CONSTRUCTING SIMULTANEOUS HECKE EIGENFORMS

It is known that there are one-to-one correspondences among the space of cusp forms, the space of homogeneous period polynomials and the space of Dedekind symbols with polynomial reciprocity laws. We add one more space, the space of quantum modular forms with polynomial period functions, to extend results from Fukuhara. Also, we consider Hecke operators on the space of quantum modular forms and construct new quantum modular forms.

Series expansion of the period function and representations of Hecke operators

Newforms in the Kohnen plus space

Ahlgren and Samart relate three cusp forms with complex multiplication to certain weakly holomorphic modular forms using p-adic bounds related to their Fourier coefficients. In these three examples, their result strengthens a theorem of Guerzhoy, Kent, and Ono which pairs certain CM forms with weakly holomorphic modular forms via p-adic limits. Ahlgren and Samart use only the theory of modular forms and Hecke operators, whereas Guerzhoy, Kent, and Ono use the theory of harmonic Maass forms. Here we extend Ahlgren and Samart's work to all cases where the cusp form space is one-dimensional and has trivial Nebentypus. Along the way, we obtain a duality result relating two families of weakly holomorphic modular forms that arise naturally in each case.

For primes $$p>3$$ we give a formula for the traces of pth Hecke operators on space of cusp forms of weight k and level 1 involving p-adic hypergeometric functions. We also express the traces of Frobenius for a certain family of elliptic curves in terms of p-adic hypergeometric function. Finally, we derive certain relations for Ramanujan’s $$\tau $$ -function and p-adic hypergeometric functions.

Representations by some quinary quadratic forms of level 8

Effective equidistribution of eigenvalues of Hecke operators

In this paper, we introduce a new way of studying derivatives of L -functions of cusp forms by associating to them cocycles analogous, in function, to the period polynomial. The main result gives a description of the effect of Hecke operators on these cocycles.

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the hypergeometric <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 3 Baseline upper F 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"/> <mml:mn>3</mml:mn> </mml:msub> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{}_3F_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> function over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript p"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {F}_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose three numerator parameters are quadratic characters and whose two denominator parameters are trivial characters. In 1992, Koike posed the problem of evaluating <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at the argument <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 slash 4"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">1/4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This problem was solved by Ono in 1998. Ten years later, Evans and Greene extended Ono’s result by evaluating an infinite family of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 3 Baseline upper F 2 left-parenthesis 1 slash 4 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"/> <mml:mn>3</mml:mn> </mml:msub> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>4</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{}_3F_2(1/4)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript q"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {F}_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of Jacobi sums. Here we present five new <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 3 Baseline upper F 2 left-parenthesis 1 slash 4 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"/> <mml:mn>3</mml:mn> </mml:msub> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>4</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{}_3F_2(1/4)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript q"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {F}_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (involving characters of orders 3, 4, 6, and 8) which are conjecturally evaluable in terms of eigenvalues for Hecke eigenforms of weights 2 and 3. There is ample numerical evidence for these evaluations. We motivate our conjectures by proving a connection between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 3 Baseline upper F 2 left-parenthesis 1 slash 4 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"/> <mml:mn>3</mml:mn> </mml:msub> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>4</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{}_3F_2(1/4)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and twisted sums of traces of the third symmetric power of twisted Kloosterman sheaves.

Structure of the space of polyharmonic Maass forms with an application to L-values

Let f be a newform of weight k ≥ 3 with Fourier coefficients in a number field K . We show that the universal deformation ring of the mod λ Galois representation associated to f is unobstructed, and thus isomorphic to a power series ring in three variables over the Witt vectors, for all but finitely many primes λ of K . We give an explicit bound on such λ for the 6 known cusp forms of level 1, trivial character, and rational Fourier coefficients. We also prove a somewhat weaker result for weight 2.

In this computational paper we verify a truncated version of the Buzzard–Calegari conjecture on the Newton polygon of the Hecke operator $$T_2$$ for all large enough weights. We first develop a formula for computing p-adic valuations of exponential sums, which we then implement to compute 2-adic valuations of traces of Hecke operators acting on spaces of cusp forms. Finally, we verify that if Newton polygon of the Buzzard–Calegari polynomial has a vertex at $$n\le 15$$ , then it agrees with the Newton polygon of $$T_2$$ up to n.

Let X be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on X in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An interesting by-product of our analysis is the evaluation and other similar identities.

In 1973, Shimura (Ann. Math. (2) 97:440–481, 1973) introduced a family of correspondences between modular forms of half-integral weight and modular forms of even integral weight. Earlier, in unpublished work, Selberg explicitly computed a simple case of this correspondence pertaining to those half-integral weight forms which are products of Jacobi’s theta function and level one Hecke eigenforms. Cipra (J. Number Theory 32(1):58–64, 1989) generalized Selberg’s work to cover the Shimura lifts where the Jacobi theta function may be replaced by theta functions attached to Dirichlet characters of prime power modulus, and where the level one Hecke eigenforms are replaced by more generic newforms. Here we generalize Cipra’s results further to cover theta functions of arbitrary Dirichlet characters multiplied by Hecke eigenforms.

Let $\phi = \sum_{r^{2} \leq 4mn}c(n,r)q^{n}\zeta^{r}$ be a Jacobi form of weight $k$ (with $k > 2$ if $\phi$ is not a cusp form) and index $m$ with integral algebraic coefficients which is an eigenfunction of all Hecke operators $T_{p}, (p,m) = 1,$ and which has at least one nonvanishing coefficient $c(n,r)$ with $r$ prime to $m$. We prove that for almost all primes $\ell$ there are infinitely many fundamental discriminants $D = r^{2}-4mn < 0$ prime to $m$ with $\nu_{\ell}(c(n,r)) = 0$, where $\nu_{\ell}$ denotes a continuation of the $\ell$-adic valuation on $\mathbb{Q}$ to an algebraic closure. As applications we show indivisibility results for special values of Dirichlet $L$-series and for the central critical values of twisted $L$-functions of even weight newforms.