Did you mean "Rational points on elliptic curves" as Noah mentioned, or Silverman's book with the title you gave?

I wish the term "symmetric algebra" weren't overloaded like this, especially since trace is only invariant under cyclic shifts. Maybe something like "E[1]-self-dual"? (btw, I'm not asking you to wage war on the responsible parties)

For the benefit of other people like me who got a headache trying to understand the one-sentence proof: The Tate curve is a p-adic analytic description of an elliptic curve as the group quotient Q_p^*/{q^n}, where q is some power of p times a number with unit norm. Since the curve has potentially multiplicative reduction, this power of p is nonzero. The l-torsion is then generated by l-th roots of unity together with an l-th root of q. The question is whether this l-th root of q has the same p-adic norm as an integer power of p. \Delta is a power series that looks like q. [out of space]

Sheaves like to have their cohomology taken, while cosheaves (aka coefficient systems) like to have their homology taken. If you do combinatorial manipulations, you are likely to be working with coefficient systems.

You might want to use \Delta^{op} instead of \Delta.

Restatement: Does the canonical map from F_k to its profinite completion merge any conjugacy classes?

I believe if you translate Hartshorne's construction of his + functor, by replacing elements of stalks with equivalence classes of representing pairs (U,s), you get everyone else's + functor. Taking H^0 and passing to the limit of refinements is the same as taking compatible functions to the union of stalks. The compatibility for these functions becomes interesting exactly when you have a cover by disjoint open sets. (Note also the bizarre statement of Hartshorne Chapter 2, exercise 1.1.)

I guess the short answer would be that the set of all morphisms is a disjoint union of morphism sets between pairs of objects (this is often an explicit axiom in the definition of category), so the freeness of the G-action on objects translates to a free action on morphisms. I may be missing something - are there subtleties in your definition of quotient category C/G?

This looks like a proof, conditional on the existence of infinitely many primes p satisfying the conditions that (p-1)/2 is prime, and the smallest prime power congruent to 1 mod p has size p^(2-epsilon).

This isn't really my milieu, but I don't think you get odd weight forms if your level structure admits the -1 automorphism. If you have odd weight forms, I guess you could say that geometric fibers of the divisor at infinity are representable, but it doesn't seem to be a log statement.

I think you can get a theorem with BNC if there are infinitely many primes of the form p'm+1 where p' is a prime of size p^(1-epsilon). If you restrict to p'=(p-1)/2, you're asking for Sophie Germain primes, and these are not known to be infinite in number.

Google: no hits for "log spin structure" or "logarithmic spin structure". For notation, maybe \pi_*(\omega^1_{E/X})? The thing inside the parentheses implicitly involves the log structures.

It's basically the same prime powers q, but for q congruent to -1 mod 4, we get with (q-1)/2 instead of (q+1)/2.

You can't get a lower bound of precisely n^3, since the L_2 series (i.e., PSL(2,Fp)) is an infinite sequence of counterexamples. If p is 1 mod 4, the group has order (p^2-1)p/2, and irreps of dimension 1, p, p+1, p-1, and (p+1)/2.