Modularity of the Abelian Surface of Conductor 277 David S. Yuen Lake Forest College joint work with Armand Brumer Fordham University Cris Poor Fordham University John Voight Dartmouth College Modular Forms and Curves of Low Genus: Computational Aspects ICERM, September 2015 David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 1 / 25
Outline Outline of talk 1. Paramodular Conjecture and evidence. 2. The abelian surface of conductor 277 and the paramodular form f 277 . 3. Computing eigenvalues by specialization. 4. Making floating point calculations rigorous. 5. Results. David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 2 / 25
Paramodular Conjecture and Evidence Paramodular Conjecture All abelian surfaces A / Q are paramodular Paramodular Conjecture (Brumer and Kramer 2009) Let N ∈ N . There is a bijection between 1. isogeny classes of abelian surfaces A / Q with conductor N and endomorphisms End Q ( A ) = Z , 2. lines of Hecke eigenforms f ∈ S 2 ( K ( N )) new that have rational eigenvalues and are not Gritsenko lifts from J cusp 2 , N . In this correspondence we have L ( A , s , Hasse-Weil ) = L ( f , s , spin ) . David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 3 / 25
Paramodular Conjecture and Evidence Background evidence Do the arithmetic and automorphic data match up? Looks like it. 1997: Brumer makes a (short) list of N < 1000 that could possibly be the conductor of an abelian surface A / Q . Theorem (PY 2009) Let p < 600 be prime. If p �∈ { 277 , 349 , 353 , 389 , 461 , 523 , 587 } then S 2 ( K ( p )) consists entirely of Gritsenko lifts. This list of primes { 277 , . . . , 587 } exactly matches Brumer’s “Yes there is an abelian surface” list for prime levels. This is a lot of evidence for the Paramodular Conjecture because prime levels p < 600 that don’t have abelian surfaces over Q also don’t have paramodular cusp forms beyond the Gritsenko lifts. David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 4 / 25
Paramodular Conjecture and Evidence Background evidence Proof. We can inject the weight two space into weight four spaces: � � J cusp 1) For g 1 , g 2 ∈ Grit ⊆ S 2 ( K ( p )), we have the injection: 2 , p S 2 ( K ( p )) ֒ → { ( H 1 , H 2 ) ∈ S 4 ( K ( p )) × S 4 ( K ( p )) : g 2 H 1 = g 1 H 2 } f �→ ( g 1 f , g 2 f ) 2) The dimensions of S 4 ( K ( p )) are known by Ibukiyama; we still have to span S 4 ( K ( p )) by computing products of Gritsenko lifts, traces of theta series and by smearing with Hecke operators. 3) Millions of Fourier coefficients mod 109 later, dim S 2 ( K ( p )) ≤ dim { ( H 1 , H 2 ) ∈ S 4 ( K ( p )) × S 4 ( K ( p )) : g 2 H 1 = g 1 H 2 } 4) When the dimension might be bigger, this method also gives a candidate nonlift as a quotient of a known weight 4 form divided by a known weight 2 form. David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 5 / 25
Paramodular Conjecture and Evidence Equality of L -series complete examples Equality of L -series modularity examples The lift to a paramodular Hecke eigenform of a certain Hilbert modular form over a real quadratic field by Johnson-Leung and Roberts (2012) is modular with respect to a certain abelian surface. Some work of Demb´ el´ e and Kumar is related to this. An example has conductor 193 2 . David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 6 / 25
Paramodular Conjecture and Evidence Equality of L -series complete examples Equality of L -series modularity examples The lift to a paramodular Hecke eigenform of a certain Hilbert modular form over a real quadratic field by Johnson-Leung and Roberts (2012) is modular with respect to a certain abelian surface. Some work of Demb´ el´ e and Kumar is related to this. An example has conductor 193 2 . For a similar but different example, constructing a lift from Bianchi modular forms, Berger, Demb´ el´ e, Pacetti, Sengun used an imaginary quadratic number field. An example is conductor N = 223 2 . David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 6 / 25
Abelian Surface and Modular Form of Conductor 277 The form and the surface Conductor 277 The form and the surface (Theorem PY 2009) dim S 2 ( K (277)) = 11 but dim J cusp 2 , 277 = 10. There is a Hecke eigenform f 277 ∈ S 2 ( K (277)) that is not a Gritsenko lift. A 277 is the Jacobian of the hyperelliptic curve y 2 + y = x 5 + 5 x 4 + 8 x 3 + 6 x 2 + 2 x David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 7 / 25
