On the Iwasawa invariants of BDP Selmer groups and BDP<i>p</i>-adic<i>L</i>-functions
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Abstract
Abstract Let p be an odd prime. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>f</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {f_{1}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>f</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> {f_{2}} be weight 2 cuspidal Hecke eigenforms with isomorphic residual Galois representations at p . Greenberg–Vatsal and Emerton–Pollack–Weston showed that if p is a good ordinary prime for the two forms, the Iwasawa invariants of their p -primary Selmer groups and p -adic L -functions over the cyclotomic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {\mathbb{Z}_{p}} -extension of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℚ</m:mi> </m:math> {\mathbb{Q}} are closely related. The goal of this article is to generalize these results to the anticyclotomic setting. More precisely, let K be an imaginary quadratic field where p splits. Suppose that the generalized Heegner hypothesis holds with respect to both <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>f</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(f_{1},K)} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>f</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(f_{2},K)} . We study relations between the Iwasawa invariants of the BDP Selmer groups and the BDP p -adic L -functions of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>f</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {f_{1}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>f</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> {f_{2}} .
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