Local behaviour of Galois representations

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Local behaviour of Galois representations Devika Sharma Weizmann Institute of Science, Israel 23rd June, 2017 Devika Sharma (Weizmann) 23rd June, 2017 1 / 14

The question Let p be a prime. Let f ř 8 ně1 a npf qq n be a normalized eigenform in Skě2 new pn, ɛq. Devika Sharma (Weizmann) 23rd June, 2017 2 / 14

The question Let p be a prime. Let f ř 8 ně1 a npf qq n be a normalized eigenform in Skě2 new pn, ɛq. Let ρ f : G Q ÝÑ GL 2 pk f,pq. Devika Sharma (Weizmann) 23rd June, 2017 2 / 14

The question Let p be a prime. Let f ř 8 ně1 a npf qq n be a normalized eigenform in Skě2 new pn, ɛq. Let ρ f : G Q ÝÑ GL 2 pk f,pq. Let G p Ă G Q be the decomposition group at p. Devika Sharma (Weizmann) 23rd June, 2017 2 / 14

The question Let p be a prime. Let f ř 8 ně1 a npf qq n be a normalized eigenform in Skě2 new pn, ɛq. Let ρ f : G Q ÝÑ GL 2 pk f,pq. Let G p Ă G Q be the decomposition group at p. Assume f is p-ordinary, i.e., p a p pf q. Devika Sharma (Weizmann) 23rd June, 2017 2 / 14

The question Let p be a prime. Let f ř 8 ně1 a npf qq n be a normalized eigenform in Skě2 new pn, ɛq. Let ρ f : G Q ÝÑ GL 2 pk f,pq. Let G p Ă G Q be the decomposition group at p. Assume f is p-ordinary, i.e., p a p pf q. Then Wiles showed, η ν ρ f Gp 0 η 1, with η 1 unramified. Devika Sharma (Weizmann) 23rd June, 2017 2 / 14

The question Let p be a prime. Let f ř 8 ně1 a npf qq n be a normalized eigenform in Skě2 new pn, ɛq. Let ρ f : G Q ÝÑ GL 2 pk f,pq. Let G p Ă G Q be the decomposition group at p. Assume f is p-ordinary, i.e., p a p pf q. Then Wiles showed, η ν ρ f Gp 0 η 1, with η 1 unramified. Natural to ask, when does ρ f Gp split? Devika Sharma (Weizmann) 23rd June, 2017 2 / 14

Current status Guess: ρ f Gp splits ðñ f has complex multiplication (CM). Devika Sharma (Weizmann) 23rd June, 2017 3 / 14

Current status Guess: ρ f Gp splits ðñ f has complex multiplication (CM). f has CM ùñ ρ f Gp splits. Devika Sharma (Weizmann) 23rd June, 2017 3 / 14

Current status Guess: ρ f Gp splits ðñ f has complex multiplication (CM). f has CM ùñ ρ f Gp splits. f is non-cm?? ùñ ρ f Gp is non-split. Devika Sharma (Weizmann) 23rd June, 2017 3 / 14

We ask for more Devika Sharma (Weizmann) 23rd June, 2017 4 / 14

We ask for more Every p-ordinary form f is part of a family of modular forms where f f k. H f : tf k0 : k 0 ě 1u Devika Sharma (Weizmann) 23rd June, 2017 4 / 14

We ask for more Every p-ordinary form f is part of a family of modular forms where f f k. H f : tf k0 : k 0 ě 1u Theorem For p 3, every member of a non-cm family H f is non-split Devika Sharma (Weizmann) 23rd June, 2017 4 / 14

We ask for more Every p-ordinary form f is part of a family of modular forms where f f k. H f : tf k0 : k 0 ě 1u Theorem For p 3, every member of a non-cm family H f is non-split if condition pc 1q condition pc 2q are satisfied. Devika Sharma (Weizmann) 23rd June, 2017 4 / 14

Set up Let p 3. Devika Sharma (Weizmann) 23rd June, 2017 5 / 14

Set up Let p 3. Let f P S 2 pnq correspond to an elliptic curve E over Q. Devika Sharma (Weizmann) 23rd June, 2017 5 / 14

Set up Let p 3. Let f P S 2 pnq correspond to an elliptic curve E over Q. Let ρ : ρ f : G S Ñ GL 2 pf p q be the corresponding residual representation. Devika Sharma (Weizmann) 23rd June, 2017 5 / 14

