Related papers: Split-CM points and central values of Hecke L-seri…

In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an…

We introduce the relationship between congruent numbers and elliptic curves, and compute the conductor of the elliptic curve $y^2 = x^3 - n^2 x$ associated with it. Furthermore, we prove that its $L$-series coefficient $a_m = 0$ when $m…

The aim of this paper is two-fold. First, we define symplectic maps between Hitchin systems related to holomorphic bundles of different degrees. We call these maps the Symplectic Hecke Correspondence (SHC) of the corresponding Higgs…

This paper reviews a result of Jensen on characters of some tilting modules for $\SL_3(k)$, where $k$ has characteristic at least five and fills in some gaps in the proof of this result. We then apply the result to finding some…

Ramanujan's theory of elliptic functions to alternative bases connects modular forms with hypergeometric series and has led to applications such as the modularity of certain hypergeometric Galois representations. In this paper, we relate…

Let X be a product of Drinfeld modular curves over a general base ring A of odd characteristic. We classify those subvarieties of X which contain a Zariski-dense set of CM points. This is an analogue of the Andr\'e-Oort conjecture. As an…

In this paper an explicit formula is given for a sequence of numbers. The positivity of this sequence of numbers implies that zeros in the critical strip of the Euler product of Hecke polynomials, which are associated with the space of cusp…

Two integral structures on the Q-vector space of modular forms of weight two on X_0(N) are compared at primes p exactly dividing N. When p=2 and N is divisible by a prime that is 3 mod 4, this comparison leads to an algorithm for computing…

We show that elliptic curves with complex multiplication (CM) naturally emerge in the spectral geometry of Hermitian one-matrix models in the two-cut phase. Focusing on a symmetric quartic potential, we derive the corresponding genus-one…

Let q be a prime and -D < -4 be an odd fundamental discriminant such that q splits in Q(\sqrt{-D}). For f a weight zero Hecke-Maass newform of level q and h the weight one theta series of level D corresponding to an ideal class group…

We give a new perspective of Heegaard splittings in terms square complexes and Guirardel's notion of a \textit{core} which allows for combinatorial measurement of the obstruction to being a connect sum of Heegaard diagrams. A Heegaard…

For a cuspidal Hecke eigenform $F$ for $Sp_n(Z)$ and a Dirichlet character $\chi$ let $L(s,F,\chi,St)$ be the standard $L$-function of $F$ twisted by $\chi$. Boecherer showed the boundedness of denominators of the algebraic part of…

In the present article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight $2,4$ and 6. We define Hecke operators on them, find some analytic relations between these…

Let K be a field of characteristic different from 2 and C an elliptic curve over K given by a Weierstrass equation. To divide an element of the group C by 2, one must solve a certain quartic equation. We characterise the quartics arising…

For the mixed CM point (a,b,c) = (1/6, 1/3, 1), define A_n^{mix} := 108^n [z^n] _2F_1(1/6, 1/3; 1; z)^3. For every split prime p >= 7, p == 1 mod 3, and every m >= 1, we prove unconditionally A_{mp}^{mix} == A_m^{mix} mod p^4. The exponent…

In this paper, we study moments of central values of cubic Hecke $L$-functions in $\mathbb{Q}(i)$, and establish quantitative non-vanishing result for those values.

We classify the irreducible components of the Hilbert scheme of $n$ points on non-reduced algebraic plane curves, and give a formula for the multiplicities of the irreducible components. The irreducible components are indexed by partitions…

We study the degree of the special cubic fourfolds in the Hilbert scheme of cubic fourfolds via a computation of the generating series of Heegner divisors of even lattice of signature (2, 20).

Let $p$ be a prime number, and let $\Delta_1,\Delta_2 < 0$ be two coprime fundamental discriminants. When $p$ splits in $\mathbb{Q}(\sqrt{\Delta_1})$ and $\mathbb{Q}(\sqrt{\Delta_2})$ the height pairings of the corresponding CM divisors on…

For $\chi_k$ a self$-$dual primitive Dirichlet character mod $k$ several reduced identities of Dirichlet $L-$functions $L_k(s):=L(s,\chi_k)$, expressed as linear combinations of Hurwitz $\zeta$ functions, are found for $s=2,3$ and some…