Discriminants of Hecke algebras

William A. Stein

September 28, 1998

Using an implementation of the modular symbols algorithm described in Cremona’s book Algorithms for modular elliptic curves I computed, for each prime N between 2 and 577, an integer D_{N} which is divisible by the discriminant of the Hecke algebra 𝕋_{N} associated to weight 2 cusp forms of level N for Γ_{0}(N). For these N we give a table of factored D_{N}.

The Hecke algebra 𝕋 = 𝕋_{N} is an order in a product E = ∏ E_{t} of totally real number fields. The discriminant disc(𝕋) is the product of the discriminants of the number fields E_{i}, multiplied by the square of the index of 𝕋 in its normalization.

Fix a prime number N and let S(N) be the space of weight 2 cusp forms of level N for Γ_{0}(N). For p not equal to N let T_{p} be the p-th Hecke operator, and let d_{p} be the discriminant of the characteristic polynomial of T_{p} acting on S(N). Consider the sequence of integers

d_{2}, gcd(d_{2}, d_{3}), gcd(d_{2}, d_{3}, d_{5}), …, gcd(d_{2}, d_{3}, …, d_{q}), …

where we omit p if p = N. Since each term divides its predecessor, this sequence must eventually stabilize at some limit Δ_{N}. Since each term is divisible by the discriminant of 𝕋, this discriminant divides Δ_{N}.

I wrote a program which computes the above sequence until it repeats some value D_{N} for 15 terms. The result of that computation is given in table 1, which can be found at the end of this document.

It is interesting to note that N = 389 is the only case in our tables for which N|D_{N}. I have checked up to N = 14537 and found no other cases in which this occurs. Whether or not this ever occurs is of interest to Ribet as this hypothesis plays a role in his paper, Torsion points on J_{0}(N) and galois representations.

Another problem is to determine, for each N in table 1, whether the primes dividing D_{N} are exactly the same as the primes dividing disc(𝕋). I have checked that this is the case for N ≤ 73. If the ring 𝕋 ⊗ 𝔽_{p} is not reduced then p| disc(𝕋). This ring can’t be reduced if T_{q} is not diagonalizable (modulo p) for some prime q not equal to N. However, this sufficient condition is not always necessary as the case N = 37 illustrates. Here 2 ramifies in the Hecke algebra even though the Hecke operators T_{q} with q ≠ 2 act semisimply modulo 2.

Factored D_{N} for N ≤ 577.

\begin{matrix}N & D_{N} \\ 11 & 1 \\ 13 & 0 \\ 17 & 1 \\ 19 & 1 \\ 23 & 5 \\ 29 & 2^{3} \\ 31 & 5 \\ 37 & 2^{2} \\ 41 & 2^{2}.37 \\ 43 & 2^{5} \\ 47 & 19.103 \\ 53 & 2^{4}.37 \\ 59 & 2^{7}.31.557 \\ 61 & 2^{4}.37 \\ 67 & 2^{4}.5^{4} \\ 71 & 3^{4}.257^{2} \\ 73 & 2^{4}.3^{2}.5.13 \\ 79 & 2^{4}.83.983 \\ 83 & 2^{8}.197.11497 \\ 89 & 2^{6}.5^{3}.6689 \\ 97 & 2^{6}.7^{2}.2777 \\ 101 & 2^{8}.17568767 \end{matrix} \begin{matrix}N & D_{N} \\ 103 & 2^{8}.5.17.411721 \\ 107 & 2^{12}.5.7.1667.19079 \\ 109 & 2^{10}.7^{2}.7537 \\ 113 & 2^{10}.3^{4}.7^{2}.11^{2}.107 \\ 127 & 2^{12}.3^{4}.7.86235899 \\ 131 & 2^{19}.5.46141.75619573 \\ 137 & 2^{10}.5^{2}.29.401.895241 \\ 139 & 2^{14}.3^{2}.7^{2}.997.2151701 \\ 149 & 2^{12}.7^{2}.234893.1252037 \\ 151 & 2^{18}.7^{2}.11.67^{2}.257.439867 \\ 157 & 2^{13}.61.397.48795779 \\ 163 & 2^{15}.3^{2}.65657.82536739 \\ 167 & 2^{16}.5.8269.5103536431379173 \\ 173 & 2^{14}.5^{2}.7.29.5608385124289 \\ 179 & 2^{22}.3^{4}.7^{2}.313.137707.536747147 \\ 181 & 2^{16}.5^{2}.7.61.397.595051637 \\ 191 & 2^{8}.3^{3}.5.382146223.319500117632677 \\ 193 & 2^{14}.5.11^{2}.17.103.401.4153.680059 \\ 197 & 2^{18}.5^{2}.61.397.35217676193989 \\ 199 & 2^{16}.3.5^{3}.29.31.71^{2}.347.947.37316093 \\ 211 & 2^{20}.3.5.7^{4}.41^{2}.43.229.52184516509 \\ 223 & 2^{36}.7^{2}.19.103.3995922697473293141 \end{matrix}