Set up Let p 3. Let f P S 2 pnq correspond to an elliptic curve E over Q. Let ρ : ρ f : G S Ñ GL 2 pf p q be the corresponding residual representation. The Galois action on the p-torsion points on E also gives ρ. Devika Sharma (Weizmann) 23rd June, 2017 5 / 14

Set up Let p 3. Let f P S 2 pnq correspond to an elliptic curve E over Q. Let ρ : ρ f : G S Ñ GL 2 pf p q be the corresponding residual representation. The Galois action on the p-torsion points on E also gives ρ. Assume that Devika Sharma (Weizmann) 23rd June, 2017 5 / 14

Set up Let p 3. Let f P S 2 pnq correspond to an elliptic curve E over Q. Let ρ : ρ f : G S Ñ GL 2 pf p q be the corresponding residual representation. The Galois action on the p-torsion points on E also gives ρ. Assume that p N, Devika Sharma (Weizmann) 23rd June, 2017 5 / 14

Set up Let p 3. Let f P S 2 pnq correspond to an elliptic curve E over Q. Let ρ : ρ f : G S Ñ GL 2 pf p q be the corresponding residual representation. The Galois action on the p-torsion points on E also gives ρ. Assume that p N, ρ is surjective. Devika Sharma (Weizmann) 23rd June, 2017 5 / 14

Set up Let p 3. Let f P S 2 pnq correspond to an elliptic curve E over Q. Let ρ : ρ f : G S Ñ GL 2 pf p q be the corresponding residual representation. The Galois action on the p-torsion points on E also gives ρ. Assume that p N, ρ is surjective. This implies that H f is a non-cm family. Devika Sharma (Weizmann) 23rd June, 2017 5 / 14

Notation Devika Sharma (Weizmann) 23rd June, 2017 6 / 14

Notation H : field cut out by the projective image of ρ Devika Sharma (Weizmann) 23rd June, 2017 6 / 14

Notation H : field cut out by the projective image of ρ Cl H : class group of H, and Ă Cl H : Cl H {Cl p H Devika Sharma (Weizmann) 23rd June, 2017 6 / 14

Notation H : field cut out by the projective image of ρ Cl H : class group of H, and Cl Ă H : Cl H {Cl p H E : global units of H, and E r : E{E p Devika Sharma (Weizmann) 23rd June, 2017 6 / 14

Notation H : field cut out by the projective image of ρ Cl H : class group of H, and Cl Ă H : Cl H {Cl p H E : global units of H, and E r : E{E p U p : ź P p U P, where U P are local units at P p in H Devika Sharma (Weizmann) 23rd June, 2017 6 / 14

Notation H : field cut out by the projective image of ρ Cl H : class group of H, and Cl Ă H : Cl H {Cl p H E : global units of H, and E r : E{E p U p : ź P p U P, where U P are local units at P p in H ru p : U p {U p p Devika Sharma (Weizmann) 23rd June, 2017 6 / 14

Notation H : field cut out by the projective image of ρ Cl H : class group of H, and Cl Ă H : Cl H {Cl p H E : global units of H, and E r : E{E p U p : ź P p U P, where U P are local units at P p in H ru p : U p {U p p ru p,0 Ă r U p Devika Sharma (Weizmann) 23rd June, 2017 6 / 14

Notation H : field cut out by the projective image of ρ Cl H : class group of H, and Cl Ă H : Cl H {Cl p H E : global units of H, and E r : E{E p U p : ź P p U P, where U P are local units at P p in H ru p : U p {U p p ru p,0 Ă r U p Remark The spaces Ă Cl H, r E, r U p and r U p,0 are all F p rg Q s-modules. Devika Sharma (Weizmann) 23rd June, 2017 6 / 14

Notation Definition Let W be the space M 2 pf p q with the conjugation action of G Q via ρ. Devika Sharma (Weizmann) 23rd June, 2017 7 / 14

Notation Definition Let W be the space M 2 pf p q with the conjugation action of G Q via ρ. Let W 0 be the submodule of trace 0 matrices in W. Devika Sharma (Weizmann) 23rd June, 2017 7 / 14

Notation Definition Let W be the space M 2 pf p q with the conjugation action of G Q via ρ. Let W 0 be the submodule of trace 0 matrices in W. For a finite dimensional F p rg Q s-module V, Devika Sharma (Weizmann) 23rd June, 2017 7 / 14

