Symmetric square L-values and dihedral congruences for cusp forms

AbstractLet p≡3(mod4) be a prime, and k=(p+1)/2. In this paper we prove that two things happen if and only if the class number h(−p)>1. One is the non-integrality at p of a certain trace of normalised critical values of symmetric square L-functions, of cuspidal Hecke eigenforms of level one and weight k. The other is the existence of such a form g whose Hecke eigenvalues satisfy “dihedral” congruences modulo a divisor of p (e.g. p=23, k=12, g=Δ). We use the Bloch–Kato conjecture to link these two phenomena, using the Galois interpretation of the congruences to produce global torsion elements which contribute to the denominator of the conjectural formula for an L-value. When h(−p)=1, the trace turns out always to be a p-adic unit

Lifting puzzles and congruences of Ikeda and Ikeda-Miyawaki lifts

We show how many of the congruences between Ikeda lifts and non-Ikeda lifts, proved by Katsurada, can be reduced to congruences involving only forms of genus 1 and 2, using various liftings predicted by Arthur's multiplicity conjecture. Similarly, we show that conjectured congruences between Ikeda–Miyawaki lifts and non-lifts can often be reduced to congruences involving only forms of genus 1, 2 and 3

Lifting puzzles and congruences of Ikeda and Ikeda-Miyawaki lifts

We show how many of the congruences between Ikeda lifts and non-Ikeda lifts, proved by Katsurada, can be reduced to congruences involving only forms of genus 1 and 2, using various liftings predicted by Arthur's multiplicity conjecture. Similarly, we show that conjectured congruences between Ikeda–Miyawaki lifts and non-lifts can often be reduced to congruences involving only forms of genus 1, 2 and 3

Quadratic Q-curves, units and Hecke L-values

Abstract We show that if K is a quadratic field, and if there exists a quadratic Q-curve E/K of prime degree N, satisfying weak conditions, then any unit u of OK satisfies a congruence ur ≡ 1 (mod N), where r = g.c.d.(N − 1, 12). If K is imaginary quadratic, we prove a congruence, modulo a divisor of N, between an algebraic Hecke character ψ˜ and, roughly speaking, the elliptic curve. We show that this divisor then occurs in a critical value L(ψ , ˜ 2), by constructing a non-zero element in a Selmer group and applying a theorem of Kato

Ramanujan-style congruences of local origin

We prove that if a prime ℓ>3 divides pk−1, where p is prime, then there is a congruence modulo ℓ, like Ramanujan's mod 691 congruence, for the Hecke eigenvalues of some cusp form of weight k and level p. We relate ℓ to primes like 691 by viewing it as a divisor of a partial zeta value, and see how a construction of Ribet links the congruence with the Bloch–Kato conjecture (theorem in this case). This viewpoint allows us to give a new proof of a recent theorem of Billerey and Menares. We end with some examples, including where p=2 and ℓ is a Mersenne prime

Eisenstein congruences for split reductive groups

We present a general conjecture on congruences between Hecke eigenvalues of parabolically induced and cuspidal automorphic representations of split reductive groups, modulo divisors of critical values of certain L-functions. We examine the consequences in several special cases and use the Bloch–Kato conjecture to further motivate a belief in the congruences

Automorphic forms on Feit’s Hermitian lattices

We consider the genus of 20 classes of unimodular Hermitian lattices of rank 12 over the Eisenstein integers. This set is the domain for a certain space of algebraic modular forms. We find a basis of Hecke eigenforms, and guess global Arthur parameters for the associated automorphic representations, which recover the computed Hecke eigenvalues. Congruences between Hecke eigenspaces, combined with the assumed parameters, recover known congruences for classical modular forms, and support new instances of conjectured Eisenstein congruences for U(2,2) automorphic forms

GL2xGSp2 L-values and Hecke eigenvalue congruences

We find experimental examples of congruences of Hecke eigenvalues between automorphic representations of groups such as GSp2(A), SO(4, 3)(A) and SO(5, 4)(A), where the prime modulus should, for various reasons, appear in the algebraic part of a critical “tensor-product” L-value associated to cuspidal automorphic representations of GL2(A) and GSp2 (A). Using special techniques for evaluating L-functions with few known coefficients, we compute sufficiently good approximations to detect the anticipated prime divisors

Kurokawa–Mizumoto congruences and degree-8 L-values

Let f be a Hecke eigenform of weight k, level 1, genus 1. Let E2,1k(f) be its genus-2 Klingen–Eisenstein series. Let F be a genus-2 cusp form whose Hecke eigenvalues are congruent modulo q to those of E2,1k(f), where q is a “large” prime divisor of the algebraic part of the rightmost critical value of the symmetric square L-function of f. We explain how the Bloch–Kato conjecture leads one to believe that q should also appear in the denominator of the “algebraic part” of the rightmost critical value of the tensor product L-function L(s, f⊗ F) , i.e. in an algebraic ratio obtained from the quotient of this with another critical value. Using pullback of a genus-5 Siegel–Eisenstein series, we prove this, under weak conditions

The representation of integers by binary additive forms

Let a, b and n be integers with ≥ 3. We show that, in the sense of natural density, almost all integers represented by the binary form ax n − by n are thus represented essentially uniquely. By exploiting this conclusion, we derive an asymptotic formula for the total number of integers represented by such a form. These conclusions augment earlier work of Hooley concerning binary cubic and quartic forms, and generalise or sharpen work of Hooley, Greaves, and Skinner and Wooley concerning sums and differences of two nth powers.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42597/1/10599_2004_Article_129125.pd