\begin{matrix}N & D_{N} \\ 227 & 2^{37}.3^{2}.5^{3}.7^{4}.13^{2}.29.31^{2}.13591.57139.273349 \\ 229 & 2^{32}.107.17467.39555937.53625889 \\ 233 & 2^{22}.3^{7}.53.139.653.4127.24989.8388019 \\ 239 & 2^{12}.7^{2}.2833.51817.97423.1174779433.8920940047 \\ 241 & 2^{23}.97.1489.20857.651474368435017 \\ 251 & 2^{28}.5^{2}.29.373.8768135668531.2006012696666681 \\ 257 & 2^{65}.29.479.71711.409177.654233.32354821 \\ 263 & 2^{20}.11.61.397.15631853.34867513.97092067.252746489 \\ 269 & 2^{22}.3^{2}.43.151.27767.65657.5550873754172978311 \\ 271 & 2^{24}.3^{2}.1367.6091.592661.1132673.14171513.172450541 \\ 277 & 2^{22}.5^{2}.19.29.37.137^{2}.92767.1530091.25531570859 \end{matrix}

N D_{N}
281 2^{22}.3.5.181.857.8388019.2647382149.1778899342669
283 2^{46}.349.1297.413713.73199099.5832488839
293 2^{26}.3^{2}.29.233.2351^{2}.69763.42711913589792108923
307 2^{50}.3^{6}.5^{5}.11^{2}.13^{3}.107^{2}.457.3697.21577.974513.568380457
311 2^{16}.5^{2}.29.3013091897.2106873009119126062143259000543887593
313 2^{24}.5.41^{2}.8619587.9614923.130838023.2164322751511
317 2^{26}.7.367.3217.660603043.14989400036918065702697531
331 2^{38}.3^{2}.53^{2}.229.1399.21911.205493.6363601.584461573862449
337 2^{28}.113.593.2791.2963615537.747945736667.4122851467451
347 2^{61}.5.7^{2}.19^{2}.331.349.479.617.1797330450291217.918291275915301361
349 2^{28}.13.103.1118857.72318613.6771977049413.1313981654817031
353 2^{34}.3^{2}.5.127^{2}.229.114641.551801.12611821.7779730837.24314514437
359 2^{36}.3^{6}.2777.16512254293.64542630435970307.2171776478013633068927
367 2^{44}.7.81421.251387.418175501.15354151381.13144405392643360366681
373 2^{32}.7.11^{3}.23.199.673.2143.1542194372227.72819251148518000363297
379 2^{34}.59.317.421.278329.5698591.2117788336277.2851210737989187265253
383 2^{32}.5.11^{2}.13.72893.3151861.16141144314299.

178236551484825400362837637090811

389 2^{63}.3^{4}.5^{6}.31^{2}.37.389.3881.215517113148241.477439237737571441
397 2^{57}.23^{2}.31^{2}.97.317.761^{2}.302609750073209.83566618884497478937
401 2^{96}.5^{2}.19.163.293^{2}.811.1218675071.71742740351.388881803749.