Notation Definition Let W be the space M 2 pf p q with the conjugation action of G Q via ρ. Let W 0 be the submodule of trace 0 matrices in W. For a finite dimensional F p rg Q s-module V, let V Ad : sum of all J-H factors isomorphic to W 0. Devika Sharma (Weizmann) 23rd June, 2017 7 / 14

Conditions The conditions are pc1q Cl Ă Ad H is trivial, and Devika Sharma (Weizmann) 23rd June, 2017 8 / 14

Conditions The conditions are pc1q Cl Ă Ad H is trivial, and pc2q the composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero. Devika Sharma (Weizmann) 23rd June, 2017 8 / 14

Conditions The conditions are pc1q Cl Ă Ad H is trivial, and pc2q the composition r E Ad Ñ r U Ad p Corollary r U Ad p,0 is non-zero. Let f P S 2 pnq, for N ď 1, 000, as above. Then every member of H f is non-split. Devika Sharma (Weizmann) 23rd June, 2017 8 / 14

Conditions The conditions are pc1q Cl Ă Ad H is trivial, and pc2q the composition r E Ad Ñ r U Ad p Corollary r U Ad p,0 is non-zero. Let f P S 2 pnq, for N ď 1, 000, as above. Then every member of H f is non-split. Remark When N 118, hypothesis pc2q fails. Devika Sharma (Weizmann) 23rd June, 2017 8 / 14

Conditions The conditions are pc1q Cl Ă Ad H is trivial, and pc2q the composition r E Ad Ñ r U Ad p Corollary r U Ad p,0 is non-zero. Let f P S 2 pnq, for N ď 1, 000, as above. Then every member of H f is non-split. Remark When N 118, hypothesis pc2q fails. We give an alternative argument to deal with such cases. Devika Sharma (Weizmann) 23rd June, 2017 8 / 14

Conditions pc1q and pc2q Devika Sharma (Weizmann) 23rd June, 2017 9 / 14

Conditions pc1q and pc2q Note that W 0 is an irreducible GL 2 pf p q-module, while the conditions (C1) and (C2) are over PGL 2 pf p q. This is because scalars act trivially on W 0. Devika Sharma (Weizmann) 23rd June, 2017 9 / 14

Conditions pc1q and pc2q Note that W 0 is an irreducible GL 2 pf p q-module, while the conditions (C1) and (C2) are over PGL 2 pf p q. This is because scalars act trivially on W 0. This works well in computations as the order of GL 2 pf p q is 48, where as PGL 2 pf p q is 24! Devika Sharma (Weizmann) 23rd June, 2017 9 / 14

Condition (C1) Devika Sharma (Weizmann) 23rd June, 2017 10 / 14

Condition (C1) ĂCl Ad H is trivial Devika Sharma (Weizmann) 23rd June, 2017 10 / 14

Condition (C1) ĂCl Ad H is trivial The field H is the splitting field of the 3-division polynomial Φ 3. Devika Sharma (Weizmann) 23rd June, 2017 10 / 14

Condition (C1) ĂCl Ad H is trivial The field H is the splitting field of the 3-division polynomial Φ 3. It can be explicitly generated by composing Φ 3, with the polynomial x 3 disc E and x 2 ` 3. Devika Sharma (Weizmann) 23rd June, 2017 10 / 14

Condition (C1) ĂCl Ad H is trivial The field H is the splitting field of the 3-division polynomial Φ 3. It can be explicitly generated by composing Φ 3, with the polynomial x 3 disc E and x 2 ` 3. In most examples, p 3 Cl H ùñ pc1q holds. Devika Sharma (Weizmann) 23rd June, 2017 10 / 14

Condition (C1) ĂCl Ad H is trivial The field H is the splitting field of the 3-division polynomial Φ 3. It can be explicitly generated by composing Φ 3, with the polynomial x 3 disc E and x 2 ` 3. In most examples, p 3 Cl H ùñ pc1q holds. When p 3 Cl H, we compute the J-H factors to deduce that pc1q holds. Devika Sharma (Weizmann) 23rd June, 2017 10 / 14

