Drinfeld Modular Forms of Arbitrary Rank

This monograph provides a foundation for the theory of Drinfeld modular forms of arbitrary rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and is subdivided into three chapters. In the first chapter, we develop the analytic theory. Most of the work goes into defining and studying the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u"> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding="application/x-tex">u</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-expansion of a weak Drinfeld modular form, whose coefficients are weak Drinfeld modular forms of rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r minus 1"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">r-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Based on that we give a precise definition of when a weak Drinfeld modular form is holomorphic at infinity and thus a Drinfeld modular form in the proper sense. In the second chapter, we compare the analytic theory with the algebraic one that was begun in a paper of the third author. For any arithmetic congruence subgroup and any integral weight we establish an isomorphism between the space of analytic modular forms with the space of algebraic modular forms defined in terms of the Satake compactification. From this we deduce the important result that this space is finite dimensional. In the third chapter, we construct and study some examples of Drinfeld modular forms. In particular, we define Eisenstein series, as well as the action of Hecke operators upon them, coefficient forms and discriminant forms. In the special case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals double-struck upper F Subscript q Baseline left-bracket t right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <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:mo stretchy="false">[</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A=\mathbb {F}_q[t]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we show that all modular forms for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper G normal upper L Subscript r Baseline left-parenthesis normal upper Gamma left-parenthesis t right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mi>r</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {GL}_r(\Gamma (t))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are generated by certain weight one Eisenstein series, and all modular forms for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper G normal upper L Subscript r Baseline left-parenthesis upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mi>r</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {GL}_r(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper S normal upper L Subscript r Baseline left-parenthesis upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mi>r</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {SL}_r(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are generated by certain coefficient forms and discriminant forms. We also compute the dimensions of the spaces of such modular forms.