Hybrid bounds for twisted L-functions

Weyl-type Hybrid Subconvexity Bounds for Twisted L-functions and Heegner Points on Shrinking Sets

Let q be odd and squarefree, and let χq be the quadratic Dirichlet character of conductor q. Let uj be a Hecke-Maass cusp form on Γ0(q) with spectral parameter tj . By an extension of work of Conrey and Iwaniec, we show L(uj ×χq, 1/2) ≪ε (q(1 + |tj |))1/3+ε, uniformly in both q and tj . A similar bound holds for twists of a holomorphic Hecke cusp form of large weight k. Furthermore, we show tha...

Hybrid Bounds for Twists of Gl(3) L-functions

Let π be a Hecke-Maass cusp form for SL(3,Z) and χ = χ1χ2 a Dirichlet character with χi primitive modulo Mi. Suppose that M1, M2 are primes such that max{(M |t|)1/3+2δ/3,M2/5|t|−9/20,M1/2+2δ|t|−3/4+2δ}(M |t|)ε < M1 < min{(M |t|)2/5, (M |t|)1/2−8δ}(M |t|)−ε for any ε > 0, where M = M1M2, |t| ≥ 1 and 0 < δ < 1/52. Then we have L ( 1 2 + it, π ⊗ χ ) ≪π,ε (M |t|).

Twisted L-Functions and Monodromy

Convexity bounds for L-functions

We prove a sharp convexity bound for L-functions, at the centre of the critical strip, under conditions weaker than those for the Selberg Class. 2000 Mathematics Subject Classification: Primary 11M41

L-functions of Twisted Legendre Curves

Let K be a global field of char p and let Fq be the algebraic closure of Fp in K. For an elliptic curve E/K with non-constant j-invariant, the L-function L(T,E/K) is a polynomial in 1+T ·Z[T ]. For any N > 1 invertible in K and finite subgroup T ⊂ E(K) of order N , we compute the mod N reduction of L(T,E/K) and determine an upper-bound for the order of vanishing at 1/q, the so-called analytic r...