If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves over $k$ has a formula involving $\lfloor p/12 \rfloor$ and the residue class of $p$ mod 12, described in Chapter V of Silverman's The Arithmetic of Elliptic Curves. If we weight these curves by the reciprocals of the orders of their automorphism groups, we obtain the substantially simpler Eichler-Deuring mass formula: $\frac{p-1}{24}$. For example, when $p=2$, the unique supersingular curve $y^2+y=x^3$ has endomorphisms given by the Hurwitz integers (a maximal order in the quaternions), and its automorphism group is therefore isomorphic to the binary tetrahedral group, which has order 24.
Silverman gives the mass formula as an exercise, and it's pretty easy to derive from the formula in the text. The proof of the complicated formula uses the Legendre form (hence only works away from 2), and the appearance of the $p/12$ boils down to the following two facts:
- Supersingular values of $\lambda$ are precisely the roots of the Hasse polynomial, which is separable of degree $\frac{p-1}2$.
- The $\lambda$-line is a 6-fold cover of the $j$-line away from $j=0$ and $j=1728$ (so the roots away from these values give an overcount by a factor of 6).
Question: Is there a proof of the Eichler-Deuring formula in the literature that avoids most of the case analysis, e.g., by using a normal form of representable level?
I suppose any nontrivial level structure will probably require some special treatment for the prime(s) dividing that level. Even so, it would be neat to see any suitably holistic enumeration, in particular, one that doesn't need to single out special $j$-invariants.
(This question has been troubling me for a while, but Greg's question inspired me to actually write it down.)
Best Answer
One argument (maybe not of the kind you want) is to use the fact that the wt. 2 Eisenstein series on $\Gamma_0(p)$ has constant term (p-1)/24.
More precisely: if $\{E_i\}$ are the s.s. curves, then for each $i,j$, the Hom space $L_{i,j} := Hom(E_i,E_j)$ is a lattice with a quadratic form (the degree of an isogeny), and we can form the corresponding theta series $$\Theta_{i,j} := \sum_{n = 0}^{\infty} r_n(L_{i,j})q^n,$$ where as usual $r_n(L_{i,j})$ denotes the number of elements of degree $n$. These are wt. 2 forms on $\Gamma_0(p)$.
There is a pairing on the $\mathbb Q$-span $X$ of the $E_i$ given by $\langle E_i,E_j\rangle = $ # $Iso(E_i,E_j),$ i.e. $$\langle E_i,E_j\rangle = 0 \text{ if } i \neq j\text{ and equals # }Aut(E_i) \text{ if }i = j,$$ and another formula for $\Theta_{i,j}$ is $$\Theta_{i,j} := 1 + \sum_{n = 1}^{\infty} \langle T_n E_i, E_j\rangle q^n,$$ where $T_n$ is the $n$th Hecke correspondence.
Now write $x := \sum_{j} \frac{1}{\text{#}Aut(E_j)} E_j \in X$. It's easy to see that for any fixed $i$, the value of the pairing $\langle T_n E_i,x\rangle$ is equal to $\sum_{d |n , (p,d) = 1} d$. (This is just the number of $n$-isogenies with source $E_i,$ where the target is counted up to isomorphism.) Now $$\sum_{j} \frac{1}{\text{#}Aut(E_j)} \Theta_{i,j} = \bigg{(}\sum_{j} \frac{1}{\text{#}Aut(E_j)}\bigg{)} + \sum_{n =1}^{\infty} \langle T_n E_i, x\rangle q^n = \bigg{(}\sum_{j}\frac{1}{\text{#}Aut(E_j)}\bigg{)} + \sum_{n = 1}^{\infty} \bigg{(}\sum_{d | n, (p,d) = 1} d\bigg{)}q^n.$$
Now the LHS is modular of wt. 2 on $\Gamma_0(p)$, thus so is the RHS. Since we know all its Fourier coefficients besides the constant term, and they coincide with those of the Eisenstein series, it must be the Eisenstein series. Thus we know its constant term as well, and that gives the mass formula.
(One can replace the geometric aspects of this argument, involving s.s. curves and Hecke correspondences, with pure group theory/automorphic forms: namely the set $\{E_i\}$ is precisely the idele class set of the multiplicative group $D^{\times}$, where $D$ is the quat. alg. over $\mathbb Q$ ramified at $p$ and $\infty$. This formula, writing the Eisenstein series as a sum of theta series, is then a special case of the Seigel--Weil formula, I believe, which in general, when you pass to constant terms, gives mass formulas of the type you asked about.)