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Glossary |
This glossary explains the meaning of many of the terms in tables, and explains caveats that one should keep in mind.
DETAILS
| Associated Abelian Varieties | Shimura constructed an isogeny class of abelian varieties associated to any newform f of weight at least 2. This is that isogeny class. |
| Component Group Orders | The order [#PhiA,p1, #PhiA,p2, ...,] of the component groups of A at the prime divisors p1, p2, etc., of the level. The pi are in increasing order. Note that these are not, in general, the Tamagawa numbers; instead they are multiples of the Tamagawa numbers. In general, there is no known algorithm to compute the order of the component group of A when p2 divides the level, so the data is incomplete in that case. |
| Congruence Groups | |
| Congruence Modulus | |
| Cuspidal Subspace | The subspace of modular forms that vanish at all of the cusps. |
| Cusps for Gamma0(N) | Representatives for the equivalence class of Q union {oo} under the action of Gamma0(N). |
| Eisenstein Subspace | The orthogonal complement of the cuspidal subspace. |
| Gamma1(N)-optimal quotient | The abelian variety in the isogeny class which is an optimal quotient of J1(N) or X1(N). |
| Isogeny Class | Collection of abelian varieties that are isogenous to each other, where an isogeny is a surjective homomorphism of finite degree. The members of the isogeny class are labeled by integers 1, 2, etc., or they are the "Gamma1-optimal quotient". The isogeny class 1 is the optimal quotient of J0(N). Note that it could be that the Gamma1-optimal quotient could be isomorphic to one of the other abelian varieties, but we give it separately, since certain invariants of the Gamma1-optimal quotient, viewed as a quotient of J1(N) may be different than the invariants of the same abelian variety viewed as a quotient of J0(N). |
| L | Large integers in python print with an L after them. If you see any integers in the tables with a trailing L, this just signifies that they are large. (I try to automatically remove all such occurrences, so if you see any, let me know.) |
| Modular Degree | If A is an abelian variety attached to a newform of weight 2, then the modular degree of A is the square root of the map from the dual of A to A induced by viewing A as a quotient of J0(N) |
| Modular Kernel | If A is an abelian variety, viewed as an optimal quotient of a Jacobian J, then the modular kernel is the kernel of the map from the dual of A to A induced by the canonical self-duality of J. More precisely, the map from J --> A dualizes to give a map A^ --> J^, and J^=J, so we obtain a map A^-->A, and the modular kernel is the kernel of this map. The modular kernel is a finite abelian group and is represented in the database by giving a sequence [n1,n2,...] of integers, so that ni divides ni+1, and the kernel is isomorphic to Z/n1Z + ... Note that no information about the structure of the kernel as a Galois module is stored in the database. If A is the Gamma1(N)-optimal quotient, then J=J1(N). | .
| Newform | A newform is a cuspidal modular form that does not come from lower level, which is normalized so that the coefficient of q is one. Newforms are the building blocks of the database, since all spaces of modular forms can be assembled from newforms, and all modular abelian varieties (over Q) are isogenous to products of abelian varieties attached to newforms. |
| Number of Terms Computed | This is the number of terms of the q-expansion that are stored in the database. |
| Order of Character | The order of the Dirichlet character eps:(Z/NZ)*--->C* associated to the space. |
| q-Expansion of Newform | This is the q-expansion at infinity of the given modular form. (The database currently has no information about q-expansions at other cusps.) The links on the right allow you to compute the expansion to some precision, typically between 10 and 100, or download the expansion in a computer-readable format. (Right click and select save-as, to download.) |
| Reductions of newform to characteristic p | The coefficients of the q-expansion of a newform are algebraic integers that generate a number field K. Let p be a prime. This table lists the reductions of these coefficients modulo each of the maximal ideals of K lying over p. |
| Subgroup of J0(N) Generated by Cusps | The subgroup of the Jacobian of X 0(N) generated by the divisor classes of differences of cusps for Gamma0(N). |
| Torsion (divisor of order) | This is some divisor of the order of the torsion subgroup. It is not well-defined, in that if I could find some way to compute the order of the torsion subgroup, this divisor would equal that order. Usually it is computed by finding the subgroup of A(Q) generated by rational cusps. |
| Torsion Multiple | This is a multiple of the order of the torsion subgroup of the given abelian variety, which is computed by taking a GCD of the number of points on the abelian variety over various finite fields. It is an invariant of the isogeny class of the abelian variety. |
| Torsion Multiple Bound | The torsion multiple bound is an integer B so that only primes up to B are used in computing the torsion bound. |
| Torsion Order | The order of the group of rational torsion points on A. |
| Trivial Character | The Dirichlet character eps:(Z/NZ)*--->C* that sends everything to 1. |