An Upper Bound on Conductors for Pairs

Sparse equidistribution problems, period bounds and subconvexity

“…7.2 is essentially equivalent to the subconvexity result proved in Sec. 6. A rather striking point is that a similar logical dependence (although manifested very differently) is present in the work of Michel.…”

“…We make the relevant notation clear in that section. 6 Recall that |x|v, for a complex place v and x ∈ Fv, is the square of the usual absolute value …”

Standard zero-free regions for Rankin–Selberg L-functions via sieve theory

“…4 We emphasize the importance of the cuspidality condition in (1) and (2) in the above remark, which rules out the divisibility of L(s, π) by the L-function of a quadratic character. For example, if π is a dihedral form on GL 2 over F, induced by a Hecke character χ of a quadratic field extension E, then L(s, π) = L(s, χ).…”

“…The proof of the prime number theorem due to de la Valleé-Poussin gives a zero-free region for the Riemann zeta function ζ(s) of the form σ > 1 − c log(|t| + 3) for s = σ + it, and this generalises to a zero-free region for L(s, π) of the form 4 log(q(π)(|t| + 3)) (1.1)…”

Second order average estimates on local data of cusp forms

“…It may come as a surprise to some that despite the recent breakthroughs in certain cases of subconvexity, it is still not known in complete generality and under no assumptions that L(s, π × π) satisfies the standard convexity bound. Molteni [16] went some way toward this goal by showing that for π any cusp form on GL n , as long as |α π (p, i)| Np 1/4 (1) for all but finitely many primes p and 1 i n then L(s, π × π) satisfies the standard convexity bound (see his Hypothesis (R )). At present, however, bounds of this quality are known only for cusp forms on GL 2 (A), where we have |α π (p, i)| Np 1/9 [11].…”