Mathematics

In this paper we give a classification of the asymptotic expansion of the q -expansion of reciprocals of Eisenstein series E k of weight k for the modular group $\func{SL}_2(\mathbb{Z})$. For k??2 even, this extends results of Hardy and Ramanujan, and Berndt, Bialek and Yee, utilizing the Circle Method on the one hand, and results of Petersson, and Bringmann and Kane, developing a theory of meromorphic Poincar{é} series on the other. We follow a uniform approach, based on the zeros of the Eisenstein series with the largest imaginary part. These special zeros provide information on the singularities of the Fourier expansion of 1/ E k (z) with respect to q= e 2?iz .

Bhargava, Hanke, and Shankar have recently shown that the asymptotic average 2 -torsion subgroup size of the family of class groups of monogenized cubic fields with positive and negative discriminants is 3/2 and 2 , respectively. In this paper, we provide strong computational evidence for these asymptotes. We then develop a pair of novel conjectures that predicts, for p prime, the asymptotic average p -torsion subgroup size in class groups of monogenized cubic fields.

We compute asymptotic estimates for the Fourier coefficients of two mixed mock modular forms, which come from Bailey pairs derived by Lovejoy and Osburn. To do so, we employ the circle method due to Wright and a modified Tauberian theorem. We encounter cancellation in our estimates for one of the mock theta functions due to the auxiliary function θ n,p arising from the splitting of Hickerson and Mortenson. We deal with this by using higher order asymptotic expansions for the Jacobi theta functions.

The Attractor Conjecture for Calabi-Yau moduli spaces predicts the algebraicity of the moduli values of certain isolated points picked out by Hodge-theoretic conditions. We provide a family of counterexamples to the Attractor Conjecture in all suitably high, odd dimensions conditional on the Zilber-Pink conjecture.

We prove a summation formula for pairs of quadratic spaces following the conjectures of Braverman-Kazhdan, Lafforgue, Ngô and Sakellaridis. We also compute the local factors where all the data are unramified.

We prove asymptotic formulas for cyclicity of reductions of elliptic curves over the rationals in a family of curves having specified torsion. These results agree with established conditional results and with average results taken over larger families. As a key tool, we prove an analogue of a result of Vl?duţ that estimates the number of elliptic curves over a finite field with a specified torsion point and cyclic group structure.

In this paper, we prove a function field-analogue of Poonen-Rains heuristics on the average size of p -Selmer group. Let E be an elliptic curve defined over Z[t] . Then E is also defined over F q for any q of prime power. We show that for large enough q , the average size of the p -Selmer groups over the family of quadratic twists of E over F q [t] is equal to p+1 for all but finitely many primes p . Namely, if we twist the curve in F q [t] by polynomials of fixed degree n and let both n and q approach to infinity, then the average rank of p -Selmer group converges to p+1 .

Let f,g,h be three normalized cusp newforms of weight 2k on ? 0 (N) which are eigenforms of Hecke operators. We use Ichino's period formula combined with a relative trace formula to show exact averages of L(3k??,f?g?h) . We also present some applications of the average formulas to the nonvanishing problem, giving a lower bound on the number of nonvanishing central L -values when one of the forms is fixed.

It is a folklore conjecture that the Möbius function exhibits cancellation on shifted primes; that is, ∑ p≤X μ(p+h) = o(π(X)) as X→∞ for any fixed shift h>0 . We prove the conjecture on average for shifts h≤H , provided logH/loglogX→∞ . We also obtain results for shifts of prime k -tuples, and for higher correlations of Möbius with von Mangoldt and divisor functions. Our argument combines sieve methods with a refinement of Matomäki, Radziwiłł, and Tao's work on an averaged form of Chowla's conjecture.

Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, three of the present authors derived representations for certain sums of products of trigonometric functions as double series of Bessel functions. These series are generalized in the present paper by introducing the novel notion of balanced derivatives, leading to further theorems. As we will see below, the regions of convergence in the unbalanced case are entirely different than those in the balanced case. From this viewpoint, it is remarkable that Ramanujan had the intuition to formulate entries that are, in our new terminology, "balanced". If x denotes the number of products of the trigonometric functions appearing in our sums, in addition to proving the identities mentioned above, theorems and conjectures for upper and lower bounds for the sums as x?��? are established.