34393898968391

409 2^{32}.3^{3}.17.1667.1741.2341.537071.14884451.18631199.

1334964067081334453235547

419 2^{55}.17.43.113.151.167.971.493657.20375986548898473293.

53097073649092855361102575237

421 2^{34}.3.31.557.4729.825403.857459.144211946777593109.

2328579379136648917067

431 2^{91}.3^{4}.5^{6}.11.19^{2}.29.31.43.197.257.6947^{2}.37619.

29252013842927.806505757406715084824003

433 2^{68}.3^{7}.7^{2}.37^{2}.101.379.1439.3613.18719.2792477.77087971.

5830108671536745647

439 2^{66}.3^{2}.5.31^{2}.173.84179.85667.16794662617.

513841517138871835091506167235408934202857

443 2^{88}.3^{2}.7^{2}.31^{2}.499.6899.48508479390300197.

2817219327571188909266947704801865987

449 2^{40}.3.7^{2}.101.44933757980789.188247485945671.

653016225615601.1431966252229376199841

457 2^{36}.5.31^{2}.653.3169.38983093.52621913.33122975406370693.

5653726203394180386934181 4

N D_{N}
461 2^{73}.5.7^{2}.19^{3}.97^{2}.80750473.3104029729.607263139073.

3729490905341009668647473177

463 2^{62}.113.311.9929.568201.132502583.1474412920219.

2770309905285622039024420194209857723

467 2^{71}.17^{2}.1212648089519.32432206859088781.

6296651104824906148358708614333895055221783

479 2^{32}.13.17.1861.4021.28745083.41556253.1202203127423.

201529385024397103.7037463122648759781611869895003

487 2^{72}.3^{16}.5^{4}.13^{2}.17^{2}.19^{4}.59.103^{2}.109.257.

623519211698413571686763.15408475904697077364866629

491 2^{104}.5^{6}.19^{2}.43.131.479.887.5650859.

54796097920639362740205317747356273097682333252495603721

499 2^{69}.3^{11}.5.71^{2}.167.495613.25224990196319.

573452584782809.277143583167463430555979797274731

503 2^{78}.3^{2}.5^{4}.11^{2}.19^{3}.257.821.2003^{2}.13597.45587.

384479819.8659024393.20115672029938390602701696607766073563

509 2^{71}.3^{3}.13.157.971.1277.4567.3691783.42330311.

1157039662523351992921397.6331071860925306189417509

521 2^{42}.23.53.67.929.13877.531096383.19526270957.1089951135204631559833.

14340527343875384245648725589439

523 2^{91}.3.5.41^{3}.59.149^{2}.1201.279121937.8371971617.

9059602909494267071628228952878552757512056969593

541 2^{46}.3^{2}.5.13.277.307.591581.1940573213.221136462575339.

1453183329662653.18044474614550745414465332996771

547 2^{105}.7^{3}.73.103^{2}.5501.11783.16097.43781.1152631.146768003.

9959758037.91268351929.102277460687.106666343972273

557 2^{46}.7^{4}.13^{2}.4787.252163.16849164271275021852893.

53296770296923102812608983.2381022539751738307256162767

563 2^{139}.5^{2}.13^{4}.37^{2}.61^{2}.37591.52667.155083.301703.938251.

46706589087295134421.299128314984453465128592656821021

569 2^{46}.73.449531828286229614392569.189316003.

257022598600391962761793946239.2294643649486046267496627432517

571 2^{166}.3^{12}.5^{8}.7^{4}.13^{2}.17.37^{3}.41^{2}.79^{2}.127^{2}.181.211.293.709.

1579^{2}.1667^{2}.12030433.807024744595934649052018211

577 2^{131}.3^{12}.5^{4}.13^{3}.59^{2}.61^{2}.257.163753.

41340850017998228328234516909328723846661. 85934741209775683850815667