<p>
Let
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 1 comma pi 2">
<mml:semantics>
<mml:mrow>
<mml:msub>
<mml:mi>
π
</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>
π
</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\pi _1,\pi _2</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
be irreducible admissible generic tempered representations of
<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 2 Baseline left-parenthesis upper F 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:mn>2</mml:mn>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>F</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathrm {GL}_2(F)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
for some finite extension
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F slash bold upper Q Subscript p">
<mml:semantics>
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo>/</mml:mo>
</mml:mrow>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">Q</mml:mi>
</mml:mrow>
<mml:mi>p</mml:mi>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">F/\mathbf {Q}_p</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
of odd residue characteristic. Inspired by work of Loeffler and previous work of the author on unramified zeta-integrals, we introduce a natural general notion of
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis pi 1 times pi 2 right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mi>
π
</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>
×
</mml:mo>
<mml:msub>
<mml:mi>
π
</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">(\pi _1\times \pi _2)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-
<italic>integral</italic>
data at which the Rankin-Selberg zeta-integral can be evaluated. We then establish an integral refinement of Jacquet-Langland’s GCD-result for this zeta-integral, when evaluated at
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis pi 1 times pi 2 right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mi>
π
</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
<mml:mo>
×
</mml:mo>
<mml:msub>
<mml:mi>
π
</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">(\pi _1\times \pi _2)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-integral data. This is compatible with the notion of integrality coming from the Fourier coefficients of newforms of even integral weights. Our approach relies on a reinterpretation of the Rankin-Selberg zeta-integral, and works of Assing and Saha on values of
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p">
<mml:semantics>
<mml:mi>p</mml:mi>
<mml:annotation encoding="application/x-tex">p</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-adic Whittaker new vectors.
</p>
<p>
Our objective in the present work is to develop a fairly complete arithmetic theory of critical
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p">
<mml:semantics>
<mml:mi>p</mml:mi>
<mml:annotation encoding="application/x-tex">p</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-adic
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L">
<mml:semantics>
<mml:mi>L</mml:mi>
<mml:annotation encoding="application/x-tex">L</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-functions on the eigencurve. To this end, we carry out the following tasks:
</p>
<p>
We give an “étale” construction of Bellaïche’s
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p">
<mml:semantics>
<mml:mi>p</mml:mi>
<mml:annotation encoding="application/x-tex">p</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-adic
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L">
<mml:semantics>
<mml:mi>L</mml:mi>
<mml:annotation encoding="application/x-tex">L</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-functions at a
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta">
<mml:semantics>
<mml:mi>
θ
</mml:mi>
<mml:annotation encoding="application/x-tex">\theta</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-critical point on the cuspidal Coleman–Mazur–Buzzard eigencurve.
</p>
<p>
We introduce the algebraic counterparts of these objects (which arise as appropriately defined Selmer complexes) and develop Iwasawa theory in this context, including a definition of an Iwasawa theoretic
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="script">L</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathscr L</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-invariant
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Subscript upper I w Superscript c r">
<mml:semantics>
<mml:msubsup>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="script">L</mml:mi>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi>I</mml:mi>
<mml:mi>w</mml:mi>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi>c</mml:mi>
<mml:mi>r</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:annotation encoding="application/x-tex">\mathscr {L}^{cr}_{Iw}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
.
</p>
<p>We formulate the (punctual) critical main conjectures, and study its relationship with its slope-zero counterpart. Along the way, we also develop descent theory (paralleling Perrin-Riou’s work).</p>
<p>
We introduce what we call
<italic>thick</italic>
(Iwasawa theoretic) fundamental line and the
<italic>thick</italic>
Selmer complex to counter Bellaïche’s secondary
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p">
<mml:semantics>
<mml:mi>p</mml:mi>
<mml:annotation encoding="application/x-tex">p</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-adic
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L">
<mml:semantics>
<mml:mi>L</mml:mi>
<mml:annotation encoding="application/x-tex">L</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-functions. This allows us to formulate an infinitesimal thickening of the Iwasawa main conjecture, and we observe that it implies both slope-zero and punctual critical main conjectures, but it seems stronger than both.
</p>
<p>
We establish an
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O Subscript script upper X">
<mml:semantics>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi>
</mml:mrow>
</mml:msub>
<mml:annotation encoding="application/x-tex">\mathcal {O}_\mathcal {X}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-adic leading term formula for the two-variable
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p">
<mml:semantics>
<mml:mi>p</mml:mi>
<mml:annotation encoding="application/x-tex">p</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-adic
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L">
<mml:semantics>
<mml:mi>L</mml:mi>
<mml:annotation encoding="application/x-tex">L</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-function over the affinoid neighborhood
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X equals upper S p m left-parenthesis script upper O Subscript script upper X Baseline right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi>
</mml:mrow>
<mml:mo>=</mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi>S</mml:mi>
<mml:mi>p</mml:mi>
<mml:mi>m</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi>
</mml:mrow>
D. Benois, Kazim Buyukboduk· Memoirs of the American Math...· 3 citations· ⚡1
<p>
We combine the relative trace formula with analytic methods to obtain zero density estimates for
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L">
<mml:semantics>
<mml:mi>L</mml:mi>
<mml:annotation encoding="application/x-tex">L</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-functions in various families of automorphic representations 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 left-parenthesis m right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="normal">G</mml:mi>
<mml:mi mathvariant="normal">L</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>m</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathrm {GL}(m)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
that show their strength close to the critical line. Applications include strong bounds for the average analytic rank of these
<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L">
<mml:semantics>
<mml:mi>L</mml:mi>
<mml:annotation encoding="application/x-tex">L</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>
-functions at the central point and average equidistribution results for the imaginary parts of the zeros.
</p>
Valentin Blomer, Jesse Thorner· Transactions of the American...· 0 citations
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