Condition (C1) ĂCl Ad H is trivial The field H is the splitting field of the 3-division polynomial Φ 3. It can be explicitly generated by composing Φ 3, with the polynomial x 3 disc E and x 2 ` 3. In most examples, p 3 Cl H ùñ pc1q holds. When p 3 Cl H, we compute the J-H factors to deduce that pc1q holds. This uses PARI/GP! Devika Sharma (Weizmann) 23rd June, 2017 10 / 14

Condition pc 2q Devika Sharma (Weizmann) 23rd June, 2017 11 / 14

Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Devika Sharma (Weizmann) 23rd June, 2017 11 / 14

Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Facts: r E Ad» W 0 Devika Sharma (Weizmann) 23rd June, 2017 11 / 14

Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Facts: r E Ad» W 0 r U Ad p» 5W 0 Devika Sharma (Weizmann) 23rd June, 2017 11 / 14

Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Facts: r E Ad» W 0 r U Ad p» 5W 0 r U Ad p,0» W 0 Devika Sharma (Weizmann) 23rd June, 2017 11 / 14

Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Facts: r E Ad» W 0 r U Ad p» 5W 0 r U Ad p,0» W 0 To check that pc2q holds, Devika Sharma (Weizmann) 23rd June, 2017 11 / 14

Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Facts: r E Ad» W 0 r U Ad p» 5W 0 r U Ad p,0» W 0 To check that pc2q holds, we find e P E satisfying Devika Sharma (Weizmann) 23rd June, 2017 11 / 14

Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Facts: r E Ad» W 0 r U Ad p» 5W 0 r U Ad p,0» W 0 To check that pc2q holds, we find e P E satisfying I pp pq fixes e, for some P p, and Devika Sharma (Weizmann) 23rd June, 2017 11 / 14

Condition pc 2q The composition r E Ad Ñ r U Ad p r U Ad p,0 is non-zero Facts: r E Ad» W 0 r U Ad p» 5W 0 r U Ad p,0» W 0 To check that pc2q holds, we find e P E satisfying I pp pq fixes e, for some P p, and e P 1 ` P n, for n ď 2 Devika Sharma (Weizmann) 23rd June, 2017 11 / 14

Elliptic curves satisfying conditions pc 1q and pc 2q A f Af pa, bq for A f Cl H e e lies in 89.a1 89 p 1323, 28134q 2 e 2 4 e2 5 e 2 6 e 7e8 2e 2 9 1 ` P 2 155.a1 5 5 31 p12528, 443664q 2 3 e2 4e4 3 e4 4 e6 5 e 4 7 e4 8 e2 9 1 ` P 1 155.b1 5 2 31 p 1323, 65178q 158.b1 2 2 79 p 4563, 111726q 2 3 e1 2e 2e 3 e e 1 5 6 1 ` P 2 8 158.c1 2 20 79 p 544347, 153226998q 158.e2 2 79 2 p 11691, 416934q... 994.b2 2 2 7 10 71 p 1509219, 324105570q 2 3 4 13 3 e 5 e e 2 6 7 e2 8 e 2 9 e 2 10 1 ` P 1 994.e2 2 7 2 71 2 p 27243, 711450q Devika Sharma (Weizmann) 23rd June, 2017 12 / 14

Exception: Example 118 Devika Sharma (Weizmann) 23rd June, 2017 13 / 14

Exception: Example 118 In this example, Cl H 2 implies that pc1q holds, Devika Sharma (Weizmann) 23rd June, 2017 13 / 14

Exception: Example 118 In this example, Cl H 2 implies that pc1q holds, while Condition pc2q fails! Devika Sharma (Weizmann) 23rd June, 2017 13 / 14

Exception: Example 118 In this example, Cl H 2 implies that pc1q holds, while Condition pc2q fails! Alternative condition pc2 1 q: This involves showing that a particular totally ramified Z{3-extension K 3 over Q 3 is distinct from the cyclotomic Z{3-extension over Q 3. Devika Sharma (Weizmann) 23rd June, 2017 13 / 14

Exception: Example 118 In this example, Cl H 2 implies that pc1q holds, while Condition pc2q fails! Alternative condition pc2 1 q: This involves showing that a particular totally ramified Z{3-extension K 3 over Q 3 is distinct from the cyclotomic Z{3-extension over Q 3. Checking the alternative condition includes explicitly computing the norm subgroup corresponding to K 3 {Q 3. This uses PARI-GP extensively. Devika Sharma (Weizmann) 23rd June, 2017 13 / 14

Thank You. Devika Sharma (Weizmann) 23rd June, 2017 14 / 14

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