Abelian Surface and Modular Form of Conductor 277 The form and the surface Conductor 277 The form and the surface (Theorem PY 2009) dim S 2 ( K (277)) = 11 but dim J cusp 2 , 277 = 10. There is a Hecke eigenform f 277 ∈ S 2 ( K (277)) that is not a Gritsenko lift. A 277 is the Jacobian of the hyperelliptic curve y 2 + y = x 5 + 5 x 4 + 8 x 3 + 6 x 2 + 2 x Magma will compute lots of Euler factors for L ( A 277 , s , H-W ) David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 7 / 25
Abelian Surface and Modular Form of Conductor 277 The form and the surface Conductor 277 The form and the surface (Theorem PY 2009) dim S 2 ( K (277)) = 11 but dim J cusp 2 , 277 = 10. There is a Hecke eigenform f 277 ∈ S 2 ( K (277)) that is not a Gritsenko lift. A 277 is the Jacobian of the hyperelliptic curve y 2 + y = x 5 + 5 x 4 + 8 x 3 + 6 x 2 + 2 x Magma will compute lots of Euler factors for L ( A 277 , s , H-W ) By contrast, it takes much more work to compute Euler factors for L ( f 277 , s , spin ), and one of the main goals of this talk is to present one method for computing high eigenvalues of f 277 . David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 7 / 25
Abelian Surface and Modular Form of Conductor 277 The form and the surface Conductor 277 The form and the surface (Theorem PY 2009) dim S 2 ( K (277)) = 11 but dim J cusp 2 , 277 = 10. There is a Hecke eigenform f 277 ∈ S 2 ( K (277)) that is not a Gritsenko lift. A 277 is the Jacobian of the hyperelliptic curve y 2 + y = x 5 + 5 x 4 + 8 x 3 + 6 x 2 + 2 x Magma will compute lots of Euler factors for L ( A 277 , s , H-W ) By contrast, it takes much more work to compute Euler factors for L ( f 277 , s , spin ), and one of the main goals of this talk is to present one method for computing high eigenvalues of f 277 . In 2009, we computed the 2, 3 and 5 Euler factors of L ( f 277 , s , spin ) and they agree with those of L ( A 277 , s , H-W ). David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 7 / 25
Abelian Surface and Modular Form of Conductor 277 Proving existence of f 277 How can we prove a weight two nonlift cusp form exists? David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 8 / 25
Abelian Surface and Modular Form of Conductor 277 Proving existence of f 277 How can we prove a weight two nonlift cusp form exists? Proof (2009). 1) We have a candidate f = H 1 / g 1 ∈ M mero ( K ( p )). 2 2) Find a weight four cusp form F ∈ S 4 ( K ( p )) and prove F g 2 1 = H 2 1 in S 8 ( K ( p )) . � 2 � H 1 is holomorphic, so is f = H 1 Since F = . g 1 g 1 David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 8 / 25
Abelian Surface and Modular Form of Conductor 277 Proving existence of f 277 How can we prove a weight two nonlift cusp form exists? Proof (2009). 1) We have a candidate f = H 1 / g 1 ∈ M mero ( K ( p )). 2 2) Find a weight four cusp form F ∈ S 4 ( K ( p )) and prove F g 2 1 = H 2 1 in S 8 ( K ( p )) . � 2 � H 1 is holomorphic, so is f = H 1 Since F = . g 1 g 1 Proof (new). We make a Borcherds product in S 2 ( K ( p )) and show the Borcherds product is not a Gritsenko lift. David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 8 / 25
Abelian Surface and Modular Form of Conductor 277 f 277 Formula for f 277 One presentation of f 277 is f 277 = ( − 14 G 2 1 − 20 G 8 G 2 + 11 G 9 G 2 + 6 G 2 2 − 30 G 7 G 10 + 15 G 9 G 10 + 15 G 10 G 1 − 30 G 10 G 2 − 30 G 10 G 3 + 5 G 4 G 5 + 6 G 4 G 6 + 17 G 4 G 7 − 3 G 4 G 8 − 5 G 4 G 9 − 5 G 5 G 6 + 20 G 5 G 7 − 5 G 5 G 8 − 10 G 5 G 9 − 3 G 2 6 + 13 G 6 G 7 + 3 G 6 G 8 − 10 G 6 G 9 − 22 G 2 7 + G 7 G 8 + 15 G 7 G 9 + 6 G 2 8 − 4 G 8 G 9 − 2 G 2 9 + 20 G 1 G 2 − 28 G 3 G 2 + 23 G 4 G 2 + 7 G 6 G 2 − 31 G 7 G 2 + 15 G 5 G 2 + 45 G 1 G 3 − 10 G 1 G 5 − 2 G 1 G 4 − 13 G 1 G 6 − 7 G 1 G 8 + 39 G 1 G 7 − 16 G 1 G 9 − 34 G 2 3 + 8 G 3 G 4 + 20 G 3 G 5 + 22 G 3 G 6 + 10 G 3 G 8 + 21 G 3 G 9 − 56 G 3 G 7 − 3 G 2 4 ) / ( − G 4 + G 6 + 2 G 7 + G 8 − G 9 + 2 G 3 − 3 G 2 − G 1 ) . for some ten Gritsenko lifts G 1 , . . . , G 10 . David Yuen et al Modularity of Abelian Surface A277 ICERM September 2015 9 / 25
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