 = 2 L 1 11 = (1 + T)2 (1 4T + 11T 2 ) f 11 = 2 L 1 37 = (1 T) ( T 4 ) f 37 = 1 N = #points of Ȳ/F p k for p k Verification of (FEq) via Dokchitser package in sage: Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
41 Overview and loose ends Hyperelliptic curves All L p, f p computable for g = 2, 3, 4, 5, 6 with reasonable effort Limitation: point count for good p N = p f p/2 bad p Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
42 Overview and loose ends Hyperelliptic curves All L p, f p computable for g = 2, 3, 4, 5, 6 with reasonable effort Limitation: point count for good p N = p f p/2 bad p (FEq) verified for hundreds of examples, N up to [B15] Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
43 Overview and loose ends Hyperelliptic curves All L p, f p computable for g = 2, 3, 4, 5, 6 with reasonable effort Limitation: point count for good p N = p f p/2 bad p (FEq) verified for hundreds of examples, N up to [B15] Superelliptic curves Y : y n = f (x) Find algorithms for L p and f p for wider range of curves Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
44 Overview and loose ends Hyperelliptic curves All L p, f p computable for g = 2, 3, 4, 5, 6 with reasonable effort Limitation: point count for good p N = p f p/2 bad p (FEq) verified for hundreds of examples, N up to [B15] Superelliptic curves Y : y n = f (x) Find algorithms for L p and f p for wider range of curves Find (all) curves of certain class with small N (in preparation) Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
45 References Thank you! [BW15] [BBW15] [B15] [D04] Computing L-functions and semistable reduction of superelliptic curves, Irene Bouw and Stefan Wewers. To appear in Glasgow Math. J. The functional equation for L-functions of hyperelliptic curves, Michel Börner, Irene I. Bouw, Stefan Wewers, arxiv: Examples of [BBW15], M.B., Computing special values of motivic L-functions, Tim Dokchitser, Exper. Math. 13, No. 2 (2004), Michel Börner (Ulm University) The functional equation for L-functions of hyperelliptic curves September 30, / 12
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