On mod <i>p</i> singular modular forms

Abstract

Abstract We show that a Siegel modular form with integral Fourier coefficients in a number field K , for which all but finitely many coefficients (up to equivalence) are divisible by a prime ideal <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔭</m:mi> </m:math> ${\mathfrak{p}}$ of K , is a constant modulo <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔭</m:mi> </m:math> ${\mathfrak{p}}$ . Moreover, we define a notion of mod <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔭</m:mi> </m:math> ${\mathfrak{p}}$ singular modular form and discuss a relation between its weight and the corresponding prime p . We discuss some examples of mod <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔭</m:mi> </m:math> ${\mathfrak{p}}$ singular modular forms arising from Eisenstein series and from theta series attached to lattices with automorphisms. Finally, we apply our results to properties mod <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔭</m:mi> </m:math> ${\mathfrak{p}}$ of Klingen–Eisenstein series.

Locations

• Forum Mathematicum - View - PDF